Chapter 5 Resource Masters - Commack Schools

Chapter 5 Resource Masters

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Copyright © by The McGraw-Hill Companies, Inc.

All rights reserved. The contents, or parts thereof, may be reproduced in print form for non-profit educational use with Glencoe Algebra 1, provided such reproductions bear copyright notice, but may not be reproduced in any form for any other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, network storage or transmission, or broadcast for distance learning.

Send all inquiries to:McGraw-Hill Education8787 Orion PlaceColumbus, OH 43240

ISBN: 978-0-07-661384-7 MHID: 0-07-661384-4

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 DOH 16 15 14 13 12 11

CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks in both English and Spanish.

MHID ISBNStudy Guide and Intervention Workbook 0-07-660292-3 978-0-07-660292-6Homework Practice Workbook 0-07-660291-5 978-0-07-660291-9

Spanish VersionHomework Practice Workbook 0-07-660294-X 978-0-07-660294-0

Answers For Workbooks The answers for Chapter 5 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

ConnectED All of the materials found in this booklet are included for viewing, printing, and editing at connected.mcgraw-hill.com.

Spanish Assessment Masters (MHID: 0-07-660289-3, ISBN: 978-0-07-660289-6) These masters contain a Spanish version of Chapter 5 Test Form 2A and Form 2C.

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Teacher’s Guide to Using the Chapter 5 Resource Masters .........................................iv

Chapter Resources Chapter 5 Student-Built Glossary ...................... 1Chapter 5 Anticipation Guide (English) ............. 3Chapter 5 Anticipation Guide (Spanish) ............ 4

Lesson 5-1Solving Inequalities by Addition and SubtractionStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ...................................... 9Enrichment ...................................................... 10

Lesson 5-2Solving Inequalities by Multiplication and DivisionStudy Guide and Intervention ...........................11Skills Practice .................................................. 13Practice ........................................................... 14Word Problem Practice .................................... 15Enrichment ...................................................... 16

Lesson 5-3Solving Multi-Step InequalitiesStudy Guide and Intervention .......................... 17Skills Practice .................................................. 19Practice ........................................................... 20Word Problem Practice .................................... 21Enrichment ...................................................... 22

Lesson 5-4Solving Compound InequalitiesStudy Guide and Intervention .......................... 23Skills Practice .................................................. 25Practice ........................................................... 26Word Problem Practice .................................... 27Enrichment ...................................................... 28

Lesson 5-5Inequalities Involving Absolute ValueStudy Guide and Intervention .......................... 29Skills Practice .................................................. 31Practice ........................................................... 32Word Problem Practice .................................... 33Enrichment ...................................................... 34Graphing Calculator Activity ............................ 35

Lesson 5-6Graphing Inequalities in Two VariablesStudy Guide and Intervention .......................... 36Skills Practice .................................................. 38Practice ........................................................... 39Word Problem Practice .................................... 40Enrichment ...................................................... 41Spreadsheet Activity ........................................ 42

AssessmentStudent Recording Sheet ................................ 43Rubric for Scoring Extended Response .......... 44Chapter 5 Quizzes 1 and 2 ............................. 45Chapter 5 Quizzes 3 and 4 ............................. 46Chapter 5 Mid-Chapter Test ............................ 47Chapter 5 Vocabulary Test ............................... 48Chapter 5 Test, Form 1 .................................... 49Chapter 5 Test, Form 2A ................................. 51Chapter 5 Test, Form 2B ................................. 53Chapter 5 Test, Form 2C ................................. 55Chapter 5 Test, Form 2D ................................. 57Chapter 5 Test, Form 3 .................................... 59Chapter 5 Extended Response Test ................ 61Standardized Test Practice .............................. 62

Answers ........................................... A1–A31

Contents

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Teacher’s Guide to Using the Chapter 5 Resource Masters

The Chapter 5 Resource Masters includes the core materials needed for Chapter 5. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing, printing, and editing at connectED.mcgraw-hill.com.

Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 5-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer an historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator, TI-Nspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.

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Assessment OptionsThe assessment masters in the Chapter 5 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended Response This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 7 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choice questions and is intended for use with below grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages with answers appearing in bold, black.

• Full-size answer keys are provided for the assessment masters.

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Chapter 5 1 Glencoe Algebra 1

Student-Built Glossary

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 5. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary TermFound

on PageDefi nition/Description/Example

boundary

closed half-plane

compound inequality

half-plane

intersection

open half-plane

set-builder notation

union

5

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Anticipation GuideLinear Inequalities

Before you begin Chapter 5

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

STEP 1A, D, or NS

StatementSTEP 2A or D

1. According to the Addition Property of Inequalities, adding any number to each side of a true inequality will result in a true inequality.

2. The inequality m + 23 ≥ 35 can be solved by adding 23 to each side.

3. 16 is no greater than the difference of a number and 12 can be written as 16 ≤ n – 12.

4. If both sides of r − 12

< 4 are multiplied by 12, the result is r < 48.

5. The result of dividing both sides of the inequality –2y ≥ 10 by –2 is y ≥ –5.

6. To solve an inequality involving multiplication, such as 9t > 27, division is used.

7. To solve the inequality 8x – 2 < 70, first divide by 8 and then add 2.

8. A compound inequality is an inequality containing more than one variable.

9. On a number line, a closed dot is used for an inequality containing the symbol ≥ or ≤.

10. If ⎪t⎥ < 8, then t equals all numbers between 0 and 8.11. On the graph of y > 2x – 3, the solution set will be all

numbers above the graph of the line y = 2x – 3.

After you complete Chapter 5

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

Step 1

5

Step 2

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Capítulo 5 4 Álgebra 1 de Glencoe

Antes de comenzar el Capítulo 5

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a)).

PASO 1A, D o NS

EnunciadoPASO 2A o D

1. Según la propiedad de adición de la desigualdad, sumar cualquier número a cada lado de una desigualdad verdadera dará como resultado una desigualdad verdadera.

2. La desigualdad m + 23 ≥ 35 se puede resolver al sumar 23 a cada lado.

3. 16 no es más que la diferencia entre un número y 12 se puede escribir como 16 ≤ n – 12.

4. Si ambos lados de r − 12

< 4 se multiplican por 12, el resultado es r < 48.

5. El resultado de dividir ambos lados de la desigualdad –2y ≥ 10 entre –2 es y ≥ –5.

6. Para resolver una desigualdad que implica multiplicación, como 9t > 27, se usa la división.

7. Para resolver la desigualdad 8x – 2 < 70, primero divide entre 8 y luego suma 2.

8. Una desigualdad compuesta es una desigualdad que contiene más de una variable.

9. En una recta numérica, un punto cerrado se usa para una desigualdad con el símbolo ≥ o ≤.

10. Si ⎪t⎥ < 8, entonces t igual a todos los números entre 0 y 8.

11. En la gráfica de y > 2x – 3, el conjunto solución será el de todos los números arriba de la gráfica de la recta y = 2x – 3.

Después de completar el Capítulo 5

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

Paso 1

Paso 2

5 Ejercicios preparatoriosDesigualdades lineales

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Study Guide and InterventionSolving Inequalities by Addition and Subtraction

Solve Inequalities by Addition Addition can be used to solve inequalities. If any number is added to each side of a true inequality, the resulting inequality is also true.

Addition Property of Inequalities For all numbers a, b, and c, if a > b, then a + c > b + c, and if a < b, then a + c and < are replaced with ≥ and ≤.

Solve x - 8 ≤ -6. Then graph the solution. x - 8 ≤ -6 Original inequality

x - 8 + 8 ≤ -6 + 8 Add 8 to each side.

x ≤ 2 Simplify.

The solution in set-builder notation is {x|x ≤ 2}.Number line graph:

-4 -3 -2 -1 0 1 2 3 4

Solve 4 - 2a > -a. Then graph the solution. 4 - 2a > -a Original inequality

4 - 2a + 2a > -a + 2a Add 2a to each side.

4 > a Simplify.

a < 4 4 > a is the same as a < 4.

The solution in set-builder notation is {a|a < 4}.Number line graph:

-2 -1 0 1 2 3 4 5 6

ExercisesSolve each inequality. Check your solution, and then graph it on a number line.

1. t - 12 ≥ 16 2. n - 12 < 6 3. 6 ≤ g - 3

26 27 28 29 30 31 32 33 34

14 1512 13 16 17 18 19 20

7 8 9 10 11 12 13 14 15

4. n - 8 < -13 5. -12 > -12 + y 6. -6 > m - 8

-9-10 -8 -7 -6 -5 -4 -3 -2

-3-4 -2 -1 0 1 2 3 4

-4 -2 -1 0 1 2 3-3 4

Solve each inequality. Check your solution.

7. -3x ≤ 8 - 4x 8. 0.6n ≥ 12 - 0.4n 9. -8k - 12 < - 9k

10. -y - 10 > 15 - 2y 11. z - 1 − 3 ≤ 4 −

3 12. -2b > -4 - 3b

Define a variable, write an inequality, and solve each problem. Check your solution.

13. A number decreased by 4 is less than 14.

14. The difference of two numbers is more than 12, and one of the numbers is 3.

15. Forty is no greater than the difference of a number and 2.

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Study Guide and Intervention (continued)

Solving Inequalities by Addition and Subtraction

Solve Inequalities by Subtraction Subtraction can be used to solve inequalities. If any number is subtracted from each side of a true inequality, the resulting inequality is also true.

Subtraction Property of InequalitiesFor all numbers a, b, and c, if a > b, then a - c > b - c, and if a < b, then a - c 4 + 2a. Then graph it on a number line.

3a + 5 > 4 + 2a Original inequality

3a + 5 - 2a > 4 + 2a - 2a Subtract 2a from each side.

a + 5 > 4 Simplify.

a + 5 - 5 > 4 - 5 Subtract 5 from each side.

a > -1 Simplify.

The solution is {a�a > -1}.Number line graph:

-4 -3 -2 -1 0 1 2 3 4

ExercisesSolve each inequality. Check your solution, and then graph it on a number line. 1. t + 12 ≥ 8 2. n + 12 > -12 3. 16 ≤ h + 9

-6 -5 -4 -3 -2 -1 0 1 2

-26 -25 -24 -23 -22 -21 5 6 7 8 9 10 11 12 13

4. y + 4 > - 2 5. 3r + 6 > 4r 6. 3 − 2 q - 5 ≥ 1 −

2 q

-8 -7 -6 -5 -4 -3 -2 -1 0

2 3 4 5 6 7 91 8

210 3 4 5 6 7 8

Solve each inequality. Check your solution. 7. 4p ≥ 3p + 0.7 8. r + 1 −

4 > 3 −

8 9. 9k + 12 > 8k

10. -1.2 > 2.4 + y 11. 4y < 5y+ 14 12. 3n + 17 < 4n


13. The sum of a number and 8 is less than 12.

14. The sum of two numbers is at most 6, and one of the numbers is -2.

15. The sum of a number and 6 is greater than or equal to -4.

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Skills PracticeSolving Inequalities by Addition and Subtraction

Match each inequality to the graph of its solution.

1. x + 11 > 16 a. -8 -7 -6 -5 -4 -3 -2 -1 0

2. x - 6 < 1 b.

3. x + 2 ≤ -3 c.

4. x + 3 ≥ 1 d.

5. x - 1 < -7 e.

Solve each inequality. Check your solution, and then graph it on a number line.

6. d - 5 ≤ 1 7. t + 9 < 8

2 30 1 4 5 6 7 8

-2 -1-4 -3 0 1 2 3 4

8. a - 7 > -13 9. w - 1 < 4

-8 -7 -6 -5 -4 -3 -2 -1 0

2 30 1 4 5 6 7 8

10. 4 ≥ k + 3 11. -9 ≤ b - 4

-2 -1-4 -3 0 1 2 3 4

-8 -7 -6 -5 -4 -3 -2 -1 0

12. -2 ≥ x + 4 13. 2y < y + 2

-6 -5-8 -7 -4 -3 -2 -1 0

-2 -1-4 -3 0 1 2 3 4


14. A number decreased by 10 is greater than -5.

15. A number increased by 1 is less than 9.

16. Seven more than a number is less than or equal to -18.

17. Twenty less than a number is at least 15.

18. A number plus 2 is at most 1.

-8 -7 -6 -5 -4 -3 -2 -1 0

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Match each inequality with its corresponding graph.

1. -8 ≥ x - 15 a. -6 -5 -4 -3 -2 -1 0 1 2

2. 4x + 3 < 5x b. 876543210

3. 8x > 7x - 4 c. -8 -7 -6 -5 -4 -3 -2 -1 0

4. 12 + x ≤ 9 d. 2 3 4 5 6 7 810

Solve each inequality. Check your solution, and then graph it on a number line.

5. r - (-5) > -2 6. 3x + 8 ≥ 4x

-8 -7 -6 -5 -4 -3 -2 -1 0

4 52 3 6 7 8 9 10

7. n - 2.5 ≥ -5 8. 1.5 < y + 1

-4 -3 -2 -1 0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4

9. z + 3 > 2 − 3 10. 1 −

2 ≤ c - 3 −

4

-4 -3 -2 -1 0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4


11. The sum of a number and 17 is no less than 26.

12. Twice a number minus 4 is less than three times the number.

13. Twelve is at most a number decreased by 7.

14. Eight plus four times a number is greater than five times the number.

15. ATMOSPHERIC SCIENCE The troposphere extends from the Earth’s surface to a height of 6–12 miles, depending on the location and the season. If a plane is flying at an altitude of 5.8 miles, and the troposphere is 8.6 miles deep in that area, how much higher can the plane go without leaving the troposphere?

16. EARTH SCIENCE Mature soil is composed of three layers, the uppermost being topsoil. Jamal is planting a bush that needs a hole 18 centimeters deep for the roots. The instructions suggest an additional 8 centimeters depth for a cushion. If Jamal wants to add even more cushion, and the topsoil in his yard is 30 centimeters deep, how much more cushion can he add and still remain in the topsoil layer?

PracticeSolving Inequalities by Addition and Subtraction

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Word Problem PracticeSolving Inequalities by Addition and Subtraction

1. SOUND The loudest insect on Earth is the African cicada. It produces sounds as loud as 105 decibels at 20 inches away. The blue whale is the loudest mammal on Earth. The call of the blue whale can reach levels up to 83 decibels louder than the African cicada. How loud are the calls of the blue whale?

2. GARBAGE The amount of garbage that the average American adds to a landfill daily is 4.6 pounds. If at least 2.5 pounds of a person’s daily garbage could be recycled, how much will still go into a landfill?

3. SHOPPING Tyler has $75 to spend at the mall. He purchases a music video for $14.99 and a pair of jeans for $18.99. He also spent $4.75 for lunch. Tyler still wants to purchase a video game. How much money can he spend on a video game?

4. SUPREME COURT The first Chief Justice of the U.S. Supreme Court, John Jay, served 2079 days as Chief Justice. He served 10,463 days fewer than John Marshall, who served as Supreme Court Chief Justice for the longest period of time. How many days must the current Supreme Court Chief Justice John Roberts serve to surpass John Marshall’s record of service?

5. WEATHER Theodore Fujita of the University of Chicago developed a classification of tornadoes according to wind speed and damage. The table shows the classification system.

Level NameWind Speed Range

(mph)

F0 Gale 40 –72

F1 Moderate 73–112

F2 Signifi cant 113–157

F3 Severe 158–206

F4 Devastating 207–260

F5 Incredible 261–318

F6 Inconceivable 319–379

Source: National Weather Service

a. Suppose an F3 tornado has winds that are 162 miles per hour. Write and solve an inequality to determine how much the winds would have to increase before the F3 tornado becomes an F4 tornado.

b. A tornado has wind speeds that are at least 158 miles per hour. Write and solve an inequality that describes how much greater these wind speeds are than the slowest tornado.

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Enrichment

Triangle InequalitiesRecall that a line segment can be named by the letters of its endpoints. Line segment AB (written as

−− AB ) has points A and B for

endpoints. The length of AB is written without the bar as AB.

AB > BC m∠ A < m∠ B

The statement on the left above shows that −−

AB is shorter than −−−

BC . The statement on the right above shows that the measure of angle A is less than that of angle B.

These three inequalities are true for any triangle ABC, no matter how long the sides.

a. AB + BC > ACb. If AB > AC, then m∠C > m∠B.c. If m∠C > m∠B, then AB > AC.

B

A C

Use the three triangle inequalities for these problems.

1. List the sides of triangle DEF in order of increasing length. D

F E60° 35°

85°

2. In the figure at the right, which line segment is the shortest?

JM

K

L

65°

60°65°55°

65°50°

3. Explain why the lengths 5 centimeters, 10 centimeters, and 20 centimeters could not be used to make a triangle.

4. Two sides of a triangle measure 3 inches and 7 inches. Between which two values must the third side be?

5. In triangle XYZ, XY = 15, YZ = 12, and XZ = 9. Which angle has the greatest measure? Which has the least?

6. List the angles ∠A, ∠C, ∠ABC, and ∠ABD, in order of increasing size. C

A

DB

13

15

12

5

9

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Study Guide and InterventionSolving Inequalities by Multiplication and Division

Solve Inequalities by Multiplication If each side of an inequality is multiplied by the same positive number, the resulting inequality is also true. However, if each side of an inequality is multiplied by the same negative number, the direction of the inequality must be reversed for the resulting inequality to be true.

Multiplication Property of Inequalities

For all numbers a, b, and c, with c ≠ 0,

1. if c is positive and a > b, then ac > bc;

if c is positive and a < b, then ac < bc;

2. if c is negative and a > b, then ac < bc;

if c is negative and a < b, then ac > bc.


Solve - y −

8 ≤ 12.

- y −

8 ≥ 12 Original inequality

(-8) (-

y −

8 ) ≤ (-8)12 Multiply each side by -8; change ≥ to ≤.

y ≤ -96 Simplify.

The solution is { y � y ≤ -96}.

Solve 3 −

4 k < 15.

3 − 4 k < 15 Original inequality

( 4 − 3 ) 3 −

4 k < ( 4 −

3 ) 15 Multiply each side by 4 −

3 .

k < 20 Simplify.

The solution is {k � k < 20}.

ExercisesSolve each inequality. Check your solution.

1. y −

6 ≤ 2 2. -

n − 50

> 22 3. 3 − 5 h ≥ -3 4. -

p −

6 < -6

5. 1 − 4 n ≥ 10 6. -

2 − 3 b < 1 −

3 7. 3m −

5 < -

3 − 20

8. -2.51 ≤ - 2h − 4

9. g −

5 ≥ -2 10. - 3 −

4 > -

9p −

5 11. n −

10 ≥ 5.4 12. 2a −

7 ≥ -6


13. Half of a number is at least 14.

14. The opposite of one-third a number is greater than 9.

15. One fifth of a number is at most 30.

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Solving Inequalities by Multiplication and Division

Solve Inequalities by Division If each side of a true inequality is divided by the same positive number, the resulting inequality is also true. However, if each side of an inequality is divided by the same negative number, the direction of the inequality symbol must be reversed for the resulting inequality to be true.

Division Property of Inequalities

For all numbers a, b, and c with c ≠ 0,

1. if c is positive and a > b, then a − c > b − c ; if c is positive and a < b, then a − c b, then a − c b − c .


Solve -12y ≥ 48.

-12y ≥ 48 Original inequality

-12y

− -12

≤ 48 − -12

Divide each side by -12 and change ≥ to ≤.

y ≤ -4 Simplify.

The solution is { y �

y ≤ -4}.


1. 25g ≥ -100 2. -2x ≥ 9 3. -5c > 2 4. -8m < -64

5. -6k < 1 − 5 6. 18 < -3b 7. 30 < -3n 8. -0.24 < 0.6w

9. 25 ≥ -2m 10. -30 > -5p 11. -2n ≥ 6.2 12. 35 < 0.05h

13. -40 > 10h 14. - 2 −

3n ≥ 6 15. -3 -x −

2

Define a variable, write an inequality, and solve each problem. Then check your solution.

17. Four times a number is no more than 108.

18. The opposite of three times a number is greater than 12.

19. Negative five times a number is at most 100.

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Skills PracticeSolving Inequalities by Multiplication and Division

Match each inequality with its corresponding statement.

1. 3n < 9 a. Three times a number is at most nine.

2. 1 − 3 n ≥ 9 b. One third of a number is no more than nine.

3. 3n ≤ 9 c. Negative three times a number is more than nine.

4. -3n > 9 d. Three times a number is less than nine.

5. 1 − 3 n ≤ 9 e. Negative three times a number is at least nine.

6. -3n ≥ 9 f. One third of a number is greater than or equal to nine.


7. 14g > 56 8. 11w ≤ 77 9. 20b ≥ -120 10. -8r < 16

11. -15p ≤ -90 12. x − 4 < 9 13. a −

9 ≥ -15 14. -

p −

7 > -9

15. - t −

12 ≥ 6 16. 5z < -90 17. -13m > -26 18. k −

5 ≤ -17

19. -y < 36 20. -16c ≥ -224 21. - h − 10

≤ 2 22. 12 > d − 12


23. Four times a number is greater than -48.

24. One eighth of a number is less than or equal to 3.

25. Negative twelve times a number is no more than 84.

26. Negative one sixth of a number is less than -9.

27. Eight times a number is at least 16.

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Match each inequality with its corresponding statement.

1. -4n ≥ 5 a. Negative four times a number is less than five.

2. 4 − 5 n > 5 b. Four fifths of a number is no more than five.

3. 4n ≤ 5 c. Four times a number is fewer than five.

4. 4 − 5 n ≤ 5 d. Negative four times a number is no less than five.

5. 4n < 5 e. Four times a number is at most five.

6. -4n < 5 f. Four fifths of a number is more than five.


7. - a − 5

< -14 8. -13h ≤ 52 9. b − 16

≥ -6 10. 39 > 13p

11. 2 − 3 n > -12 12. -

5 − 9 t < 25 13. -

3 − 5 m ≤ -6 14. 10 −

3 k ≥ -10

15. -3b ≤ 0.75 16. -0.9c > -9 17. 0.1x ≥ -4 18. -2.3 < j −

4

19. -15y < 3 20. 2.6v ≥ -20.8 21. 0 > -0.5u 22. 7 − 8 f ≤ -1


23. Negative three times a number is at least 57.

24. Two thirds of a number is no more than -10.

25. Negative three fifths of a number is less than -6.

26. FLOODING A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will exceed flood stage. How long can the river rise at this rate without exceeding flood stage?

27. SALES Pet Supplies makes a profit of $5.50 per bag on its line of natural dog food. If the store wants to make a profit of no less than $5225 on natural dog food, how many bags of dog food does it need to sell?

5-2 PracticeSolving Inequalities by Multiplication and Division

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1. PIZZA Tara and friends order a pizza. Tara eats 3 of the 10 slices and pays $4.20 for her share. Assuming that Tara has paid at least her fair share, write an ineqality for how much the pizza could have cost.

2. AIRLINES On average, at least 25,000 pieces of luggage are lost or misdirected each day by United States airlines. Of these, 98% are located by the airlines within 5 days. From a given day’s lost luggage, at least how many pieces of luggage are still lost after 5 days?

3. SCHOOL Gil earned these scores on the first three tests in biology this term: 86, 88, and 78. What is the lowest score that Gil can earn on the fourth and final test of the term if he wants to have an average of at least 83?

4. EVENT PLANNING The Downtown Community Center does not charge a rental fee as long as a rentee orders a minimum of $5000 worth of food from the center. Antonio is planning a banquet for the Quarterback Club. If he is expecting 225 people to attend, what is the minimum he will have to spend on food per person to avoid paying a rental fee?

5. PHYSICS The density of a substance determines whether it will float or sink in a liquid. The density of water is 1 gram per milliliter. Any object with a greater density will sink and any object with a lesser density will float. Density is given by the formula d =

m − v , where m is mass and v is volume. Here is a table of common chemical solutions and their densities.

Solution Density (g/mL)

concentrated calcium chloride 1.40

70% isopropyl alcohol 0.92

Source: American Chemistry Council

a. Plastics vary in density when they are manufactured; therefore, their volumes are variable for a given mass. A tablet of polystyrene (a manufactured plastic) sinks in water and in alcohol solution and floats in calcium chloride solution. The tablet has a mass of 0.4 gram. What is the most its volume can be?

b. What is the least its volume can be?

Word Problem PracticeSolving Inequalities by Multiplication and Division

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Enrichment

Quadratic InequalitiesLike linear inequalities, inequalities with higher degrees can also be solved. Quadratic inequalities have a degree of 2. The following example shows how to solve quadratic inequalities.

Solve (x + 3)(x - 2) > 0.

Step 1 Determine what values of x will make the left side 0. In other words, what values of x will make either x + 3 = 0 or x - 2 = 0?

x = -3 or 2

Step 2 Plot these points on a number line. Above the number line, place a + if x + 3 is positive for that region or a - if x + 3 is negative for that region. Next, above the signs you have just entered; do the same for x – 2.

Step 3 Below the chart, enter the product of the two signs. Your sign chart should look like the following:

5 6-6 43210-1-5-4-3-2

x - 2

x + 3

(x - 2)(x + 3)

The final positive regions correspond to values for which the quadratic expression is greater than 0. So, the answer is

x < -3 or x > 2.

ExercisesSolve each inequality.

1. (x - 1)(x + 2) > 0 2. (x + 5)(x + 2) > 0

3. (x - 1)(x - 5) < 0 4. (x + 2)(x - 4) ≤ 0

5. (x – 3)(x + 2) ≥ 0 6. (x + 3)(x - 4) ≤ 0

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Study Guide and InterventionSolving Multi-Step Inequalities

Solve Multi-Step Inequalities To solve linear inequalities involving more than one operation, undo the operations in reverse of the order of operations, just as you would solve an equation with more than one operation.

Solve 6x - 4 ≤ 2x + 12.6x - 4 ≤ 2x + 12 Original inequality

6x - 4 - 2x ≤ 2x + 12 - 2x Subtract 2x from

each side.

4x - 4 ≤ 12 Simplify.

4x - 4 + 4 ≤ 12 + 4 Add 4 to each side.

4x ≤ 16 Simplify.

4x − 4 ≤

16 −

4 Divide each side by 4.

x ≤ 4 Simplify.

The solution is {x � x ≤ 4}.

Solve 3a - 15 > 4 + 5a.

3a - 15 > 4 + 5a Original inequality

3a - 15 - 5a > 4 + 5a - 5a Subtract 5a from

each side.

-2a - 15 > 4 Simplify.

-2a - 15 + 15 > 4 + 15 Add 15 to each side.

-2a > 19 Simplify.

-2a − -2

< 19 − -2

Divide each side by -2

and change > to <.

a < -9 1 − 2 Simplify.

The solution is {a � a < -9 1 − 2 } .


1. 11y + 13 ≥ -1 2. 8n - 10 < 6 - 2n 3. q −

7 + 1 > -5

4. 6n + 12 < 8 + 8n 5. -12 - d > -12 + 4d 6. 5r - 6 > 8r - 18

7. -3x + 6 − 2 ≤ 12 8. 7.3y - 14.4 > 4.9y 9. -8m - 3 < 18 - m

10. -4y - 10 > 19 - 2y 11. 9n - 24n + 45 > 0 12. 4x - 2 − 5 ≥ -4


13. Negative three times a number plus four is no more than the number minus eight.

14. One fourth of a number decreased by three is at least two.

15. The sum of twelve and a number is no greater than the sum of twice the number and -8.

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Solving Multi-Step Inequalities

Solve Inequalities Involving the Distributive Property When solving inequalities that contain grouping symbols, first use the Distributive Property to remove the grouping symbols. Then undo the operations in reverse of the order of operations, just as you would solve an equation with more than one operation.

Solve 3a - 2(6a - 4) > 4 - (4a + 6).

3a - 2(6a - 4) > 4 - (4a + 6) Original inequality

3a - 12a + 8 > 4 - 4a - 6 Distributive Property

-9a + 8 > -2 - 4a Combine like terms.

-9a + 8 + 4a > -2 - 4a + 4a Add 4a to each side.

-5a + 8 > -2 Combine like terms.

-5a + 8 - 8 > -2 - 8 Subtract 8 from each side.

-5a > -10 Simplify.

a < 2 Divide each side by -5 and change > to <.

The solution in set-builder notation is {a � a < 2}.


1. 2(t + 3) ≥ 16 2. 3(d - 2) - 2d > 16 3. 4h - 8 < 2(h - 1)

4. 6y + 10 > 8 - (y + 14) 5. 4.6(x - 3.4) > 5.1x 6. -5x - (2x + 3) ≥ 1

7. 3(2y - 4) - 2(y + 1) > 10 8. 8 - 2(b + 1) < 12 - 3b 9. -2(k - 1) > 8(1+ k)

10. 0.3( y - 2) > 0.4(1 + y) 11. m + 17 ≤ -(4m - 13)

12. 3n + 8 ≤ 2(n - 4) - 2(1 - n) 13. 2(y - 2) > -4 + 2y

14. k - 17 ≤ -(17 - k) 15. n - 4 ≤ - 3(2 + n)


16. Twice the sum of a number and 4 is less than 12.

17. Three times the sum of a number and six is greater than four times the number decreased by two.

18. Twice the difference of a number and four is less than the sum of the number and five.

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Skills PracticeSolving Multi-Step Inequalities

Justify each indicated step.

1. 3 − 4 t - 3 ≥ -15

3 − 4 t - 3 + 3 ≥ -15 + 3 a. ?

3 − 4 t ≥ -12

4 − 3 (

3 − 4 ) t ≥ 4 −

3 (-12) b. ?

t ≥ -16

2. 5(k + 8) - 7 ≤ 23 5k + 40 - 7 ≤ 23 a. ?

5k + 33 ≤ 235k + 33 - 33 ≤ 23 - 33 b. ?

5k ≤ -10 5k − 5 ≤ -10 −

5 c. ?

k ≤ -2


3. -2b + 4 > -6 4. 3x + 15 ≤ 21 5. d − 2 - 1 ≥ 3

6. 2 − 5 a - 4 < 2 7. -

t − 5 + 7 > -4 8. 3 −

4 j - 10 ≥ 5

9. - 2 − 3 f + 3 < -9 10. 2p + 5 ≥ 3p - 10 11. 4k + 15 > -2k + 3

12. 2(-3m - 5) ≥ -28 13. -6(w + 1) < 2(w + 5) 14. 2(q - 3) + 6 ≤ -10


15. Four more than the quotient of a number and three is at least nine.

16. The sum of a number and fourteen is less than or equal to three times the number.

17. Negative three times a number increased by seven is less than negative eleven.

18. Five times a number decreased by eight is at most ten more than twice the number.

19. Seven more than five sixths of a number is more than negative three.

20. Four times the sum of a number and two increased by three is at least twenty-seven.

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PracticeSolving Multi-Step Inequalities

Justify each indicated step. 1. x > 5x - 12 −

8

8x > (8) 5x - 12 − 8 a. ?

8x > 5x - 12

8x - 5x > 5x - 12 - 5x b. ?

3x > -12

3x − 3 > -12 −

3 c. ?

x > -4

2. 2(2h + 2) < 2(3h + 5) - 124h + 4 < 6h + 10 - 12 a. ?4h + 4 < 6h - 2

4h + 4 - 6h < 6h - 2 - 6h b. ?-2h + 4 < -2

-2h + 4 - 4 < -2 - 4 c. ?-2h < -6

-2h − -2

> -6 − -2

d. ?

h > 3


3. -5 - t − 6 ≥ -9 4. 4u - 6 ≥ 6u - 20 5. 13 > 2 −

3 a - 1

6. w + 3 − 2 < -8 7.

3f - 10 −

5 > 7

8. h ≤ 6h + 3 −

5 9. 3(z + 1) + 11 < -2(z + 13)

10. 3r + 2(4r + 2) ≤ 2(6r + 1) 11. 5n - 3(n - 6) ≥ 0


12. A number is less than one fourth the sum of three times the number and four.

13. Two times the sum of a number and four is no more than three times the sum of the number and seven decreased by four.

14. GEOMETRY The area of a triangular garden can be no more than 120 square feet. The base of the triangle is 16 feet. What is the height of the triangle?

15. MUSIC PRACTICE Nabuko practices the violin at least 12 hours per week. She practices for three fourths of an hour each session. If Nabuko has already practiced 3 hours in one week, how many sessions remain to meet or exceed her weekly practice goal?

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Word Problem PracticeSolving Multi-Step Inequalities

1. BEACHCOMBING Jay has lost his mother’s favorite necklace, so he will rent a metal detector to try to find it. A rental company charges a one-time rental fee of $15 plus $2 per hour to rent a metal detector. Jay has only $35 to spend. What is the maximum amount of time he can rent the metal detector?

2. AGES Bobby, Billy, and Barry Smith are each one year apart in age. The sum of their ages is greater than the age of their father, who is 60. How old can the oldest brother can be?

3. TAXI FARE Jamal works in a city and sometimes takes a taxi to work. The taxicabs charge $1.50 for the first 1−

5 mile

and $0.25 for each additional 1−5

mile.

Jamal has only $3.75 in his pocket. What is the maximum distance he can travel by taxi if he does not tip the driver?

4. PLAYGROUND The perimeter of a rectangular playground must be no greater than 120 meters, because that is the total length of the materials available for the border. The width of the playground cannot exceed 22 meters. What are the possible lengths of the playground?

5. MEDICINE Clark’s Rule is a formula used to determine pediatric dosages of over-the-counter medicines.

weight of child ( lb) −

150 × adult dose = child

dose

a. If an adult dose of acetaminophen is 1000 milligrams and a child weighs no more than 90 pounds, what is the recommended child’s dose?

b. This label appears on a child’s cold medicine. What is the adult minimum dosage in milliliters?

Weight (lb) Age (yr) Dose

under 48 under 6 call a doctor

48-95 6-11 2 tsp or 10 mL

c. What is the maximum adult dosage in milliliters?

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Enrichment

Carlos Montezuma During his lifetime, Carlos Montezuma (1866–1923) was one of the most influential Native Americans in the United States. He was recognized as a prominent physician and was also a passionate advocate of the rights of Native American peoples. The exercises that follow will help you learn some interesting facts about Dr. Montezuma’s life.

Solve each inequality. The word or phrase next to the equivalent inequality will complete the statement correctly.

1. -2k > 10 2. 5 ≥ r - 9 Montezuma was born in the state He was a Native American of the

of ? . Yavapais, who are a ? people.

a. k < -5 Arizona a. r ≤ -4 Navajo

b. k > -5 Montana b. r ≥ -4 Mohawk

c. k > 12 Utah c. r ≤ 14 Mohave-Apache

3. -y ≤ -9 4. -3 + q > 12 Montezuma received a medical As a physician, Montezuma’s field of

degree from ? in 1889. specialization was ? .

a. y ≥ 9 Chicago Medical College a. q > -4 heart surgery

b. y ≥ -9 Harvard Medical School b. q > 15 internal medicine

c. y ≤ 9 Johns Hopkins University c. q < -15 respiratory diseases

5. 5 + 4x - 14 ≤ x 6. 7 - t < 7 + t For much of his career, he maintained In addition to maintaining his medical

a medical practice in ? . practice, he was also a(n) ? .

a. x ≤ 9 New York City a. t > 7 director of a blood bank

b. x ≤ 3 Chicago b. t > 0 instructor at a medical college

c. x ≥ -9 Boston c. t < -7 legal counsel to physicians

7. 3a + 8 ≥ 4a - 10 8. 6n > 8n - 12 Montezuma founded, wrote, and Montezuma testified before a

edited ? , a monthly newsletter committee of the United States that addressed Native American Congress concerning his work in concerns. treating ? .

a. a ≤ -2 Yavapai a. n < 6 appendicitis

b. a ≥ 18 Apache b. n > -6 asthma

c. a ≤ 18 Wassaja c. n > -10 heart attacks

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Study Guide and InterventionSolving Compound Inequalities

Inequalities Containing and A compound inequality containing and is true only if both inequalities are true. The graph of a compound inequality containing and is the intersection of the graphs of the two inequalities. Every solution of the compound inequality must be a solution of both inequalities.

Graph the solution set of x < 2 and x ≥ -1. Graph x < 2.

Graph x ≥ -1.

Find the intersection.

The solution set is {x � -1 ≤ x < 2}.

Solve -1 < x + 2 < 3. Then graph the solution set.

-1 < x + 2 and x + 2 < 3-1 - 2 < x + 2 - 2 x + 2 - 2 < 3 - 2

-3 < x x < 1

Graph x > -3.

Graph x < 1.

Find the intersection.

The solution set is {x � -3 < x < 1}.

ExercisesGraph the solution set of each compound inequality.

1. b > -1 and b ≤ 3 2. 2 ≥ q ≥ -5 3. x > -3 and x ≤ 4

4. -2 ≤ p < 4 5. -3 < d and d< 2 6. -1 ≤ p ≤ 3

Solve each compound inequality. Then graph the solution set.

7. 4 < w + 3 ≤ 5 8. -3 ≤ p - 5 < 2

9. -4 < x + 2 ≤ -2 10. y - 1 < 2 and y + 2 ≥ 1

11. n - 2 > -3 and n + 4 < 6 12. d - 3 < 6d + 12 < 2d + 32

-2 -1-3 0 1 2 3

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3 4 5-3-4 -2 -1 0 1 2 3 4

-3-4 -2 -1 0 1 2 3 4-7 -6 -5 -4 -3 -2 -1 0 1

0 1 2 3 4 5 6 7 8-3-4 -2 -1 0 1 2 3 4

-3-4 -2 -1 0 1 2 3 4-3-4 -2 -1 0 1 2 3 4-3 -2 -1 0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4-4 -3-6 -5 -2 -1 0 1 2-4 -3 -2 -1 0 1 2 3 4

-2 -1-4 -3 0 1 2

-3-4 -2 -1 0 1 2

-3-4 -2 -1 0 1 2

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Solving Compound Inequalities

Inequalities Containing or A compound inequality containing or is true if one or both of the inequalities are true. The graph of a compound inequality containing or is the union of the graphs of the two inequalities. The union can be found by graphing both inequalities on the same number line. A solution of the compound inequality is a solution of either inequality, not necessarily both.

Solve 2a + 1 < 11 or a > 3a + 2. Then graph the solution set.

2a + 1 < 11 or a > 3a + 2 2a + 1 - 1 < 11 - 1 a - 3a > 3a - 3a + 2 2a < 10 -2a > 2

2a −

2 < 10 −

2

-2a − -2

< 2 − -2

a < 5 a < -1

-2 -1 0 1 2 3 4 5 6 Graph a < 5.

-2 -1 0 1 2 3 4 5 6 Graph a < -1.

-2 -1 0 1 2 3 4 5 6 Find the union.

The solution set is {a � a < 5}.

ExercisesGraph the solution set of each compound inequality. 1. b > 2 or b ≤ -3 2. 3 ≥ q or q ≤ 1 3. y ≤ -4 or y > 0

4. 4 ≤ p or p < 8 5. -3 < d or d < 2 6. -2 ≤ x or 3 ≤ x


7. 3 < 3w or 3w ≥ 9 8. -3p + 1 ≤ -11 or p < 2

9. 2x + 4 ≤ 6 or x ≥ 2x - 4 10. 2y + 2 < 12 or y - 3 ≥ 2y

11. 1 − 2 n > -2 or 2n - 2 < 6 + n 12. 3a + 2 ≥ 5 or 7 + 3a < 2a + 6

0-1-2-3-4 1 2 3 40-1-2-3-4 1 2 3 4

0 1 2 3 4 5 6 7 8-2 -1 0 1 2 3 4 5 6

0 1 2 3 4 5 6 7 8-3-4 -2 -1 0 1 2 3 4

-3-4 -2 -1 0 1 2 3 40-1-2-3-4 1 2 3 40-1-2 1 2 3 4 5 6

-3-4-5 -2 -1 0 1 2 3-3-4 -2 -1 0 1 2 3 4-3-4 -2 -1 0 1 2 3 4

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Skills PracticeSolving Compound Inequalities

Graph the solution set of each compound inequality.

1. b > 3 or b ≤ 0 2. z ≤ 3 and z ≥ -2

3. k > 1 and k > 5 4. y < -1 or y ≥ 1

Write a compound inequality for each graph.

5. -2 -1-4 -3 0 1 2 3 4

6.

7. 8.


9. m + 3 ≥ 5 and m + 3 < 7 10. y - 5 < -4 or y - 5 ≥ 1

11. 4 < f + 6 and f + 6 < 5 12. w + 3 ≤ 0 or w + 7 ≥ 9

13. -6 1


15. A number plus one is greater than negative five and less than three.

16. A number decreased by two is at most four or at least nine.

17. The sum of a number and three is no more than eight or is more than twelve.

-3-4 -2 -1 0 1 2 3 4-2 -1 0 1 2 3 4 5 6

-3-4 -2 -1 0 1 2 3 4-4 -3 -2 -1 0 1 2 3 4

-2 -1 0 1 2 3 4 5 6-2 -1 0 1 2 3 4 5 6

-4 -3 -2 -1 0 1 2 3 4

-3-4 -2 -1 0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4-3-4 -2 -1 0 1 2 3 4

-2 -1 0 1 2 3 4 5 6

-4 -3 -2 -1 0 1 2 3 4

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0 1 2 3 4 5 6 7 8

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PracticeSolving Compound Inequalities

Graph the solution set of each compound inequality.

1. -4 ≤ n ≤ 1 2. x > 0 or x < 3

3. g < -3 or g ≥ 4 4. -4 ≤ p ≤ 4

Write a compound inequality for each graph.

5. 6.

7. 8.


9. k - 3 < -7 or k + 5 ≥ 8 10. -n < 2 or 2n - 3 > 5

11. 5 < 3h + 2 ≤ 11 12. 2c - 4 > -6 and 3c + 1 < 13


13. Two times a number plus one is greater than five and less than seven.

14. A number minus one is at most nine, or two times the number is at least twenty-four.

15. METEOROLOGY Strong winds called the prevailing westerlies blow from west to east in a belt from 40° to 60° latitude in both the Northern and Southern Hemispheres.

a. Write an inequality to represent the latitude of the prevailing westerlies.

b. Write an inequality to represent the latitudes where the prevailing westerlies are not located.

16. NUTRITION A cookie contains 9 grams of fat. If you eat no fewer than 4 and no more than 7 cookies, how many grams of fat will you consume?

-3-4 -2 -1 0 1 2 3 4

-2 -1-4 -3-6 -5 0 1 2

-2 -1 0 1 2 3 4 5 6-2 -1-4 -3 0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4-3-4 -2 -1 0 1 2 3 4

-2-3-4-5-6 -1 0 1 2-2 -1 0 1 2 3 4 5 6

-2 -1 0 1 2 3 4 5 6-4 -3 -2 -1 0 1 2 3 4

-2 -1-4 -3 0 1 2 3 4

0-1-2-3-4 1 2 3 4

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Word Problem PracticeSolving Compound Inequalities

1. WEATHER Ken saw this graph in the newspaper weather forecast. It shows the predicted temperature range for the following day. Write an inequality to represent the number line graph.

2. POOLS The pH of a person’s eyes is 7.2. Therefore, the ideal pH for the water in a swimming pool is between 7.0 and 7.6. Write a compound inequality to represent pH levels that could cause physical discomfort to a person’s eyes.

3. STORE SIGNS In Randy’s town, street-side signs themselves must be exactly 8 feet high. When mounted on poles, the signs must be shorter than 20 feet or taller than 35 feet so that they do not interfere with the power and phone lines. Write a compound inequality to represent the possible height of the poles.

4. HEALTH The human heart circulates from 770,000 to 1,600,000 gallons of blood through a person’s body every year. How many gallons of blood does the heart circulate through the body in one day?

5. HEALTH Body mass index (BMI) is a measure of weight status. The BMI of a person over 20 years old is calculated using the following formula.

BMI = 703 × weight in pounds

−− (height in inches)2

The table below shows the meaning of different BMI measures.

Source: Centers for Disease Control

a. Write a compound inequality to represent the normal BMI range.

b. Write a compound inequality to represent an adult weight that is within the healthy BMI range for a person 6 feet tall.

62° 64° 66° 68° 70°60°58°56°54°52°50°F

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BMI Weight Status

less than 18.5 underweight

18.5 – 24.9 normal

25 – 29.9 overweight

more than 30 obese


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Enrichment

Some Properties of InequalitiesThe two expressions on either side of an inequality symbol are sometimes called the first and second members of the inequality.

If the inequality symbols of two inequalities point in the same direction, the inequalities have the same sense. For example, a d have opposite senses.

In the problems on this page, you will explore some properties of inequalities.

Three of the four statements below are true for all numbers a and b (or a, b, c, and d). Write each statement in algebraic form. If the statement is true for all numbers, prove it. If it is not true, give an example to show that it is false.

1. Given an inequality, a new and equivalent inequality can be created by interchanging the members and reversing the sense.

2. Given an inequality, a new and equivalent inequality can be created by changing the signs of both terms and reversing the sense.

3. Given two inequalities with the same sense, the sum of the corresponding members are members of an equivalent inequality with the same sense.

4. Given two inequalities with the same sense, the difference of the corresponding members are members of an equivalent inequality with the same sense.

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Study Guide and Intervention Inequalities Involving Absolute Value

Inequalities Involving Absolute Value (<) When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving < (or ≤).

Remember that inequalities with and are related to intersections.

Solve |3a + 4| < 10. Then graph the solution set.

Write � 3a + 4 � < 10 as 3a + 4 < 10 and 3a + 4 > -10. 3a + 4 < 10 and 3a + 4 > -10 3a + 4 - 4 < 10 - 4 3a + 4 - 4 > -10 - 4 3a < 6 3a > -14 3a −

3 < 6 −

3 3a −

3 > -14 −

3

a < 2 a > -4 2 − 3

The solution set is {a � -4 2 − 3 < a < 2} .

ExercisesSolve each inequality. Then graph the solution set.

1. � y � < 3 2. � x - 4 � < 4 3. � y + 3 � ≤ 2

-3-4 -2 -1 0 1 2 3 4 0 1 2 3 4 5 6 7 8-8 -7 -6 -5 -4 -3 -2 -1 0

4. � b + 2 � ≤ 3 5. � w - 2 � ≤ 5 6. � t + 2 � ≤ 4

-6 -5 -4 -3 -2 -1 0 1 2 -8 -6 -4 -2 0 2 4 6 8 -8 -6 -4 -2 0 2 4 6 8

7. � 2x � ≤ 8 8. � 5y - 2 � ≤ 7 9. � p - 0.2 � < 0.5

-3-4 -2 -1 0 1 2 3 4 -3-4 -2 -1 0 1 2 3 4 -0.8 -0.4 0 0.4 0.8

If � x � < n, then x > -n and x < n.

Now graph the solution set.

-2 -1-4-5 -3 0 1 2 3

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Inequalities Involving Absolute Value

Solve Absolute Value Inequalities (>) When solving inequalities that involve absolute value, there are two cases to consider for inequalities involving > (or ≥). Remember that inequalities with or are related to unions.

Solve � 2b + 9 � > 5. Then graph the solution set.

Write � 2b + 9 � > 5 as � 2b + 9 � > 5 or � 2b + 9 � < -5.

2b + 9 > 5 or 2b + 9 < -5

2b + 9 - 9 > 5 - 9 2b + 9 - 9 < -5 - 9

2b > -4 2b < -14

2b − 2 > -4 −

2 2b −

2 < -14 −

2

b > -2 b < -7

The solution set is {b � b > -2 or b < -7} .

ExercisesSolve each inequality. Then graph the solution set.

1. � c - 2 � > 6 2. � x - 3 � > 0 3. � 3f + 10 � ≥ 4

20-2-4-6 4 6 8 10 0-1-2-3-4 1 2 3 4 -6 -5 -4 -3 -2 -1 0 1 2

4. � x � ≥ 2 5. � x � ≥ 3 6. � 2x + 1 � ≥ -2

-4 -3 -2 -1 0 1 2 3 4 0-1-2-3-4 1 2 3 4 0-1-2-3-4 1 2 3 4

7. � 2d - 1 � ≥ 4 8. � 3 - (x - 1) � ≥ 8 9. � 3r + 2 � > -5

-4 -3 -2 -1 0 1 2 3 4 -4 -2 0 2 4 6 8 10 12 0-1-2-3-4 1 2 3 4

5-5

-9 -8 -7 -6 -5 -4 -3 -2 0-1

Now graph the solution set.

Example

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Skills PracticeInequalities Involving Absolute Value

Match each open sentence with the graph of its solution set.

1. � x � > 2 a. -2-3-4-5 -1 0 1 2 3 4 5

2. � x - 2 � ≤ 3 b. -2-3-4-5 -1 0 1 2 3 4 5

3. � x + 1 � < 4 c. -2-3-4 -1 0 1 2 3 4 5 6

Express each statement using an inequality involving absolute value.

4. The weatherman predicted that the temperature would be within 3° of 52°F.

5. Serena will make the B team if she scores within 8 points of the team average of 92.

6. The dance committee expects attendance to number within 25 of last year’s 87 students.

Solve each inequality. Then graph the solution set.

7. � x + 1 � < 0 8. � c - 3 � < 1

9. � n + 2 � ≥ 1 10. � t + 6 � > 4

11. � w - 2 � < 2 12. � k - 5 � ≤ 4

-8 -7-10 -9 -6 -5 -4 -3 -2 -1 0-2 -1-4-5-6 -3 0 1 2 3 4

-3 -1 0 1 2 3 4 5 6-2 7-3-4-5-6 -2 -1 0 1 2 3 4

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-3-4 -2 -1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10

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PracticeInequalities Involving Absolute Value

Match each open sentence with the graph of its solution set.

1. � x - 3 � ≥ 1 a.

2. � 2x + 1 � < 5 b.

3. � 5 - x � ≥ 3 c.

Express each statement using an inequality involving absolute value.

4. The height of the plant must be within 2 inches of the standard 13-inch show size.

5. The majority of grades in Sean’s English class are within 4 points of 85.

Solve each inequality. Then graph the solution set.

6. |2z - 9| ≤ 1 7. |3 - 2r| > 7

8. |3t + 6| < 9 9. |2g - 5| ≥ 9

Write an open sentence involving absolute value for each graph.

10. 11.

12. 13.

14. RESTAURANTS The menu at Jeanne’s favorite restaurant states that the roasted chicken with vegetables entree typically contains 480 Calories. Based on the size of the chicken, the actual number of Calories in the entree can vary by as many as 40 Calories from this amount.

a. Write an absolute value inequality to represent the situation.

b. What is the range of the number of Calories in the chicken entree?

-2-3 -1 0 1 2 3 4 5 6 7-2-3-4-5-6-7-8 -1 0 1 2

1 2 3 4 5 6 7 8 9 10 11

-2 -1 0 1 2 3 4 5 6 7 8-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5-5 -4 -3 -2 -1 0 1 2 3 4 5

-2-3-4-5 -1 0 1 2 3 4 5

-2 -1 0 1 2 3 4 5 6 7 8

-2-3-4-5 -1 0 1 2 3 4 5

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-2-3-4-5-6-7-8 -1 0 1 2

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Word Problem PracticeInequalities Involving Absolute Value

1. SPEEDOMETERS The government requires speedometers on cars sold in the United States to be accurate within ±2.5% of the actual speed of the car. If your speedometer reads 60 miles per hour while you are driving on a highway, what is the range of possible actual speeds at which your car could be traveling?

2. BAKING Pete is making muffins for a bake sale. Before he starts baking, he goes online to research different muffin recipes. The recipes that he finds all specify baking temperatures between 350°F and 400°F, inclusive. Write an absolute value inequality to represent the possible temperatures t called for in the muffin recipes Pete is researching.

3. ARCHERY In an Olympic archery event, the center of the target is set exactly 130 centimeters off the ground. To get the highest score of ten points, an archer must shoot an arrow no further than 3.05 centimeters from the exact center of the target.

a. Write an absolute value inequality to represent the possible distances dfrom the ground an archer can hit the target and still score ten points.

b. Graph the solution set of the inequality you wrote in part a.

124 126 128 130 132 134

4. CATS During a recent visit to the veterinarian’s office, Mrs. Van Allen was informed that a healthy weight for her cat is approximately 10 pounds, plus or minus one pound. Write an absolute value inequality that represents unhealthy weights w for her cat.

5. STATISTICS The most familiar statistical measure is the arithmetic mean, or average. A second important statistical measure is the standard deviation, which is a measure of how far the individual scores deviate from the mean. For example, in a recent year the mean score on the mathematics section of the SAT test was 515 and the standard deviation was 114. This means that people within one deviation of the mean have SAT math scores that are no more than 114 points higher or 114 points lower than the mean.

a. Write an absolute value inequality to find the range of SAT mathematics test scores within one standard deviation of the mean.

b. What is the range of SAT mathematics test scores ±2 standard deviation from the mean?

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Enrichment

Precision of MeasurementThe precision of a measurement depends both on your accuracy in measuring and the number of divisions on the ruler you use. Suppose you measured a length of wood to the nearest one-eighth of an inch and got a length of 6 5 −

8 inches.

The drawing shows that the actual measurement lies somewhere

between 6 9 − 16

and 6 11 − 16

inches. This measurement can be written using

the symbol ±, which is read plus or minus. It can also be written as a compound inequality.

6 5 − 8 ± 1 −

16 in. 6 9 −

16 in. ≤ m ≤ 6 11 −

16 in.

In this example, 1 − 16

inch is the absolute error. The absolute error is

one-half the smallest unit used in a measurement.

Write each measurement as a compound inequality. Use the variable m.

1. 5 1 − 2 ± 1 −

4 in. 2. 3.78 ± 0.005 kg 3. 7.11 ± 0.005 g

4. 16 ± 1 − 2 yd 5. 22 ± 0.5 cm 6. 9 −

16 ± 1 −

32 in.

For each measurement, give the smallest unit used and the absolute error.

7. 9.5 in. ≤ m ≤ 10.5 in. 8. 4 1 − 4 in. ≤ m ≤ 4 3 −

4 in.

9. 23 1 −

2 cm ≤ m ≤ 24 1 −

2 cm 10. 7.135 mm ≤ m ≤ 7.145 mm

5 6 7 8

65–8

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Graphing Calculator ActivityAbsolute Value Inequalities

The TEST menu can be used to solve and graph absolute value inequalities by using the equivalent compound inequalities related to absolute value.

ExercisesGraph and solve each inequality.

1. |x + 3| ≥ 2 2. |2x + 6| ≤ 4 3. ⎪ 2 - 4x −

5 ⎥ > 2 4. |x + 8| < -3

Graph and solve each inequality.

a. |x + 4| ≥ 8

Enter the inequality into Y1. Then enter the equivalent compound inequality into Y2 and graph to view the results. Be sure to choose appropriate settings for the view window.Keystrokes: MATH ENTER + 4 ) 2nd [TEST] 4 8

ENTER + 4 2nd [TEST] 6 (–) 8 2nd [TEST] 2

+ 4 2nd [TEST] 4 8 ENTER GRAPH .

Use TRACE to confirm the solution. When y = 1 the statement is true, and when y = 0 the statement is false. Thus, the solution is x ≤

-12 or x ≥ 4.

b. ⎪

5x + 2 −

4

⎥

≤ 7

Enter the inequality into Y1 and the equivalent compound inequalities into Y2. Then graph the solution set.Keystrokes: MATH ENTER ( 5 + 2 ) ÷ 4 )

2nd [TEST] 6 7 ENTER ( 5 + 2 ) ÷ 4 2nd

[TEST] 4 (–) 7 2nd [TEST] ENTER ( 5 + 2 )

÷ 4 2nd [TEST] 6 7 ENTER GRAPH .

The statement is true between -6 and 5.2. Thus the solution is -6 ≤ x ≤ 5.2.

[-18.8, 18.8] scl:2 by [-3.1, 3.1] scl:1

[-18.8, 18.8] scl:2 by [-3.1, 3.1] scl:1

[-18.8, 18.8] scl:2 by [-3.1, 3.1] scl:1

[-18.8, 18.8] scl:2 by [-3.1, 3.1] scl:1

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Study Guide and InterventionGraphing Inequalities in Two Variables

Graph Linear Inequalities The solution set of an inequality that involves two variables is graphed by graphing a related linear equation that forms a boundary of a half-plane. The graph of the ordered pairs that make up the solution set of the inequality fill a region of the coordinate plane on one side of the half-plane.

Graph y ≤ -3x - 2.

Graph y = -3x - 2. Since y ≤ -3x - 2 is the same as y < -3x - 2 and y = -3x - 2, the boundary is included in the solution set and the graph should be drawn as a solid line.Select a point in each half plane and test it. Choose (0, 0) and (-2, -2). y ≤ -3x - 2 y ≤ -3x - 2 0 ≤ -3(0) - 2 -2 ≤ -3(-2) - 2 0 ≤ -2 is false. -2 ≤ 6 - 2

-2 ≤ 4 is true.The half-plane that contains (-2, -2) contains the solution. Shade that half-plane.

ExercisesGraph each inequality.

1. y < 4 2. x ≥ 1 3. 3x ≤ y

4. -x > y 5. x - y ≥ 1 6. 2x - 3y ≤ 6

7. y < - 1 − 2 x - 3 8. 4x - 3y < 6 9. 3x + 6y ≥ 12

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Graphing Inequalities in Two Variables

Solve Linear Inequalities We can use a coordinate plane to solve inequalities with one variable.

Use a graph to solve 2x + 2 > -1.

Step 1 First graph the boundary, which is the related function. Replace the inequality sign with an equals sign, and get 0 on a side by itself.

2x + 2 > -1 Original inequality

2x + 2 = -1 Change < to = .

2x + 2 + 1 = -1 + 1 Add 1 to each side.

2x + 3 = 0 Simplify.

Graph 2x + 3 = y as a dashed line.

Step 2 Choose (0, 0) as a test point, substituting these values into the original inequality give us 3 > -5.

Step 3 Because this statement is true, shade the half plane containing the point (0, 0).

Notice that the x-intercept of the graph is at -1 1 − 2 . Because the half-plane

to the right of the x-intercept is shaded, the solution is x > -1 1 − 2 .

ExercisesUse a graph to solve each inequality.

1. x + 7 ≤ 5 2. x - 2 > 2 3. -x + 1 < -3

4. -x - 7 ≥ -6 5. 3x - 20 < -17 6. -2x + 11 ≥ 15

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Skills PracticeGraphing Inequalities in Two Variables

Match each inequality to the graph of its solution.

1. y - 2x < 2 a. b.

2. y ≤ -3x

3. 2y - x ≥ 4

4. x + y > 1

c. d.

Graph each inequality.

5. y < -1 6. y ≥ x - 5 7. y > 3x

8. y ≤ 2x + 4 9. y + x > 3 10. y - x ≥ 1

Use a graph to solve each inequality.

11. 1 > 2x + 5 12. 7 ≤ 3x + 4 13. - 1 − 2 < - 1 −

2 x + 1

x

y

Ox

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Ox

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PracticeGraphing Inequalities in Two Variables

Determine which ordered pairs are part of the solution set for each inequality.

1. 3x + y ≥ 6, {(4, 3), (-2, 4), (-5, -3), (3, -3)}

2. y ≥ x + 3, {(6, 3), (-3, 2), (3, -2), (4, 3)}

3. 3x - 2y < 5, {(4, -4), (3, 5), (5, 2), (-3, 4)}

Graph each inequality.

4. 2y - x < -4 5. 2x - 2y ≥ 8 6. 3y > 2x - 3

Use a graph to solve each inequality.

7. -5 ≤ x - 9 8. 6 > 2 − 3 x + 5 9. 1 −

2 > -2 x + 7 −

2

10. MOVING A moving van has an interior height of 7 feet (84 inches). You have boxes in 12 inch and 15 inch heights, and want to stack them as high as possible to fit. Write an inequality that represents this situation.

11. BUDGETING Satchi found a used bookstore that sells pre-owned DVDs and CDs. DVDs cost $9 each, and CDs cost $7 each. Satchi can spend no more than $35.

a. Write an inequality that represents this situation.

b. Does Satchi have enough money to buy 2 DVDs and 3 CDs?

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1. FAMILY Tyrone said that the ages of his siblings are all part of the solution set of y > 2x, where x is the age of a sibling and y is Tyrone’s age. Which of the following ages is possible for Tyrone and a sibling?Tyrone is 23; Maxine is 14. Tyrone is 18; Camille is 8. Tyrone is 12; Francis is 4. Tyrone is 11; Martin is 6. Tyrone is 19; Paul is 9.

2. FARMING The average value of U.S. farm cropland has steadily increased in recent years. In 2000, the average value was $1490 per acre. Since then, the value has increased at least an average of $77 per acre per year. Write an inequality to show land values above the average for farmland.

3. SHIPPING An international shipping company has established size limits for packages with all their services. The total of the length of the longest side and the girth (distance completely around the package at its widest point perpendicular to the length) must be less than or equal to 419 centimeters. Write and graph an inequality that represents this situation.

4. FUNDRAISING Troop 200 sold cider and donuts to raise money for charity. They sold small boxes of donut holes for $1.25 and cider for $2.50 a gallon. In order to cover their expenses, they needed to raise at least $100. Write and graph an inequality that represents this situation.

5. INCOME In 2006 the median yearly family income was about $48,200 per year. Suppose the average annual rate of change since then is $1240 per year.

a. Write and graph an inequality for the annual family incomes y that are less than the median for x years after 2006.

b. Determine whether each of the following points is part of the solution set. (2, 51,000) (8, 69,200) (5, 50,000) (10, 61,000)

O

Gir

th

Length

100

150

50

200

250

300

350

400

450

500g

150100

50 250 350200 300 400

450500

�

Cid

er (

gal

)Donut holes

20

30

10

40

50

60

70

80

90c

302010 50 7040 60 dO

O

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me

($10

00)

Years since 2006

46,000

48,000

44,000

50,000

52,000

54,000

56,000

58,000y

321 5 7 8 9 104 6 x

length

girth

5-6 Word Problem PracticeGraphing Inequalities in Two Variables


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Enrichment

Linear ProgrammingLinear programming can be used to maximize or minimize costs. It involves graphing a set of linear inequalities and using the region of intersection. You will use linear programming to solve the following problem.

Layne’s Gift Shoppe sells at most 500 items per week. To meet her customers’ demands, she sells at least 100 stuffed animals and 75 greeting cards. If the profit for each stuffed animal is $2.50 and the profit for each greeting card is $1.00, the equation P(a, g) = 2.50a + 1.00g can be used to represent the profit. How many of each should she sell to maximize her profit?

Write the inequalities: Graph the inequalities:a + g ≤ 500a ≥ 100g ≥ 75

Find the vertices of the triangle formed: (100, 75), (100, 400), and (425, 75) Substitute the values of the vertices into the equation found above: 2.50(100) + 1(75) = 325 2.50(100) + 1(400) = 650 2.50(425) + 1(75) = 1137.50 The maximum profit is $1137.50.

Exercises The Spirit Club is selling shirts and banners. They sell at most 400 of the two items. To meet the demands of the students, they must sell at least 50 T-shirts and 100 banners. The profit on each shirt is $4.00 and the profit on each banner is $1.50, the equation P(t, b) = 4.00t + 1.50b can be used to represent the profit. How many should they sell of each to maximize the profit?

1. Write the inequalities to represent this situation.

2. Graph the inequalities from Exercise 1.

3. Find the vertices of the figure formed.

4. What is the maximum profit the Spirit Club can make?

b

tO 100

300

400

500

100

200

200 300 400 500

g

aO 100

-300

-400

-500

-100

-200

200 300 400 500

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Spreadsheet ActivityInequalities in Two Variables

You can use a spreadsheet to determine whether ordered pairs satisfy an inequality.

ExercisesUse a spreadsheet to determine which ordered pairs are part of the solution set for each inequality.

1. 2x + 3y > 1; {(0, 3), (1, -3), (-2, -1), (6, 8)}

2. 7x - y < 8; {(1, 2), (-3, -1), (0, -10), (6, 9)}

3. y ≥ 3x; {(3, 1), (-4, 5), (-1, 0), (7, -1), (2, 7)}

4. y ≤ -4x; {(9, 3), (-3, 5), (0, 0), (12, 1), (3, 9)}

5. y > 12 - 2x; {(-3, -3), (-1, 9), (12, 13), (-4, 11)}

6. y > 2 + 6x; {(0, -4), (-4, 8), (9, 17), (-2, 18), (-5, -5)}

7. |x + 1| ≤ y; {(1, -8), (0, 4), (5, 16), (-2, -8), (11, -2)}

8. |y - 7| < x; {(5, 8), (-1, 3), (2, 19), (-6, -6), (10, -22)}

Use a spreadsheet to determine which ordered pairs from the set {(2, 3), (4, 1), (-1, 2), (0, 7), (-8, -10)} are part of the solution set for 5x - 2y > 12.

Step 1 Use columns A and B of the spreadsheet for the replacement set. Enter the x-coordinates in column A and the y-coordinates in column B.

Step 2 Column C contains the formula for the inequality. Use the names of the cells containing the x- and y-coordinates of each ordered pair to determine whether that ordered pair is part of the solution set. The formula will return TRUE or FALSE.

The solution set contains the ordered pairs for which the inequality is true. The ordered pair {(4, 1)} is part of the solution set of 5x - 2y > 12.

The spreadsheet can also evaluate inequalities involving absolute value. Enter an absolute value expression like |x| using the function ABS(x).

A

1

4

56

7

2

3

B Cy 5x - 2y > 12

=5*A2-2*B2>12

=5*A3-2*B3>12

=5*A4-2*B4>12

=5*A5-2*B5>12

=5*A6-2*B6>12

x

2

4

-1

0

-8

3

1

2

7

-10

Sheet 1 Sheet 2 Sheet 3

A1

4567

23

B Cy 5x - 2y > 12x

2

4

-1

0

-8

3

1

2

7

-10

FALSE

TRUE

FALSE

FALSE

FALSE

Sheet 1 Sheet 2 Sheet 3

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Student Recording Sheet

Use this recording sheet with pages 330 – 331 of the Student Edition.

5

Record your answers for Question 17 on the back of this paper.

Extended Response

10. 9.

10. (grid in)

11.

12.

13. (grid in)

14.

15.

16.

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Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

8. F G H J

Multiple Choice

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing each number or symbol in a box. Then fill in the corresponding circle for that number or symbol.

Short Response/Gridded Response

043_052_ALG1_A_CRM_C05_AS_661384.indd 43043_052_ALG1_A_CRM_C05_AS_661384.indd 43 12/21/10 4:38 PM12/21/10 4:38 PM

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Rubric for Scoring Extended Response

General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show how he or she arrived at the answer, the student will be awarded only 1 credit. All extended response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for all parts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.

• Students who use trial and error to solve a problem must show their method. Merely showing that the answer checks or is correct is not considered a complete response for full credit.

Exercise 17 Rubric

5

Score Specifi c Criteria

4

The inequality written in part a shows that Theresa’s savings after w weeks is 35w and that she can take a vacation after saving at least $640 (35w ≥ 640). In part b, both sides of the inequality should be divided by 35 to get w ≥ 18 2 −

7 , and it should be noted that Theresa must save for 19 weeks (the

mixed number should be rounded up to the next largest integer). In part c, the inequality 45w ≥ 640 should be solved for w (w ≥ 14 2 −

9 ) . The student

should then show that the minimum time will be decreased by 19 – 5 or 4 weeks.

3 A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.


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SCORE


For Questions 1 and 3, solve each inequality.

1. - d − 5 - 12 ≥ 8

2. 23 - t ≤ 2(t - 9) - 3(t + 2)

3. 16 < 3t - 2

4. Define a variable, write an inequality, and solve: The sum of a number and three is less than nineteen less the number.

5. MULTIPLE CHOICE Connor is mailing some letters at the post office. Stamps cost 44 cents each. He also needs to mail a package that costs $7.65. Which expression shows how many letters Connor can mail if he has $10.00 to spend in total?

A 7.65 + 0.44x > 10.00 C 7.65 + 0.44x ≥ 10.00 B 7.65 + 0.44x < 10.00 D 7.65 + 0.44x ≤ 10.00

1. Solve w + 9 ≤ -5. Then graph your solution on a number line.

2. Define a variable, write an inequality, and solve: A number decreased by 7 is at least 15.

Solve each inequality.

3. m − 13

> -6

4. -3n ≤ 84

5. MULTIPLE CHOICE Which inequality does not have the solution {x|x < -2}?

A -3x > 6 C 7x < -14

B - x − 2 < 1 D 4 −

3 x < -

8 − 3

Chapter 5 Quiz 1 (Lessons 5-1 and 5-2)

1.

2.

3.

4.

5.

Chapter 5 Quiz 2 (Lesson 5-3)

1.

-15-16-17-18 -14-13 -11-10-12

2.

3.

4.

5.

5

5

SCORE

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Chapter 5 Quiz 3(Lessons 5-4 and 5-5)

1. Solve -1 < 2x - 1 ≤ 5. Then graph the solution set.

2. MULTIPLE CHOICE Which value of x is not a solution to 3x - 1 < 5 or 7 - x ≤ 3?

A 0 B 2 C 4 D 5

3. Solve ⎪ x - 1 −

2 ⎥ ≤ 1. Then graph the solution set.

4. Solve ⎪ 2x - 1 ⎪ ≥ 3. Then graph the solution set.

5. Write an open sentence involving absolute value for the graph shown.

1. MULTIPLE CHOICE Which is not true about the graph of 2x + y ≥ 1?

A The point (2, 2) is located inside the shaded region.

B The boundary is graphed as a solid line.

C The boundary is graphed along y = -2x + 1.

D The origin is located inside the shaded region.

2. Determine whether the test point (3, 3) is in the shaded half-plane of the graph of y + 2 ≤ 3x.

3. Use a graph to solve 1 − 3 x + 4 ≤ 3.

For Questions 4 and 5, graph each inequality.

4. x < 3

5. -2(x - y) ≤ 4

Chapter 5 Quiz 4 (Lesson 5-6)

1.

2.

3.

4.

5.

5

5

-4 -3 -2 -1 0 1 2 3 4

1.

2.

3.

4.

5.

-4 54321-3 0-1-2

-4 54321-3 0-1-2

-4 54321-3 0-1-2

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SCORE

Part I Write the letter for the correct answer in the blank at the right of each question.

For Questions 1 – 5, solve each inequality.

1. r -

7−

8> 1

A {

r � �

r >

1 −

8 }

B {

r � �

r <

1 −

8 }

C {

r � �

r > 1

7 −

8 }

D {

r � �

r < 1

7 −

8 }

2. 12x + 5 ≥ 17x - 10

F {x � x ≤ -3} G {x � x ≥ 3} H {x � x ≥ -3} J {x � x ≤ 3}

3. 6m - 2(7 + 3m) > 5(2m - 3) - m

A {m � m < 1} B {

m � �

m < 1 −

9 }

C {m � m > 1} D {

m � �

m > 1 −

9 }

4. 2n −

7 ≤ 4

F {n � n ≤ 14} G {n � n ≥ 14} H {

n � �

n ≤ 8 −

7 }

J {

n � �

n ≥ 8 −

7 }

5. 3t - 2(t - 1) ≥ 5t - 4(2 + t) A

{

t � t ≤ -

5 −

7 }

C {t � all real numbers}

B {

t � t ≤ 3 −

4 }

D

Part II

6. Solve the inequality 4.2 > -11 + t. Check your solution.

7. Solve the inequality 2x – 1 > 7. Then graph the solution set.

Define a variable, write an inequality, and solve each problem.

8. For a package to qualify for a certain postage rate, the sum of its length and girth cannot exceed 85 inches. If the girth is 63 inches, how long can the package be?

9. The minimum daily requirement of vitamin C for 14-year-olds is at least 50 milligrams per day. An average-sized apple contains 6 milligrams of vitamin C. How many apples would a person have to eat each day to satisfy this requirement?

1.

2.

3.

4.

5.

6.

7.

8.

9.

Chapter 5 Mid-Chapter Test (Lessons 5-1 through 5-3)

5

1 1098762 543

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SCORE Chapter 5 Vocabulary Test

boundary

closed half-plane

compound inequality

half-plane

intersection

set-builder notation

system of inequalities

union

Choose a term from the vocabulary list above to complete the sentence.

1. An equation defines the or edge for each half-plane.

2. A containing and is true if both of the inequalities it contains are true.

3. The solution set for an inequality that contains twovariables consists of many ordered pairs which fill a region on the coordinate plane called a .

4. The graph of a compound inequality containing and is the of the graphs of the two inequalities.

5. The graph of a compound inequality containing or is the of the graphs of the two inequalities.

Define each term in your own words.

6. open half-plane

7. set-builder notation

1.

2.

3.

4.

5.

6.

7.

5


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SCORE Chapter 5 Test, Form 1

Write the letter for the correct answer in the blank at the right of each question.

For Questions 1–7, solve each inequality.

1. x - 7 > 3 A {x | x > 10} B {x | x > -4} C {x | x < 10} D {x | x < -4}

2. 3 ≥ t + 1 F {t ⎪ t ≤ 4} G {t | t ≥ 2} H {t ⎪ t ≤ 2} J {t ⎪ t ≥ 4}

3. 1 ≥ -y

− 4

A {y � � y ≥ -

1 − 4 } B {y | y ≥ -4} C {y | y ≤ 4} D {y | y ≤ 3}

4. 5m < -25 F {m | m < 125} G {m | m < -125} H {m | m > -5} J {m | m < -5}

5. -36 ≤ 3t A {t ⎪ t ≥ -12} B {t | t ≤ 12} C {t | t ≥ 12} D {t | t ≤ -12}

6. 6y - 8 > 4y + 26 F {y | y > -9} G {y ⎪ y > -17} H {y | y > 9} J {y ⎪ y > 17}

7. 3(2d - 1) ≥ 4(2d - 3) - 3 A {d | d ≥ -9} B {d ⎪ d ≤ -6} C {d | d ≥ 3} D {d ⎪ d ≤ 6}

8. Six is at least four more than a number. Which inequality represents this sentence?

F 6 ≤ n + 4 G 6 ≥ n + 4 H 4 ≤ n + 6 J 4 ≥ n + 6

9. More than eighteen students in an algebra class pass the first test. This is about three-fifths of the class. How many students are in the class?A less than 30 B less than 25 C more than 30 D 25

10. Phillip has between two hundred and three hundred baseball cards. Which inequality represents this situation?

F 200 p > 300 J p < 200 and p > 300

11. Which of the following is the graph of the solution set of m > -1 and m ≤ 1?

A C

B D

12. Which compound inequality has the solution set shown in the graph?

F x < -1 or x > 3 H x > -1 or x ≥ 3 G x > -1 or x < 3 J x ≤ -1 or x ≥ 3

-1-2-3-4 0 1 2 43

-1-2-3-4 0 1 2 43 -1-2-3-4 0 1 2 43

-1-2-3-4 0 1 2 43

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

-1-2-3-4 0 1 2 43

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13. Which of the following is the solution set of 2a + 1 > 9 or a < -1?

A {a � a < -1 or a > 4} C {a � -1 ≤ a ≤ 4}

B {a � a ≤ -1 or a ≥ 4} D {a � a < -1 or a > 5}

14. Which inequality corresponds to the graph shown?

-2 -1-3 0 1 2 3 4 5

F �x - 3� ≤ 1 H �x - 3� ≥ 1 G �x - 1� ≤ 3 J �x - 1� ≥ 3

15. Solve � x - 3 � < 2. A {x � 1 < x < 5} C {x � -1 < x < 1} B {x � -5 < x < -1} D {x � -1 < x < 5}

16. Which inequality has the solution set shown in the graph?

F y < 1 H y > 1 G y ≤ 1 J y ≥ 1

17. Which inequality has the solution set shown in the graph?

A y < -x + 2 C y < -x + 1 B y > -x + 2 D y > -x + 1

18. Determine which of the ordered pairs are a part of the solution set for the inequality graphed at the right.

F (2, 1) H (-3, -3) G (1, 3) J (-2, -3)

19. Which inequality has a solution set of {x � x > 3 or x < -3}?

A � 2x � > 6 C � 2x � ≥ 6 B � 2x � < 6 D � 2x � ≤ 6

20. Juan’s income y consists of at least $37,500 salary plus 5% commission on all of his sales x. Which inequality represents Juan’s income in one year?

F y ≤ 37,500 + 5x H y ≥ 37,500 + 0.05x G y ≥ x + 0.05(37,500) J y ≥ 37,500 + 5

Bonus If x < 0, which integer does not satisfy the inequality x + 2 < 1?

13.

14.

15.

16.

17.

18.

19.

20.

Chapter 5 Test, Form 1 (continued)

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SCORE


For Questions 1–6, solve each inequality.

1. -51 ≤ x + 38 A {x � x ≤ -13} B {x � x ≤ 89} C {x � x ≥ -89} D {x � x ≥ -13} 1.

2. m - 3 − 8 >

1 − 2

F = {m | | m > 7 −

8 } G {m |

| m < 7 − 8 } H {m |

| m < 1 − 8 } J {m |

| m > 1 − 8 } 2.

3. t − -2

> 4

A {t | t < -8} B {t | t < -2} C {t | t > 2} D {t | t > -8} 3.

4. -3.5z < 42 F {z | z > 12} G {z | z < 12} H {z | z < -12} J {z | z > -12} 4.

5. 4w - 6 > 6w - 20 A {w | w < 7} B {w | w < 2} C {w | w < -7} D {w | w < -2} 5.

6. 8r - (5r + 4) ≥ -31 F {r | r ≤ -9} G {r | r ≥ -9} H {r | r ≥ 9} J {r | r ≤ 9} 6.

7. The sum of two consecutive integers is at most 3. What is the greatest possible value for the greater integer?

A 5 B 1 C 3 D 2 7.

8. Which of the following is the graph of the solution set of y < -3 or y < 1?

F -1-2-3-4 0 1 2 43

H -1-2-3-4 0 1 2 43

G -1-2-3-4 0 1 2 43

J -1-2-3-4 0 1 2 43

8.


A -1 < n < 2 C n ≥ -1 or n < 2 B -1 ≤ n < 2 D -1 < n ≤ 2 9.

10. Which of the following is the solution set of -4 < 3t + 5 ≤ 20? F {t |-3 < t ≤ 5} H {t | t < -3} G {t | t < -3 and t ≤ 5} J {t | t < -3 or t ≥ 5} 10.

11. Which of the following is the graph of the solution set of t - 4 ≥ 4t + 8 or 3t > 14 - 4t?

A -1-2-3-4-5 0 1 2 3

C -1-2-3-4-5 0 1 2 3

B -1-2-3-4-5 0 1 2 3

D -1-2-3-4-5 0 1 2 3

11.

-1-2-3-4 0 1 2 43

Chapter 5 Test, Form 2A 5

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12. Which inequality corrresponds to the graph shown?

-2 -1 0 1 2 3 4 5 6

F � x - 2 � < 3 H � x - 2 � ≥ 3 G � x - 2 � > 3 J � x - 2 � ≤ 3

13. Which of the following is the solution set of | 2x - 3 | > 4? A {x � x < -0.5 or x > 3.5} C {x � -0.5 < x < 3.5}

B {x � x < -1 or x > 7} D {x � x < 0.5 or x > 3.5}

14. Pete’s grade on a test was within 5 points of his class average of 94. What is his range of grades on the test?F g ≤ 89 or g ≥ 99 H g ≥ 89 or g ≥ 99

G 89 ≤ g ≤ 99 J g < 99 or g < 89

15. Which ordered pair is part of the solution set of the inequality 12 + y ≤ -3x? A (-16, 3) B (1, 4) C (4, -1) D (3, -16)

16. Which inequality is graphed at the right? F y < 2x + 1 H y < 1 −

2 x + 1

G y > 2x + 1 J y > 1 − 2 x + 1

17. Taka bought a new coat and new shoes. He spent $122. Which inequality represents this situation if x represents the cost of a coat and y represents the cost of the shoes he buys?

A 122 ≤ y + x B y ≤ 122 + x C y - x ≥ 122 D y ≤ 122 - x

18. Determine which of the ordered pairs are a part of the solution of y + 1 > 1 −

2 x + 3.

F (2, 3) G (-4, 0) H (1, 2) J (-3, 1)

19. Which inequality has a solution set of {x � x > -3 or x < -4}? A � 2x + 7 � < 1 C � 2x + 7 � > -1 B � 2x + 7 � > 1 D � 2x + 7 � > -1

20. Laurie and Maya sold at most $50 worth of get-well and friendship cards. The friendship cards, x, were sold for $2 each and the get-well cards, y, were sold for $1.50 each. Which point represents a reasonable number of cards sold?

F (20, 10) G (15, 10) H (18, 20) J (10, 30)

Bonus Solve 6(|n| - 3) - 4 |n| + 5 ≤ 11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Chapter 5 Test, Form 2A (continued)

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For Questions 1 – 6, solve each inequality.

1. -13 > w + 12 A {w ⎪ w < -25} B {w ⎪ w > -25} C {w ⎪ w > -1} D {w ⎪ w < -1}

2. x - 1 −

4 ≤ -

1 −

2

F {

x �

�

x ≤ -

1 −

4 }

G {

x �

�

x ≤ -

3 −

4 }

H {

x �

�

x ≥ -

1

−

4

}

J {

x �

�

x ≥ -

3 −

4 }

3. m −

-5 < -3

A {m ⎪ m > -15} B {m ⎪ m < -15} C {m ⎪ m < 15} D {m ⎪ m > 15}

4. -1.1t ≤ 4.62 F {t ⎪ t ≤ 5.72} G {t ⎪ t ≥ 5.72} H {t ⎪ t ≤ -4.2} J {t ⎪ t ≥ -4.2}

5. 5z - 4 > 2z + 8 A {z ⎪ z > 4} B {z ⎪ z < 1} C {z ⎪ z < 4} D {z ⎪ z > 1}

6. 7 - 9r - (r + 12) ≤ 25 F {r ⎪ r ≤ -3} G {r ⎪ r ≤ -0.6} H {r ⎪ r ≥ -3} J {r ⎪ r ≥ -0.6}

7. The sum of two consecutive integers is at most 7. What is the largest possible value for the lesser integer?

A 1 B 3 C 2 D 5

8. Which of the following is the graph of the solution set of x > 0 or x < -4? F

-1-2-3-4-5-6 0 1 2 H

-1-2-3-4-5-6 0 1 2

G -1-2-3-4-5-6 0 1 2

J -1-2-3-4-5-6 0 1 2


A -2 < y < 3 C y ≥ -2 or y < 3 B -2 < y ≤ 3 D -2 ≤ y < 3

10. Which of the following is the solution set of -3 < 2x + 7 ≤ 13? F {x ⎪ -5 < x ≤ 3} H {x ⎪ x < -5} G {x ⎪ x < 3 or x > -5} J {x ⎪ -5 ≤ x < 3}

11. Which of the following is the graph of the solution set of 7a + 3 ≤ a - 15 or 5a - 3 < 8a?

A -1-2-3-4-5-6 0 1 2

C -1-2-3-4-5-6 0 1 2

B -1-2-3-4-5-6 0 1 2

D -1-2-3-4 0 1 2 43

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

-1-2-3-4 0 1 2 43

Chapter 5 Test, Form 2B5

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12. Which inequality corresponds to the graph shown?

-4 -3 -2 -1 0 1 2 3 4

F � x - 3 � > 1 H � x - 1 � > 3

G � x - 3 � < 1 J � x - 1 � < 3

13. Which of the following is the solution set of 4 - 7x ≥ 3?

A {x | | x < 1 − 7 or x > 1} C {x | | x ≤ 1 −

7 or x ≥ 1}

B {x | x is a real number.} D {x | 1 ≤ x ≤ 7}

14. Katrina’s weight is within 8 pounds of her ideal weight of 120 pounds. What is her range of weight?F x ≥ 112 or x ≥ 128 H 112 ≥ x ≥ 128 G x ≤ 112 or x ≤ 128 J 112 ≤ x ≤ 128

15. Which ordered pair is part of the solution set of the inequality 5 - y ≤ -3x?

A (2, -1) B (-2, -1) C (-3, -5) D (3, -5)

16. Which inequality is graphed?

F y ≤ 2x - 1 H y ≤ -2x - 1 G y ≥ 2x - 1 J y ≥ -2x - 1

17. Alicia has at most $196 to buy a new baseball glove and a new baseball bat. Which inequality represents this situation?

A y ≤ 196 - x B y ≤ 196 + x C 196 ≤ y + x D y - x ≥ 196

18. Determine which of the ordered pairs are a part of the solution set of y + 3 < 2x - 1.F (0, 0) G (2, 0) H (0, -4) J (2, -2)

19. Which inequality has a solution set of {x � x > 4 and x < 8}?

A ⎪ 1 − 2 x - 3⎥ < 1 C ⎪ 1 −

2 x - 1⎥ < 3

B ⎪ 1 − 2 x - 3⎥ > 1 D ⎪ 1 −

2 x - 1⎥ > 3

20. Beng and Shim have less than $30 for candle-making supplies. The moldsx cost $6 each and the wax y is $2 per pound. Which point represents a reasonable number of molds and pounds of wax they could buy?

F (3, 4) G (4, 4) H (5, 1) J (3, 6)

Bonus Solve 2 - 3x < 5(2 - x) ≤ 3(2 - x) + 10.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Chapter 5 Test, Form 2B (continued)5

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SCORE

1. Solve x - 12 > 1. Then graph your solution on a number line.


2. 7 + z < 3

3. b − 8 > -

1 − 5

4. t − 6 ≥ 14

5. -19.8 ≥ 3.6y

6. -4r < 22

7. 4x - 5 < 2x + 11

8. 5(p + 2) - 2(p - 1) ≥ 7p + 4

9. 1.3(c - 4) ≤ 2.6 + 0.7c


10. 3w < 6 and -5 < w

11. -4 ≤ n or 3n + 1 < -2

12. -4x - 8 ≥ -4 or 7x - 5 < 16

For Questions 13 and 14, solve each inequality. Then graph the solution set.

13. � 1 - x � ≤ 2

14. � 3 - 2x � ≥ 1

Chapter 5 Test, Form 2C5

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

-1-2-3-4 0 1 2 43

-1-2-3-4 0 1 2 43

-1-2-3-4 0 1 2 43

0-1-2-3-4 1 2 43

-1-2-3-4-5-6 0 1 2

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15. Solve �8x + 2� < 14.

16. Ian has $6000. He wants to buy a car within $1500 of this amount. Define a variable, write an open sentence, and find the range of car prices.

17. Graph y > - 1 − 3 x + 2.

18. Use a graph to solve 2x - 3y ≤ 6.

19. What inequality has the solution set shown in the graph?

20. EXPENSES Camille has no more than $20.00 to spend each week for lunch and bus fare. Lunch costs $3.00 each day, and bus fare is $0.75 each ride. Write an inequality for this situation. Can Camille buy lunch 5 times and ride the bus 8 times in one week?

Bonus Graph the solution set of the compound inequality 3 < � x - 4 � < 7.

Chapter 5 Test, Form 2C (continued)5

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16.

17.

18.

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SCORE

1. Solve y - 7 ≤ 5. Then graph your solution on a number line.


2. 8 + k ≥ 13

3. h − 3 < 9

4. - 2 − 3 > z −

5

5. 9.8 ≥ 2.8k

6. -3m < -18

7. 5t + 8 ≤ 3t - 3

8. 3(-w - 6) < 2(2w + 8) + 1

9. 1.9 + 1.7x < 2.1(3 + x)


10. 7w > 14 and w < 3

11. w − 3 < 1 or 3w + 5 > 11

12. 2 + 3x > 8 or 4 - 7x ≤ -17

For Questions 13 and 14, solve each inequality. Then graph the solution set.

13. � z + 4 � ≥ 7

14. � w - 1 � ≤ 4

15. Solve � 2x - 5 � < 3.

Chapter 5 Test, Form 2D5

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

-11 30

-1-2-3-4 0 1 2 43

0-1-2-3-4 1 2 3 4

-1-2-3-4 0 1 2 43

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16. Abe has $4500. He wants to buy a boat within $1300 of this amount. Define a variable, write an open sentence, and find the range of boat prices.

17. Graph y ≤ 3x.

18. Use a graph to solve 2y - 4x < 8.

19. What inequality has the solution set shown in the graph?

20. SHOPPING Matthew is shopping for shoes and socks. He has $75.00 to spend. The shoes he likes cost $28.00, and the socks cost $4.00. Write an inequality for this situation. Can Matthew buy 2 pairs of shoes and 5 pairs of socks?

Bonus Graph the solution set of the compound inequality | x + 1 | < 4 or | x + 1 | ≥ 6.

y

x

Chapter 5 Test, Form 2D (continued)5

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17.

18.

19.

20.

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SCORE

Solve each inequality. Then graph your solution on a number line.

1. m - (-3.4) ≥ 12.7

2. t + (-4) < 32


3. Negative three sevenths plus a number is at least 2.

4. A number less 15 is greater than the sum of twice the number and 8.


5. -2.6 ≥ w − 4

6. -11t < -9

7. 2 - 3b > 11 - 15b − 7

8. 5x - 3(x - 6) ≤ 0

9. -3x + 2(6x - 7) > 4(3 - 2x) + 17x - 8


10. Raul plans to spend no more than $78.00 on two shirts and a pair of jeans. He bought the two shirts for $19.89 each. How much can he spend on the jeans?

11. The sum of two consecutive positive even integers is at most 15. What are the possible pairs of integers?

12. Susan makes 10% commission on her sales. She also receives a salary of $25,600. How much must she sell to receive a total income between $32,500 and $41,900?

Chapter 5 Test, Form 35

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

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Solve each compound inequality, and graph the solution set.

13. - n − 2 < 3 or 2n - 3 > 12

14. 2(x - 14) - x < 7(x + 2) + x ≤ x + 70

For Questions 15–17, solve each inequality. Then graph the solution set.

15. |-4x + 8 | < 16

16. | 5x - 3 | ≥ 17

17. ⎪ 3 - 2x −

5 ⎥ ≥ 1

18. Graph -y ≤ 3x.

19. Use a graph to solve x + 3y > -12.

20. DOGS Each afternoon Maria walks the dogs at a local pet shelter for up to 2 hours. Maria spends 16 minutes walking a large dog and 12 minutes walking a small dog. Write an inequality for this situation. If Maria walked 9 dogs in one afternoon, what is the greatest number of large dogs that she could have walked that afternoon?

Bonus If xy < 0, determine if the compound inequality, 2x + 1 > 7 and 4 - y < 3, is true or false. Explain your reasoning.

Chapter 5 Test, Form 3 (continued)5

13.

14.

-6 80

15.

16.

17.

18.

19.

20.

B:

-1-2-3-4-5-6-7-8-9

-1-2 0 1 2 4 5 63

-1-2 0 1 2 4 5 63

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SCORE Chapter 5 Extended-Response Test

Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. Solve 10n - 7(n + 2) > 5n - 12. Explain each step in your solution.

2. Draw a line on a coordinate plane so that you can determine at least two points on the graph.

a. Write an inequality to represent one of the half planes created by the line.

b. Determine if the solution set of the inequality written for part a includes the line or not. Explain your response.

3. Let b > 2. Describe how you would determine if ab > 2a.

4. Determine if the open sentence | x - 2 | > 4 and the compound inequality -2x < 4 or x > 6 have the same solution set.

5. ARCHITECTURE An architect is designing a house for the Frazier family. In the design, she must consider the desires of the family and the local building codes. The rectangular lot on which the house will be built is 158 feet long, and 90 feet wide.

a. The building codes state that one can build no closer than 20 feet to the lot line. Write an inequality to represent the possible widths of the house along the 90-foot dimension. Solve the inequality.

b. The Fraziers requested that the rectangular house contain no less than 2800 square feet and no more than 3200 square feet of floor space. If the house has only one floor, use the maximum value for the width of the house from part a, and explain how to use an inequality to find the possible lengths.

c. The Fraziers have asked that the cost of the house be about $175,000 and are willing to deviate from this price no more than $20,000. Write an open sentence involving an absolute value and solve. Explain the meaning of the answer.

5

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SCORE

1. Which equation is not equivalent to x - 7 = 12? (Lesson 2-2)

A x - 9 = 14 B x - 10 = 9 C x = 19 D x - 3 = 16

2. Find the value of y so that the line through (2, 3) and (5, y) has a slope of -2. (Lesson 3-3)

F -3 G 3 − 2 H 9 J 9 −

2

3. Solve -8x - 15 = -31. (Lesson 2-3)

A 22 B 6 C 2 D 26

4. If f (x) = 3(x - 5), find f (4). (Lesson 1-7)

F 7 G 27 H -3 J 3

5. Which equation shows the slope-intercept form of the line passing through (0, 1) and (2, 0)? (Lesson 4-2)

A y = -2x + 1 B y = 1 − 2 x -1 C y = 2x - 1 D y = - 1 −

2 x + 1

6. Write a compound inequality for the graph shown below. (Lesson 5-4)

F -1 < x ≤ 2 H -1 ≤ x < 2 G x ≤ -1 or x > 2 J x < -1 or x ≥ 2

7. Solve - 1 − 3 h ≤ 6. (Lesson 5-2)

A h ≤ –2 B h ≤ -18 C h ≥ -2 D h ≥ -18

8. Solve h + 3 ≥ 2. (Lesson 5-1)

F h ≤ 2 G h ≥ -1 H h ≥ 5 J h ≤ -1

9. Solve 4x + 12 > 2. (Lesson 5-3)

A x > -2 1 − 2 B x > -40 C x > 2 1 −

2 D x > 3 1 −

2

10. Which of the following is an arithmetic sequence? (Lesson 3-5)

F 1, 3, 6, 10, … H 34, 35, 38, 43, … G 5, 8, 11, 14, … J 1, 4, 9, 16, …

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. F G H J

F G H J

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

5 Standardized Test Practice(Chapters 1–5)

-3 -2 -1 0 1 2 3 4 5

A B C D

A B C D

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pyr

ight

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oe/

McG

raw

-Hill

, a d

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of

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McG

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s, In

c.C

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ill, a

div

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Inc.

NAME DATE PERIOD

Ass

essm

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11. Determine which is a linear equation. (Lesson 3-1)

A 1 − x – y = 7 C 3 = xy

B x2 – 4 = y D x - y = 4

12. Find the discounted price. Pants: $24 (Lesson 2-7)

Discount: 15%F $20.40 G $3.60 H $20 J $9

13. Solve 8x - 5 = 23 + 4x. (Lesson 2-4)

A 4.5 B 7 C 23 D 5

14. Rewrite 5(a - b + c) using the Distributive Property. (Lesson 1-4)

F 5a - b + c H 5a - 5b + 5cG 5a + 5b + c J 5a + b + c

15. Write an equation that passes through (3, 2) and has a slope of -2. (Lesson 4-3)

A y = 8x - 2 C y = -2x + 7 B y = -2x + 8 D y = -2x + 2

16. Find the slope of the line that passes through (-7, 8) and (-6, 5). (Lesson 3-3)

F -3 G - 1 − 3 H 3 J -6

17. Evaluate the expression if x = 4, y = 3, and z = 2. (Lesson 1-2)

x2 + 4y + z A 27 B 22 C 20 D 30

11.

12.

13.

14.

15.

16.

17. A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

18. What is the slope of a line parallel to the line that passes through (-3, 1) and (3, 7)? (Lesson 4-4)

19. If m + 3 ≥ 14, then complete the inequality m - 6 ≥ . (Lesson 5-1)

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

0

?

5 Standardized Test Practice (continued)

Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the answer in a column box

and then shading in the appropriate circle that corresponds to that entry.


Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

NAME DATE PERIOD

PDF Pass


20. Solve a −

6 - 5 = 12. (Lesson 2-3)

21. If f (x) = x2 - 4x, find f (-3). (Lesson 1-7)

22. Solve y = 1 − 4 x - 1 if the domain is {-4, -2, 0, 2, 4}. (Lesson 1-5)

23. Write the slope-intercept form of an equation of the line that passes through (0, -4) and is parallel to the graph of 4x - y = 7. (Lesson 4-4)

24. Solve 4 − 5 a ≤ -12. (Lesson 5-2)

25. Solve the proportion 0.6 − x = 0.3 − 5 . (Lesson 2-6)

26. Graph 2x + 3y ≥ -9. (Lesson 5-6)

27. Solve ⎪3f + 2⎥ ≤ 7. Then graph the solution set. (Lesson 5-5)

28. Solve -5 ≤ 2a- 1 < 9. Then graph the solution set. (Lesson 5-4)

29. Graph the equation y = x - 4. (Lesson 3-1)

30. Mark is shopping during a computer store’s 20% sale. He is considering buying computers that range in cost from $500 to $1000.

a. How much are the computers after the 20% discount? (Lesson 5-4)

b. If sales tax is 7%, how much should Mark expect to pay? (Lesson 5-4)

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30a.

30b.

y

xO

5 Standardized Test Practice (continued)

Part 3: Short Response

Instructions: Write your answers in the space provided.

-4 -3 -2 -1 0 1 2 3 4

-3 -2 -1 0 1 2 3 4 5

y

xO

Co

pyr

ight

© G

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McG

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, a d

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of

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c.

An

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s

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Chapter 5 A1 Glencoe Algebra 1

Chapter Resources

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

3 G

lenc

oe A

lgeb

ra 1

Ant

icip

atio

n G

uide

Lin

ear

Ineq

ualiti

es

B

efor

e yo

u b

egin

Ch

ap

ter

5

•

Rea

d ea

ch s

tate

men

t.

•

Dec

ide

wh

eth

er y

ou A

gree

(A

) or

Dis

agre

e (D

) w

ith

th

e st

atem

ent.

•

Wri

te A

or

D i

n t

he

firs

t co

lum

n O

R i

f yo

u a

re n

ot s

ure

wh

eth

er y

ou a

gree

or

disa

gree

, wri

te N

S (

Not

Su

re).

ST

EP

1A

, D, o

r N

SS

tate

men

tS

TE

P 2

A o

r D

1.

Acc

ordi

ng

to t

he

Add

itio

n P

rope

rty

of I

neq

ual

itie

s, a

ddin

g an

y n

um

ber

to e

ach

sid

e of

a t

rue

ineq

ual

ity

wil

l re

sult

in

a

tru

e in

equ

alit

y.

2.

Th

e in

equ

alit

y m

+ 2

3 ≥

35

can

be

solv

ed b

y ad

din

g 23

to

each

sid

e.

3.

16 i

s n

o gr

eate

r th

an t

he

dif

fere

nce

of

a n

um

ber

and

12

can

be

wri

tten

as

16 ≤

n –

12.

4.

If b

oth

sid

es o

f r −

12

< 4

are

mu

ltip

lied

by

12, t

he

resu

lt i

s r

< 4

8.

5.

Th

e re

sult

of

divi

din

g bo

th s

ides

of

the

ineq

ual

ity

–2y

≥ 1

0 by

–2

is y

≥ –

5.

6.

To

solv

e an

in

equ

alit

y in

volv

ing

mu

ltip

lica

tion

, su

ch a

s 9t

> 2

7, d

ivis

ion

is

use

d.

7.

To

solv

e th

e in

equ

alit

y 8x

– 2

< 7

0, f

irst

div

ide

by 8

an

d th

en a

dd 2

.

8.

A c

ompo

un

d in

equ

alit

y is

an

in

equ

alit

y co

nta

inin

g m

ore

than

on

e va

riab

le.

9.

On

a n

um

ber

lin

e, a

clo

sed

dot

is u

sed

for

an i

neq

ual

ity

con

tain

ing

the

sym

bol

≥ o

r ≤

.10

. If

⎪t⎥

< 8

, th

en t

equ

als

all

nu

mbe

rs b

etw

een

0 a

nd

8.11

. O

n t

he

grap

h o

f y

> 2

x – 3

, th

e so

luti

on s

et w

ill

be a

ll

nu

mbe

rs a

bove

th

e gr

aph

of

the

lin

e y

= 2

x – 3

.

Aft

er y

ou c

omp

lete

Ch

ap

ter

5

•

Rer

ead

each

sta

tem

ent

and

com

plet

e th

e la

st c

olu

mn

by

ente

rin

g an

A o

r a

D.

•

Did

an

y of

you

r op

inio

ns

abou

t th

e st

atem

ents

ch

ange

fro

m t

he

firs

t co

lum

n?

•

For

th

ose

stat

emen

ts t

hat

you

mar

k w

ith

a D

, use

a p

iece

of

pape

r to

wri

te a

n

exam

ple

of w

hy

you

dis

agre

e.

Step

1

5 Step

2

A A A AD A AD D D D

001_

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ALG

1_A

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4:37

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Answers (Anticipation Guide and Lesson 5-1)

Lesson 5-1


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

5 G

lenc

oe A

lgeb

ra 1

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g I

neq

ualiti

es b

y A

dd

itio

n a

nd

Su

btr

acti

on

Solv

e In

equ

alit

ies

by

Ad

dit

ion

Add

itio

n c

an b

e u

sed

to s

olve

in

equ

alit

ies.

If

any

nu

mbe

r is

add

ed t

o ea

ch s

ide

of a

tru

e in

equ

alit

y, t

he

resu

ltin

g in

equ

alit

y is

als

o tr

ue.

Ad

dit

ion

Pro

per

ty o

f In

equ

alit

ies

Fo

r a

ll nu

mb

ers

a,

b,

an

d c

, if a

> b

, th

en

a +

c >

b +

c,

an

d if

a <

b,

the

n a

+ c



an

d <

are

rep

lace

d w

ith

≥ a

nd

≤.

S

olve

x -

8 ≤

-6.

T

hen

gra

ph

th

e so

luti

on.

x

- 8

≤ -

6 O

rigin

al in

equalit

y

x

- 8

+ 8

≤ -

6 +

8

Add 8

to e

ach s

ide.

x

≤ 2

S

implif

y.

Th

e so

luti

on i

n s

et-b

uil

der

not

atio

n i

s {x

|x ≤

2}.

Nu

mbe

r li

ne

grap

h:

-4

-3

-2

-1

01

23

4

S

olve

4 -

2a

> -

a. T

hen

gr

aph

th

e so

luti

on.

4

- 2

a >

-a

Ori

gin

al in

equalit

y

4

- 2

a +

2a

> -

a +

2a

Add 2

a to

each s

ide.

4

> a

S

implif

y.

a

< 4

4 >

a is t

he s

am

e a

s a

< 4

.

The

sol

utio

n in

set

-bui

lder

not

atio

n is

{a|

a <

4}.

Nu

mbe

r li

ne

grap

h:

-2

-1

01

23

45

6

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Ch

eck

you

r so

luti

on, a

nd

th

en g

rap

h i

t on

a n

um

ber

lin

e.

1. t

- 1

2 ≥

16

{t �

t ≥

28}

2. n

- 1

2 <

6 {

n �

n <

18}

3.

6 ≤

g -

3 {

g �

g ≥

9}

2627

2829

3031

3233

34

14

1512

1316

1718

1920

78

910

1112

1314

15

4. n

- 8

< -

13 {

n �

n <

-5}

5.

-12

> -

12 +

y {

y �

y <

0}

6. -

6 >

m -

8 {

m �

m <

2}

-9

-10

-8

-7

-6

-5

-4

-3

-2

-3

-4

-2

-1

01

23

4

-

4-

2-

10

12

3-

34

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

.

7. -

3x ≤

8 -

4x

8. 0

.6n

≥ 1

2 -

0.4

n

9. -

8k -

12

< -

9k

{

x �

x ≤

8}

{n

� n

≥ 1

2}

{k

� k

< 1

2}

10. -

y -

10

> 1

5 -

2y

11. z

- 1 −

3 ≤

4 −

3

12. -

2b >

-4

- 3

b

{

y �

y >

25}

{

z �

z ≤

1 2 −

3 }

{b

� b

> -

4}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k

you

r so

luti

on.

13–1

5. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

13. A

nu

mbe

r de

crea

sed

by 4

is

less

th

an 1

4. n

- 4

< 1

4; {

n �

n <

18}

14. T

he

diff

eren

ce o

f tw

o n

um

bers

is

mor

e th

an 1

2, a

nd

one

of t

he

nu

mbe

rs i

s 3.

n

- 3

> 1

2; {

n �

n >

15}

or

3 -

n >

12;

{n

� n

< -

9}

15. F

orty

is

no

grea

ter

than

th

e di

ffer

ence

of

a n

um

ber

and

2. 4

0 ≤

n -

2;

{n �

n ≥

42}

5-1

Exam

ple

1Ex

amp

le 2

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

512

/21/

10

4:37

PM

A01_A12_ALG1_A_CRM_C05_AN_661384.indd A1A01_A12_ALG1_A_CRM_C05_AN_661384.indd A1 12/21/10 6:14 PM12/21/10 6:14 PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

6 G

lenc

oe A

lgeb

ra 1

Stud

y G

uide

and

Inte

rven

tion

(c

onti

nu

ed)

So

lvin

g I

neq

ualiti

es b

y A

dd

itio

n a

nd

Su

btr

acti

on

Solv

e In

equ

alit

ies

by

Sub

trac

tio

n S

ubtr

acti

on c

an b

e us

ed t

o so

lve

ineq

ualit

ies.

If

any

num

ber

is s

ubtr

acte

d fr

om e

ach

side

of

a tr

ue i

nequ

alit

y, t

he r

esul

ting

ine

qual

ity

is a

lso

true

.

Su

btr

acti

on

Pro

per

ty o

f In

equ

alit

ies

Fo

r a

ll nu

mb

ers

a,

b,

an

d c

, if a

> b

, th

en

a -

c >

b -

c,

an

d if

a <

b,

the

n a

- c



an

d <

are

rep

lace

d w

ith

≥ a

nd

≤.

S

olve

3a

+ 5

> 4

+ 2

a. T

hen

gra

ph

it

on a

nu

mb

er l

ine.

3a

+ 5

> 4

+ 2

a O

rigin

al in

equalit

y

3a +

5 -

2a

> 4

+ 2

a -

2a

Subtr

act

2a

from

each s

ide.

a

+ 5

> 4

S

implif

y.

a

+ 5

- 5

> 4

- 5

S

ubtr

act

5 f

rom

each s

ide.

a

> -

1 S

implif

y.

Th

e so

luti

on i

s {a

�a >

-1}

.N

um

ber

lin

e gr

aph

: -

4-

3-

2-

10

12

34

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Ch

eck

you

r so

luti

on, a

nd

th

en g

rap

h i

t on

a n

um

ber

lin

e.

1. t

+ 1

2 ≥

8

2. n

+ 1

2 >

-12

3.

16

≤ h

+ 9

{

t � t

≥ -

4}

{n

� n

> -

24}

{h

� h

≥ 7

}

-6

-5

-4

-3

-2

-1

01

2

-

26-

25-

24-

23-

22-

21

5

67

89

1011

1213

4. y

+ 4

> -

2

5. 3

r +

6 >

4r

6. 3 −

2 q

- 5

≥ 1 −

2 q

{

y �

y >

-6}

{

r � r

< 6

} {

q �

q ≥

5}

-8

-7

-6

-5

-4

-3

-2

-1

0

2

34

56

79

18

21

03

45

67

8

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

. 7

. 4p

≥ 3

p +

0.7

8.

r +

1 −

4 >

3 −

8

9. 9

k +

12

> 8

k

{

p �

p ≥

0.7

} {

r � r

> 1 −

8 }

{k

� k

> -

12}

10. -

1.2

> 2

.4 +

y

11. 4

y <

5y+

14

12. 3

n +

17

< 4

n

{

y �

y <

-3.

6}

{y

� y

> -

14}

{n

� n

> 1

7}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k y

our

solu

tion

. 13

–15.

Sam

ple

an

swer

: L

et n

= t

he

nu

mb

er.

13. T

he

sum

of

a n

um

ber

and

8 is

les

s th

an 1

2. n

+ 8

< 1

2; {

n �

n <

4}

14. T

he

sum

of

two

nu

mbe

rs i

s at

mos

t 6,

an

d on

e of

th

e n

um

bers

is

-2.

n

+ (

-2)

≤ 6

; {n

� n

≤ 8

}

15. T

he s

um o

f a

num

ber

and

6 is

gre

ater

tha

n or

equ

al t

o -

4. n

+ 6

≥ -

4; {

n �

n ≥

-10

}

5-1

Exam

ple

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

612

/21/

10

4:37

PM

Lesson 5-1


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

7 G

lenc

oe A

lgeb

ra 1

Skill

s Pr

acti

ceS

olv

ing

In

eq

ualiti

es b

y A

dd

itio

n a

nd

Su

btr

acti

on

Mat

ch e

ach

in

equ

alit

y to

th

e gr

aph

of

its

solu

tion

.

1. x

+ 1

1 >

16

c a.

-

8-

7-

6-

5-

4-

3-

2-

10

2. x

- 6

< 1

e

b.

3. x

+ 2

≤ -

3 a

c.

4. x

+ 3

≥ 1

b

d.

5. x

- 1

< -

7 d

e.

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

, an

d t

hen

gra

ph

it

on a

nu

mb

er l

ine.

6. d

- 5

≤ 1

{d

� d

≤ 6

} 7.

t +

9 <

8 {

t � t

< -

1}

2

30

14

56

78

-2

-1

-4

-3

01

23

4

8. a

- 7

> -

13 {

a �

a >

-6}

9.

w -

1 <

4 {

w �

w <

5}

-8

-7

-6

-5

-4

-3

-2

-1

0

2

30

14

56

78

10. 4

≥ k

+ 3

{k

� k

≤ 1

} 11

. -9

≤ b

- 4

{b

� b

≥ -

5}

-

2-

1-

4-

30

12

34

-8

-7

-6

-5

-4

-3

-2

-1

0

12. -

2 ≥

x +

4 {

x �

x ≤

-6}

13

. 2y

< y

+ 2

{y

� y

< 2

}

-

6-

5-

8-

7-

4-

3-

2-

10

-2

-1

-4

-3

01

23

4

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m.

Ch

eck

you

r so

luti

on.

14–1

8. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

14. A

nu

mbe

r de

crea

sed

by 1

0 is

gre

ater

th

an -

5. n

- 1

0 >

-5;

{n

� n

> 5

}

15. A

nu

mbe

r in

crea

sed

by 1

is

less

th

an 9

. n

+ 1

< 9

; {n

� n

< 8

}

16. S

even

mor

e th

an a

num

ber

is l

ess

than

or

equa

l to

-18

. n

+ 7

≤ -

18; {

n �

n ≤

-25

}

17. T

wen

ty l

ess

than

a n

um

ber

is a

t le

ast

15.

n -

20

≥ 1

5; {

n �

n ≥

35}

18. A

nu

mbe

r pl

us

2 is

at

mos

t 1.

n +

2 ≤

1;

{n �

n ≤

-1}

-8

-7

-6

-5

-4

-3

-2

-1

087

65

43

21

0

43

21

0-

1-

2-

3-

4

5-1

87

65

43

21

0

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

712

/21/

10

4:37

PM

Answers (Lesson 5-1)

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

8 G

lenc

oe A

lgeb

ra 1

5-1

Mat

ch e

ach

in

equ

alit

y w

ith

its

cor

resp

ond

ing

grap

h.

1. -

8 ≥

x -

15

b

a.

-6

-5

-4

-3

-2

-1

01

2

2. 4

x +

3 <

5x

d

b.

87

65

43

21

0

3. 8

x >

7x

- 4

a

c.

-8

-7

-6

-5

-4

-3

-2

-1

0

4. 1

2 +

x ≤

9 c

d

. 2

34

56

78

10

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

, an

d t

hen

gra

ph

it

on a

nu

mb

er l

ine.

5. r

- (

-5)

> -

2 {r

| r >

- 7

} 6.

3x

+ 8

≥ 4

x {x

| x ≤

8}

-8

-7

-6

-5

-4

-3

-2

-1

0

4

52

36

78

910

7. n

- 2

.5 ≥

-5

{n | n

≥ -

2.5

} 8.

1.5

< y

+ 1

{y | y

> 0

.5}

-4

-3

-2

-1

01

23

4

-

4-

3-

2-

10

12

34

9. z

+ 3

> 2

−

3 10

. 1 −

2 ≤

c -

3 −

4

-4

-3

-2

-1

01

23

4

-

4-

3-

2-

10

12

34

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k

you

r so

luti

on.

11. T

he

sum

of

a n

um

ber

and

17 i

s n

o le

ss t

han

26.

n

+ 1

7 ≥

26;

{n

| n ≥

9}

12. T

wic

e a

nu

mbe

r m

inu

s 4

is l

ess

than

th

ree

tim

es t

he

nu

mbe

r. 2n

- 4

< 3

n;

{n | n

> -

4}

13. T

wel

ve i

s at

mos

t a

nu

mbe

r de

crea

sed

by 7

. 12

≤ n

- 7

; {n

| n ≥

19}

14. E

igh

t pl

us

fou

r ti

mes

a n

um

ber

is g

reat

er t

han

fiv

e ti

mes

th

e n

um

ber.

8 +

4n

> 5

n;

{n | n

< 8

}

15. A

TMO

SPH

ERIC

SC

IEN

CE

Th

e tr

opos

pher

e ex

ten

ds f

rom

th

e E

arth

’s s

urf

ace

to a

hei

ght

of 6

–12

mil

es, d

epen

din

g on

th

e lo

cati

on a

nd

the

seas

on. I

f a

plan

e is

fly

ing

at a

n

alti

tude

of

5.8

mil

es, a

nd

the

trop

osph

ere

is 8

.6 m

iles

dee

p in

th

at a

rea,

how

mu

ch

hig

her

can

th

e pl

ane

go w

ith

out

leav

ing

the

trop

osph

ere?

16. E

AR

TH S

CIE

NC

E M

atu

re s

oil

is c

ompo

sed

of t

hre

e la

yers

, th

e u

pper

mos

t be

ing

tops

oil.

Jam

al i

s pl

anti

ng

a bu

sh t

hat

nee

ds a

hol

e 18

cen

tim

eter

s de

ep f

or t

he

root

s. T

he

inst

ruct

ion

s su

gges

t an

add

itio

nal

8 c

enti

met

ers

dept

h f

or a

cu

shio

n. I

f Ja

mal

wan

ts t

o ad

d ev

en m

ore

cush

ion

, an

d th

e to

psoi

l in

his

yar

d is

30

cen

tim

eter

s de

ep, h

ow m

uch

m

ore

cush

ion

can

he

add

and

stil

l re

mai

n i

n t

he

tops

oil

laye

r?

Prac

tice

So

lvin

g I

neq

ualiti

es b

y A

dd

itio

n a

nd

Su

btr

acti

on

11-

14. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

no

mo

re t

han

2.8

mi

no

mo

re t

han

4 c

m

{z | z

> -

2 1 −

3 }

{c | c

≥ 1

1 −

4 }

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

812

/21/

10

4:37

PM

Lesson 5-1


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

9 G

lenc

oe A

lgeb

ra 1

Wor

d Pr

oble

m P

ract

ice

So

lvin

g I

neq

ualiti

es b

y A

dd

itio

n a

nd

Su

btr

acti

on

1.SO

UN

D T

he

lou

dest

in

sect

on

Ear

th i

s th

e A

fric

an c

icad

a. I

t pr

odu

ces

sou

nds

as

lou

d as

105

dec

ibel

s at

20

inch

es a

way

. T

he

blu

e w

hal

e is

th

e lo

ude

st m

amm

al

on E

arth

. Th

e ca

ll o

f th

e bl

ue

wh

ale

can

re

ach

lev

els

up

to 8

3 de

cibe

ls l

oude

r th

an

the

Afr

ican

cic

ada.

How

lou

d ar

e th

e ca

lls

of t

he

blu

e w

hal

e?

2. G

AR

BA

GE

Th

e am

oun

t of

gar

bage

th

at

the

aver

age

Am

eric

an a

dds

to a

lan

dfil

l da

ily

is 4

.6 p

oun

ds. I

f at

lea

st 2

.5 p

oun

ds

of a

per

son

’s d

aily

gar

bage

cou

ld b

e re

cycl

ed, h

ow m

uch

wil

l st

ill

go i

nto

a

lan

dfil

l?

3. S

HO

PPIN

G T

yler

has

$75

to

spen

d at

th

e m

all.

He

purc

has

es a

mu

sic

vide

o fo

r $1

4.99

an

d a

pair

of

jean

s fo

r $1

8.99

. He

also

spe

nt

$4.7

5 fo

r lu

nch

. Tyl

er s

till

w

ants

to

purc

has

e a

vide

o ga

me.

How

m

uch

mon

ey c

an h

e sp

end

on a

vid

eo

gam

e?

4. S

UPR

EME

CO

UR

T T

he

firs

t C

hie

f Ju

stic

e of

th

e U

.S. S

upr

eme

Cou

rt, J

ohn

Ja

y, s

erve

d 20

79 d

ays

as C

hie

f Ju

stic

e.

He

serv

ed 1

0,46

3 da

ys f

ewer

th

an J

ohn

M

arsh

all,

wh

o se

rved

as

Su

prem

e C

ourt

C

hie

f Ju

stic

e fo

r th

e lo

nge

st p

erio

d of

ti

me.

How

man

y da

ys m

ust

th

e cu

rren

t S

upr

eme

Cou

rt C

hie

f Ju

stic

e Jo

hn

R

ober

ts s

erve

to

surp

ass

Joh

n M

arsh

all’s

re

cord

of

serv

ice?

5. W

EATH

ER T

heo

dore

Fu

jita

of

the

Un

iver

sity

of

Ch

icag

o de

velo

ped

a cl

assi

fica

tion

of

torn

adoe

s ac

cord

ing

to

win

d sp

eed

and

dam

age.

Th

e ta

ble

show

s th

e cl

assi

fica

tion

sys

tem

.

Lev

elN

ame

Win

d S

pee

d R

ang

e (m

ph

)

F0

Ga

le4

0 –

72

F1

Mo

de

rate

73

–11

2

F2

Sig

nifi

ca

nt

113

–15

7

F3

Seve

re15

8–

20

6

F4

Deva

sta

tin

g2

07

–2

60

F5

Incre

dib

le2

61

–3

18

F6

Inco

nce

iva

ble

319

–3

79

Sou

rce:

Nat

iona

l Wea

ther

Ser

vice

a. S

upp

ose

an F

3 to

rnad

o h

as w

inds

th

at

are

162

mil

es p

er h

our.

Wri

te a

nd

solv

e an

in

equ

alit

y to

det

erm

ine

how

m

uch

th

e w

inds

wou

ld h

ave

to

incr

ease

bef

ore

the

F3

torn

ado

beco

mes

an

F4

torn

ado.

16

2 +

x ≥

207

; x

≥ 4

5; a

t le

ast

45 m

ph

b.

A t

orn

ado

has

win

d sp

eeds

th

at a

re a

t le

ast

158

mil

es p

er h

our.

Wri

te a

nd

solv

e an

in

equ

alit

y th

at d

escr

ibes

how

m

uch

gre

ater

th

ese

win

d sp

eeds

are

th

an t

he

slow

est

torn

ado.

40 +

y ≥

158

; y ≥

118

; at

leas

t 11

8 m

ph

less

th

an 1

88 d

ecib

els

no

mo

re t

han

2.1

po

un

ds

per

day

.

Tyle

r ca

n s

pen

d n

o m

ore

than

$36

.27

on

a v

ideo

gam

e.

x >

12,

542;

mo

re t

han

12,

542

day

s

5-1

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

912

/21/

10

4:37

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass


Lesson 5-2


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

11

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g I

neq

ualiti

es b

y M

ult

iplicati

on

an

d D

ivis

ion

Solv

e In

equ

alit

ies

by

Mu

ltip

licat

ion

If

each

sid

e of

an

in

equ

alit

y is

mu

ltip

lied

by

the

sam

e po

siti

ve n

um

ber,

the

resu

ltin

g in

equ

alit

y is

als

o tr

ue.

How

ever

, if

each

sid

e of

an

in

equ

alit

y is

mu

ltip

lied

by

the

sam

e n

egat

ive

nu

mbe

r, th

e di

rect

ion

of

the

ineq

ual

ity

mu

st

be r

ever

sed

for

the

resu

ltin

g in

equ

alit

y to

be

tru

e.

Mu

ltip

licat

ion

Pro

per

ty o

f In

equ

alit

ies

Fo

r a

ll nu

mb

ers

a,

b,

an

d c

, w

ith

c ≠

0,

1.

if c

is p

ositiv

e a

nd

a >

b,

the

n a

c >

bc;

if c

is p

ositiv

e a

nd

a <

b,

the

n a

c <

bc;

2.

if c

is n

ega

tive

an

d a

> b

, th

en

ac

 b

c.

Th

e pr

oper

ty i

s al

so t

rue

wh

en >

an

d <

are

rep

lace

d w

ith

≥ a

nd

≤.

S

olve

- y −

8 ≤

12.

- y −

8 ≥

12

Orig

inal in

equalit

y

(-8)

(- y −

8 ) ≤

(-

8)12

M

ultip

ly e

ach s

ide b

y -

8; change ≥

to ≤

.

y

≤ -

96

Sim

plif

y.

Th

e so

luti

on i

s { y

� y ≤

-96

}.

S

olve

3 −

4 k

< 1

5.

3 −

4 k

< 1

5 O

rigin

al in

equalit

y

( 4 −

3 ) 3

−

4 k <

( 4 −

3 ) 1

5 M

ultip

ly e

ach s

ide b

y 4

− 3

.

k

< 2

0 S

implif

y.

Th

e so

luti

on i

s {k

� k

< 2

0}.

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Ch

eck

you

r so

luti

on.

1. y −

6 ≤

2

2. -

n

−

50 >

22

3. 3 −

5 h

≥ -

3 4.

- p −

6 <

-6

{

y �

y ≤

12}

{

n �

n <

-11

00}

{

h �

h ≥

-5}

{

p �

p >

36}

5.

1 −

4 n

≥ 1

0 6.

- 2 −

3 b

< 1 −

3

7. 3m

−

5

< -

3 −

20

8.

-2.

51 ≤

- 2h

−

4

{

n �

n ≥

40}

{b

� b

> -

1 −

2 }

{m

� m

< -

1 −

4 }

{h

� h

≤ 5

.02}

9. g −

5 ≥

-2

10. -

3 −

4 >

- 9p

−

5

11.

n

−

10 ≥

5.4

12

. 2a

−

7

≥ -

6

{

g �

g ≥

-10

} {p

� p

>

5 −

12

} {

n �

n ≥

54}

{

a �

a ≥

-21

}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k

you

r so

luti

on.

13–1

5. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

13. H

alf

of a

nu

mbe

r is

at

leas

t 14

. 1 −

2 n

≥ 1

4; {

n �

n ≥

28}

14. T

he

oppo

site

of

one-

thir

d a

nu

mbe

r is

gre

ater

th

an 9

. -

1 −

3 n

> 9

; {n

� n

< -

27}

15. O

ne

fift

h o

f a

nu

mbe

r is

at

mos

t 30

. 1 −

5 n

≤ 3

0; {

n �

n ≤

150

}

5-2

Exam

ple

1Ex

amp

le 2

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1112

/21/

10

4:37

PM


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

10

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

Tria

ng

le I

neq

ualiti

es

Rec

all

that

a l

ine

segm

ent

can

be

nam

ed b

y th

e le

tter

s of

its

en

dpoi

nts

. Lin

e se

gmen

t A

B (

wri

tten

as

−−

AB

) h

as p

oin

ts A

an

d B

for

en

dpoi

nts

. Th

e le

ngt

h o

f A

B i

s w

ritt

en w

ith

out

the

bar

as A

B.

AB

> B

C

m

∠ A

< m

∠ B

Th

e st

atem

ent

on t

he

left

abo

ve s

how

s th

at −

−

AB

is

shor

ter

than

−−−

BC

. T

he

stat

emen

t on

th

e ri

ght

abov

e sh

ows

that

th

e m

easu

re o

f an

gle

A

is l

ess

than

th

at o

f an

gle

B.

Th

ese

thre

e in

equ

alit

ies

are

tru

e fo

r an

y tr

ian

gle

AB

C,

no

mat

ter

how

lon

g th

e si

des.

a. A

B +

BC

> A

Cb

. If

AB

> A

C, t

hen

m∠

C >

m∠

B.

c. I

f m

∠C

> m

∠B

, th

en A

B >

AC

.

B

AC

Use

th

e th

ree

tria

ngl

e in

equ

alit

ies

for

thes

e p

rob

lem

s.

1. L

ist

the

side

s of

tri

angl

e D

EF

in

ord

er o

f in

crea

sin

g le

ngt

h.

D

FE

60°

35°

85°

−−

DF , −

−

DE

, −−

EF

2. I

n t

he

figu

re a

t th

e ri

ght,

wh

ich

lin

e se

gmen

t is

th

e sh

orte

st?

JM

K

L

65°

60°

65°

55°

65°

50°

−

−

LM

3. E

xpla

in w

hy

the

len

gth

s 5

cen

tim

eter

s, 1

0 ce

nti

met

ers,

an

d 20

cen

tim

eter

s co

uld

not

be

use

d to

mak

e a

tria

ngl

e. 5

+ 1

0 is

no

t g

reat

er t

han

20.

4. T

wo

side

s of

a t

rian

gle

mea

sure

3 i

nch

es a

nd

7 in

ches

. Bet

wee

n w

hic

h t

wo

valu

es m

ust

th

e th

ird

side

be?

4 in

. an

d 1

0 in

.

5. I

n t

rian

gle

XY

Z, X

Y =

15,

YZ

= 1

2, a

nd

XZ

= 9

. Wh

ich

an

gle

has

th

e gr

eate

st m

easu

re?

Wh

ich

has

th

e le

ast?

∠ Z

; ∠

Y

6. L

ist

the

angl

es ∠

A, ∠

C, ∠

AB

C, a

nd

∠A

BD

, in

ord

er o

f in

crea

sin

g si

ze.

C ADB

13 1512

5 9

∠

AB

D, ∠

A, ∠

AB

C, ∠

C

5-1

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1012

/21/

10

4:37

PM

Answers (Lesson 5-1 and Lesson 5-2)

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

12

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(c

onti

nu

ed)

So

lvin

g I

neq

ualiti

es b

y M

ult

iplicati

on

an

d D

ivis

ion

Solv

e In

equ

alit

ies

by

Div

isio

n I

f ea

ch s

ide

of a

tru

e in

equ

alit

y is

div

ided

by

the

sam

e po

siti

ve n

um

ber,

the

resu

ltin

g in

equ

alit

y is

als

o tr

ue.

How

ever

, if

each

sid

e of

an

in

equ

alit

y is

div

ided

by

the

sam

e n

egat

ive

nu

mbe

r, th

e di

rect

ion

of

the

ineq

ual

ity

sym

bol

mu

st b

e re

vers

ed f

or t

he

resu

ltin

g in

equ

alit

y to

be

tru

e.

Div

isio

n P

rop

erty

o

f In

equ

alit

ies

Fo

r a

ll nu

mb

ers

a,

b,

an

d c

with

c ≠

0,

1.

if c

is p

ositiv

e a

nd

a >

b,

the

n a −

c >

b

−

c ; if c

is p

ositiv

e a

nd

a <

b,

the

n a −

c b

, th

en

a −

c 

b

−

c .

Th

e pr

oper

ty i

s al

so t

rue

wh

en >

an

d <

are

rep

lace

d w

ith

≥ a

nd

≤.

S

olve

-12

y ≥

48.

-12

y ≥

48

Ori

gin

al in

equalit

y

-12

y −

-

12

≤

48

−

-12

D

ivid

e e

ach s

ide b

y -

12 a

nd c

hange ≥

to ≤

.

y

≤ -

4 S

implif

y.

Th

e so

luti

on i

s {y

� y

≤ -

4}.

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Ch

eck

you

r so

luti

on.

1. 2

5g ≥

-10

0 2.

-2x

≥ 9

3.

-5c

> 2

4.

-8m

< -

64

{

g �

g ≥

-4}

{

x �

x ≤

-4

1 −

2 }

{c �

c <

- 2 −

5 }

{m

� m

> 8

}

5. -

6k <

1 −

5

6. 1

8 <

-3b

7.

30

< -

3n

8. -

0.24

< 0

.6w

{

k �

k >

- 1 −

30

} {

b �

b <

-6}

{

n �

n <

-10

} {

w �

w >

-0.

4}

9. 2

5 ≥

-2m

10

. -30

> -

5p

11. -

2n ≥

6.2

12

. 35

< 0

.05h

{m

� m

≥ -

12 1 −

2 }

{p

� p

> 6

} {

n �

n ≤

-3.

1}

{h

� h

> 7

00}

13. -

40 >

10h

14

. - 2 −

3n

≥ 6

15

. -3

 -

x −

2

{

h �

h <

-4}

{

n �

n ≤

-9}

{

p �

p >

-12

} {

x �

x >

-8}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. T

hen

ch

eck

yo

ur

solu

tion

.

17. F

our

tim

es a

nu

mbe

r is

no

mor

e th

an 1

08.

4n ≤

108

; {n

� n

≤ 2

7}

18. T

he

oppo

site

of

thre

e ti

mes

a n

um

ber

is g

reat

er t

han

12.

-3n

> 1

2; {

n �

n <

-4}

19. N

egat

ive

five

tim

es a

nu

mbe

r is

at

mos

t 10

0. -

5n ≤

10

0; {

n �

n ≥

-20

}

5-2

Exam

ple

17–1

9. S

amp

le a

nsw

er:

Let

n =

the

nu

mb

er.

001_

012_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1212

/21/

10

4:37

PM


Lesson 5-2


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

13

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceS

olv

ing

In

eq

ualiti

es b

y M

ult

iplicati

on

an

d D

ivis

ion

Mat

ch e

ach

in

equ

alit

y w

ith

its

cor

resp

ond

ing

stat

emen

t.

1. 3

n <

9 d

a.

Th

ree

tim

es a

nu

mbe

r is

at

mos

t n

ine.

2. 1 −

3 n

≥ 9

f

b. O

ne

thir

d of

a n

um

ber

is n

o m

ore

than

nin

e.

3. 3

n ≤

9 a

c.

Neg

ativ

e th

ree

tim

es a

nu

mbe

r is

mor

e th

an n

ine.

4. -

3n >

9 c

d

. Th

ree

tim

es a

nu

mbe

r is

les

s th

an n

ine.

5. 1 −

3 n

≤ 9

b

e. N

egat

ive

thre

e ti

mes

a n

um

ber

is a

t le

ast

nin

e.

6. -

3n ≥

9 e

f.

On

e th

ird

of a

nu

mbe

r is

gre

ater

th

an o

r eq

ual

to

nin

e.

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

.

7. 1

4g >

56

8. 1

1w ≤

77

9. 2

0b ≥

-12

0 10

. -8r

< 1

6

{

g �

g >

4}

{w

� w

≤ 7

} {

b �

b ≥

-6}

{

r � r

> -

2}

11. -

15p

≤ -

90

12.

x −

4 <

9

13.

a −

9 ≥

-15

14

. - p −

7 >

-9

{

p �

p ≥

6}

{x

� x

< 3

6}

{a

� a

≥ -

135}

{

p �

p <

63}

15. -

t

−

12 ≥

6

16. 5

z <

-90

17

. -13

m >

-26

18

. k −

5 ≤

-17

{

t � t

≤ -

72}

{z

� z

< -

18}

{m

� m

< 2

} {

k �

k ≤

-85

}

19. -

y <

36

20. -

16c

≥ -

224

21. -

h

−

10 ≤

2

22. 1

2 >

d

−

12

{

y �

y >

-36

} {

c �

c ≤

14}

{

h �

h ≥

-20

} {

d �

d <

144

}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m.

Ch

eck

you

r so

luti

on.

23–2

7. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

23. F

our

tim

es a

nu

mbe

r is

gre

ater

th

an -

48.

4n >

-48

; {n

� n

> -

12}

24. O

ne

eigh

th o

f a

nu

mbe

r is

les

s th

an o

r eq

ual

to

3. 1

−

8 n ≤

3;

{n �

n ≤

24}

25. N

egat

ive

twel

ve t

imes

a n

um

ber

is n

o m

ore

than

84.

-12

n ≤

84;

{n

� n

≥ -

7}

26. N

egat

ive

one

sixt

h o

f a

nu

mbe

r is

les

s th

an -

9. -

1 −

6 n

< -

9; {

n �

n >

54}

27. E

igh

t ti

mes

a n

um

ber

is a

t le

ast

16.

8n ≥

16;

{n

� n

≥ 2

}

5-2

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1312

/21/

10

4:37

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass


Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

14

Gle

ncoe

Alg

ebra

1

Mat

ch e

ach

in

equ

alit

y w

ith

its

cor

resp

ond

ing

stat

emen

t.

1. -

4n ≥

5 d

a.

Neg

ativ

e fo

ur

tim

es a

nu

mbe

r is

les

s th

an f

ive.

2. 4 −

5 n

> 5

f

b. F

our

fift

hs

of a

nu

mbe

r is

no

mor

e th

an f

ive.

3. 4

n ≤

5 e

c.

Fou

r ti

mes

a n

um

ber

is f

ewer

th

an f

ive.

4. 4 −

5 n

≤ 5

b

d. N

egat

ive

fou

r ti

mes

a n

um

ber

is n

o le

ss t

han

fiv

e.

5. 4

n <

5 c

e.

Fou

r ti

mes

a n

um

ber

is a

t m

ost

five

.

6. -

4n <

5 a

f.

Fou

r fi

fth

s of

a n

um

ber

is m

ore

than

fiv

e.

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

.

7. -

a −

5

< -

14

8. -

13h

≤ 5

2 9.

b −

16

≥ -

6 10

. 39

> 1

3p

{

a �

a >

70}

{

h �

h ≥

-4}

{

b �

b ≥

-96

} {

p �

p <

3}

11.

2 −

3 n

> -

12

12. -

5 −

9 t

< 2

5 13

. - 3 −

5 m

≤ -

6 14

. 10

−

3 k

≥ -

10

{

n �

n >

-18

} {

t � t

> -

45}

{m

� m

≥ 1

0}

{k

� k

≥ -

3}

15. -

3b ≤

0.7

5 16

. -0.

9c >

-9

17. 0

.1x

≥ -

4 18

. -2.

3 <

j −

4

{

b �

b ≥

-0.

25}

{c

� c

< 1

0}

{x

� x

≥ -

40}

{j

� j >

-9.

2}

19. -

15y

< 3

20

. 2.6

v ≥

-20

.8

21. 0

> -

0.5u

22

. 7 −

8 f

≤ -

1

{

y �

y >

- 1 −

5 }

{v

� v

≥ -

8}

{u

� u

> 0

} {

f � f

≤ -

8 −

7 }

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k

you

r so

luti

on.

23–2

5. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

23. N

egat

ive

thre

e ti

mes

a n

um

ber

is a

t le

ast

57.

-3n

≥ 5

7; {

n �

n ≤

-19

}

24. T

wo

thir

ds o

f a

nu

mbe

r is

no

mor

e th

an -

10.

2 −

3 n

≤ -

10;

{n �

n ≤

-15

}

25. N

egat

ive

thre

e fi

fth

s of

a n

um

ber

is l

ess

than

-6.

- 3 −

5 n

< -

6; {

n �

n >

10}

26. F

LOO

DIN

G A

riv

er i

s ri

sin

g at

a r

ate

of 3

in

ches

per

hou

r. If

th

e ri

ver

rise

s m

ore

than

2

feet

, it

wil

l ex

ceed

flo

od s

tage

. How

lon

g ca

n t

he

rive

r ri

se a

t th

is r

ate

wit

hou

t ex

ceed

ing

floo

d st

age?

27. S

ALE

S P

et S

upp

lies

mak

es a

pro

fit

of $

5.50

per

bag

on

its

lin

e of

nat

ura

l do

g fo

od. I

f th

e st

ore

wan

ts t

o m

ake

a pr

ofit

of

no

less

th

an $

5225

on

nat

ura

l do

g fo

od, h

ow m

any

bags

of

dog

foo

d do

es i

t n

eed

to s

ell?

5-2

Prac

tice

So

lvin

g I

neq

ualiti

es b

y M

ult

iplicati

on

an

d D

ivis

ion

no

mo

re t

han

8 h

at le

ast

950

bag

s

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1412

/21/

10

4:37

PM


Lesson 5-2


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

15

Gle

ncoe

Alg

ebra

1

1.PI

ZZA

Tar

a an

d fr

ien

ds o

rder

a p

izza

. T

ara

eats

3 o

f th

e 10

sli

ces

and

pays

$4

.20

for

her

sh

are.

Ass

um

ing

that

Tar

a h

as p

aid

at l

east

her

fai

r sh

are,

wri

te a

n

ineq

alit

y fo

r h

ow m

uch

th

e pi

zza

cou

ld

hav

e co

st.

2.A

IRLI

NES

On

ave

rage

, at

leas

t 25

,000

pi

eces

of

lugg

age

are

lost

or

mis

dire

cted

ea

ch d

ay b

y U

nit

ed S

tate

s ai

rlin

es. O

f th

ese,

98%

are

loc

ated

by

the

airl

ines

w

ith

in 5

day

s. F

rom

a g

iven

day

’s l

ost

lugg

age,

at

leas

t h

ow m

any

piec

es o

f lu

ggag

e ar

e st

ill

lost

aft

er 5

day

s?

3.SC

HO

OL

Gil

ear

ned

th

ese

scor

es o

n t

he

firs

t th

ree

test

s in

bio

logy

th

is t

erm

: 86,

88

, an

d 78

. Wh

at i

s th

e lo

wes

t sc

ore

that

G

il c

an e

arn

on

th

e fo

urt

h a

nd

fin

al t

est

of t

he

term

if

he

wan

ts t

o h

ave

an

aver

age

of a

t le

ast

83?

4.EV

ENT

PLA

NN

ING

Th

e D

own

tow

n

Com

mu

nit

y C

ente

r do

es n

ot c

har

ge a

re

nta

l fe

e as

lon

g as

a r

ente

e or

ders

a

min

imu

m o

f $5

000

wor

th o

f fo

od f

rom

th

e ce

nte

r. A

nto

nio

is

plan

nin

g a

ban

quet

fo

r th

e Q

uar

terb

ack

Clu

b. I

f h

e is

ex

pect

ing

225

peop

le t

o at

ten

d, w

hat

is

the

min

imu

m h

e w

ill

hav

e to

spe

nd

on

food

per

per

son

to

avoi

d pa

yin

g a

ren

tal

fee?

$2

2.23

5. P

HY

SIC

S T

he

den

sity

of

a su

bsta

nce

de

term

ines

wh

eth

er i

t w

ill

floa

t or

si

nk

in a

liq

uid

. Th

e de

nsi

ty o

f w

ater

is

1 gr

am p

er m

illi

lite

r. A

ny

obje

ct w

ith

a

grea

ter

den

sity

wil

l si

nk

and

any

obje

ct

wit

h a

les

ser

den

sity

wil

l fl

oat.

Den

sity

is

giv

en b

y th

e fo

rmu

la d

= m

−

v

, wh

ere

m i

s m

ass

and

v is

vol

um

e. H

ere

is a

ta

ble

of c

omm

on c

hem

ical

sol

uti

ons

and

thei

r de

nsi

ties

.

So

luti

on

Den

sity

(g

/mL

)

concentr

ate

d c

alc

ium

chlo

ride

1.40

70

% iso

pro

pyl a

lco

ho

l0

.92

Sour

ce:

Amer

ican

Che

mis

try

Cou

ncil

a. P

last

ics

vary

in

den

sity

wh

en t

hey

are

m

anu

fact

ure

d; t

her

efor

e, t

hei

r vo

lum

es a

re v

aria

ble

for

a gi

ven

mas

s.

A t

able

t of

pol

ysty

ren

e (a

m

anu

fact

ure

d pl

asti

c) s

inks

in

wat

er

and

in a

lcoh

ol s

olu

tion

an

d fl

oats

in

ca

lciu

m c

hlo

ride

sol

uti

on. T

he

tabl

et

has

a m

ass

of 0

.4 g

ram

. Wh

at i

s th

e m

ost

its

volu

me

can

be?

b.

Wh

at i

s th

e le

ast

its

volu

me

can

be?

Wor

d Pr

oble

m P

ract

ice

So

lvin

g I

neq

ualiti

es b

y M

ult

iplicati

on

an

d D

ivis

ion

5-2 at le

ast

500

pie

ces

80

x ≤

$14

0.43

5 g

0.28

6 g

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1512

/21/

10

4:37

PM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

16

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

Qu

ad

rati

c I

neq

ualiti

es

Lik

e li

nea

r in

equ

alit

ies,

in

equ

alit

ies

wit

h h

igh

er d

egre

es c

an a

lso

be s

olve

d. Q

uad

rati

c in

equ

alit

ies

hav

e a

degr

ee o

f 2.

Th

e fo

llow

ing

exam

ple

show

s h

ow t

o so

lve

quad

rati

c in

equ

alit

ies.

S

olve

(x

+ 3

)(x

- 2

) >

0.

Ste

p 1

D

eter

min

e w

hat

val

ues

of

x w

ill

mak

e th

e le

ft s

ide

0. I

n o

ther

w

ords

, wh

at v

alu

es o

f x

wil

l m

ake

eith

er x

+ 3

= 0

or

x -

2 =

0?

x

= -

3 or

2

Ste

p 2

P

lot

thes

e po

ints

on

a n

um

ber

lin

e. A

bove

th

e n

um

ber

lin

e, p

lace

a

+ i

f x

+ 3

is

posi

tive

for

th

at r

egio

n o

r a

- i

f x

+ 3

is

neg

ativ

e fo

r th

at r

egio

n. N

ext,

abo

ve t

he

sign

s yo

u h

ave

just

en

tere

d; d

o th

e sa

me

for

x –

2.

Ste

p 3

B

elow

th

e ch

art,

en

ter

the

prod

uct

of

the

two

sign

s. Y

our

sign

ch

art

shou

ld l

ook

like

th

e fo

llow

ing:

56

-6

43

21

0-

1-

5-4-

3-2

x -

2

x +

3

( x -

2)(

x +

3)

Th

e fi

nal

pos

itiv

e re

gion

s co

rres

pon

d to

val

ues

for

wh

ich

th

e qu

adra

tic

expr

essi

on i

s gr

eate

r th

an 0

. S

o, t

he

answ

er i

s

x

< -

3 or

x >

2.

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

1. (x

- 1

)(x

+ 2

) > 0

2.

(x +

5)(

x +

2) >

0

x

< -

2 o

r x >

1

x

< -

5 o

r x

> -

2

3. (x

- 1

)(x

- 5

) < 0

4.

(x +

2)(

x -

4) ≤

0

1

< x

< 5

-

2 ≤

x ≤

4

5. (x

– 3

)(x

+ 2

) ≥ 0

6.

(x +

3)(

x -

4) ≤

0x ≤

-2

or

x ≥

3

-3

≤ x

≤ 4

5-2

Exam

ple

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1612

/21/

10

4:37

PM


Lesson 5-3


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

17

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g M

ult

i-S

tep

In

eq

ualiti

es

Solv

e M

ult

i-St

ep In

equ

alit

ies

To

solv

e li

nea

r in

equ

alit

ies

invo

lvin

g m

ore

than

on

e op

erat

ion

, un

do t

he

oper

atio

ns

in r

ever

se o

f th

e or

der

of o

pera

tion

s, ju

st a

s yo

u w

ould

sol

ve

an e

quat

ion

wit

h m

ore

than

on

e op

erat

ion

.

S

olve

6x

- 4

≤ 2

x +

12.

6x -

4 ≤

2x

+ 1

2 O

rigin

al in

equalit

y

6x -

4 -

2x

≤ 2

x +

12

- 2

x Su

btr

act

2x

from

each s

ide.

4x -

4 ≤

12

Sim

plif

y.

4x -

4 +

4 ≤

12

+ 4

A

dd 4

to e

ach s

ide.

4x ≤

16

Sim

plif

y.

4x

−

4 ≤

16

−

4

Div

ide e

ach s

ide b

y 4

.

x ≤

4

Sim

plif

y.

Th

e so

luti

on i

s {x

� x

≤ 4

}.

S

olve

3a

- 1

5 >

4 +

5a

.

3a -

15

> 4

+ 5

a O

rigin

al in

equalit

y

3a -

15

- 5

a >

4 +

5a

- 5

a Su

btr

act

5a

from

each s

ide.

-2a

- 1

5 >

4

Sim

plif

y.

-2a

- 1

5 +

15

> 4

+ 1

5 A

dd 1

5 t

o e

ach s

ide.

-2a

> 1

9 S

implif

y.

-2a

−

-2 <

19

−

-2

Div

ide e

ach s

ide b

y -

2

and c

hange >

to <

.

a <

-9

1 −

2

Sim

plif

y.

Th

e so

luti

on i

s {a

� a

< -

9 1 −

2 } .

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Ch

eck

you

r so

luti

on.

1. 1

1y +

13

≥ -

1 2.

8n

- 1

0 <

6 -

2n

3.

q −

7 +

1 >

-5

{y

� y

≥ -

1 3

−

11 }

{n �

n <

1 3 −

5 }

{q

� q

> -

42}

4. 6

n +

12

< 8

+ 8

n

5. -

12 -

d >

-12

+ 4

d

6. 5

r -

6 >

8r

- 1

8

{

n �

n >

2}

{d

� d

< 0

} {

r � r

< 4

}

7.

-3x

+ 6

−

2 ≤

12

8. 7

.3y

- 1

4.4

> 4

.9y

9. -

8m -

3 <

18

- m

{

x �

x ≥

-6}

{

y �

y >

6}

{m

� m

> -

3}

10. -

4y -

10

> 1

9 -

2y

11. 9

n -

24n

+ 4

5 >

0

12.

4x -

2

−

5 ≥

-4

{y

� y

< -

14 1 −

2 }

{n

� n

< 3

} {x

� x

≥ -

4 1 −

2 }

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k y

our

solu

tion

. 13

–15.

Sam

ple

an

swer

: L

et n

= t

he

nu

mb

er.

13. N

egat

ive

thre

e ti

mes

a n

um

ber

plu

s fo

ur

is n

o m

ore

than

th

e n

um

ber

min

us

eigh

t.

-3n

+ 4

≤ n

- 8

; {n

� n

≥ 3

}

14. O

ne

fou

rth

of

a n

um

ber

decr

ease

d by

th

ree

is a

t le

ast

two.

1 −

4 n

- 3

≥ 2

; {n

� n

≥ 2

0}

15. T

he

sum

of

twel

ve a

nd

a n

um

ber

is n

o gr

eate

r th

an t

he

sum

of

twic

e th

e n

um

ber

and

-8.

1

2 +

n ≤

2n

+ (

-8)

; {n

� n

≥ 2

0}

5-3

Exam

ple

1Ex

amp

le 2

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1712

/21/

10

4:37

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

18

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

So

lvin

g M

ult

i-S

tep

In

eq

ualiti

es

Solv

e In

equ

alit

ies

Invo

lvin

g t

he

Dis

trib

uti

ve P

rop

erty

Wh

en s

olvi

ng

ineq

ual

itie

s th

at c

onta

in g

rou

pin

g sy

mbo

ls, f

irst

use

th

e D

istr

ibu

tive

Pro

pert

y to

rem

ove

the

grou

pin

g sy

mbo

ls. T

hen

un

do t

he

oper

atio

ns

in r

ever

se o

f th

e or

der

of o

pera

tion

s, ju

st a

s yo

u

wou

ld s

olve

an

equ

atio

n w

ith

mor

e th

an o

ne

oper

atio

n.

S

olve

3a

- 2

(6a

- 4

) >

4 -

(4a

+ 6

).

3a -

2(6

a -

4)

> 4

- (

4a +

6)

Ori

gin

al in

equalit

y

3a -

12a

+ 8

> 4

- 4

a -

6

Dis

trib

utive

Pro

pert

y

-9a

+ 8

> -

2 -

4a

Com

bin

e lik

e t

erm

s.

-9a

+ 8

+ 4

a >

-2

- 4

a +

4a

Add 4

a to

each s

ide.

-5a

+ 8

> -

2 C

om

bin

e lik

e t

erm

s.

-5a

+ 8

- 8

> -

2 -

8

Subtr

act

8 f

rom

each s

ide.

-5a

> -

10

Sim

plif

y.

a <

2

Div

ide e

ach s

ide b

y -

5 a

nd c

hange >

to <

.

Th

e so

luti

on i

n s

et-b

uil

der

not

atio

n i

s {a

� a

< 2

}.

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Ch

eck

you

r so

luti

on.

1. 2

(t +

3)

≥ 1

6 2.

3(d

- 2

) -

2d

> 1

6 3.

4h

- 8

< 2

(h -

1)

{

t � t

≥ 5

} {

d �

d >

22}

{

h �

h <

3}

4. 6

y +

10

> 8

- (

y +

14)

5.

4.6

(x -

3.4

) >

5.1

x 6.

-5x

- (

2x +

3)

≥ 1

{

y �

y >

-2

2 −

7 }

{x

� x

< -

31.2

8}

{x

� x

≤ -

4 −

7 }

7. 3

(2y

- 4

) -

2(y

+ 1

) >

10

8. 8

- 2

(b +

1)

< 1

2 -

3b

9. -

2(k

- 1

) >

8(1

+ k

)

{

y �

y >

6}

{b

� b

< 6

} {

k �

k <

- 3 −

5 }

10. 0

.3(y

- 2

) >

0.4

(1 +

y)

11. m

+ 1

7 ≤

-(4

m -

13)

{

y �

y <

-10

} {m

� m

≤ -

4 −

5 }

12. 3

n +

8 ≤

2(n

- 4

) -

2(1

- n

) 13

. 2(y

- 2

) >

-4

+ 2

y

{

n �

n ≥

18}

�

14. k

- 1

7 ≤

-(1

7 -

k)

15. n

- 4

≤ -

3(2

+ n

)

{

k �

k is

a r

eal n

um

ber

} {

n �

n ≤

- 1 −

2 }

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k y

our

solu

tion

.

16. T

wic

e th

e su

m o

f a

nu

mbe

r an

d 4

is l

ess

than

12.

2(n

+ 4

) <

12;

{n

� n

< 2

}

17. T

hre

e ti

mes

th

e su

m o

f a

nu

mbe

r an

d si

x is

gre

ater

th

an f

our

tim

es t

he

nu

mbe

r de

crea

sed

by t

wo.

18. T

wic

e th

e di

ffer

ence

of

a n

um

ber

and

fou

r is

les

s th

an t

he

sum

of

the

nu

mbe

r an

d fi

ve.

5-3

Exam

ple

16–1

8. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

3(n

+ 6

) >

4n

- 2

; {n

� n

< 2

0}

2(n

- 4

) <

n +

5;

{n �

n <

13}

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1812

/21/

10

4:37

PM


Lesson 5-3


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

19

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceS

olv

ing

Mu

lti-

Ste

p I

neq

ualiti

es

Ju

stif

y ea

ch i

nd

icat

ed s

tep

.

1.

3 −

4 t

- 3

≥ -

15

3 −

4 t

- 3

+ 3

≥ -

15 +

3

a.

?

3 −

4 t

≥ -

12

4 −

3 ( 3

−

4 ) t ≥

4 −

3 (-

12)

b.

?

t ≥

-16

a. A

dd

3 t

o e

ach

sid

e. b

. Mu

ltip

ly e

ach

sid

e by

4 −

3 .

2. 5

(k +

8)

- 7

≤ 2

3 5

k +

40

- 7

≤ 2

3 a.

?

5k +

33

≤ 2

35k

+ 3

3 -

33

≤ 2

3 -

33

b.

?5k

≤ -

10 5k

−

5

≤ -

10

−

5

c.

?

k ≤

-2

a. D

istr

ibu

tive

Pro

per

ty b

. Su

btr

act

33 f

rom

eac

h s

ide.

c. D

ivid

e ea

ch s

ide

by 5

.

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

.

3. -

2b +

4 >

-6

4. 3

x +

15

≤ 2

1 5.

d

−

2 - 1

≥ 3

{

b �

b <

5}

{x

� x

≤ 2

} {

d �

d ≥

8}

6.

2 −

5 a

- 4

< 2

7.

- t −

5 +

7 >

-4

8. 3 −

4 j

- 1

0 ≥

5

{

a �

a <

15}

{

t � t

< 5

5}

{j

� j ≥

20}

9. -

2 −

3 f

+ 3

< -

9 10

. 2p

+ 5

≥ 3

p -

10

11. 4

k +

15

> -

2k +

3

{

f � f

> 1

8}

{p

� p

≤ 1

5}

{k

� k

> -

2}

12. 2

(-3m

- 5

) ≥

-28

13

. -6(

w +

1)

< 2

(w +

5)

14. 2

(q -

3)

+ 6

≤ -

10

{

m �

m ≤

3}

{w

� w

> -

2}

{q

� q

≤ -

5}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m.

Ch

eck

you

r so

luti

on.

15–2

0. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

. 15

. Fou

r m

ore

than

th

e qu

otie

nt

of a

nu

mbe

r an

d th

ree

is a

t le

ast

nin

e.

n

−

3 + 4

≥ 9

;

{n

� n

≥ 1

5} 16

. Th

e su

m o

f a

nu

mbe

r an

d fo

urt

een

is

less

th

an o

r eq

ual

to

thre

e ti

mes

th

e n

um

ber.

n

+ 1

4 ≤

3n

; {n

� n

≥ 7

} 17

. Neg

ativ

e th

ree

tim

es a

nu

mbe

r in

crea

sed

by s

even

is

less

th

an n

egat

ive

elev

en.

-

3n +

7 <

-11

; {n

� n

> 6

} 18

. Fiv

e ti

mes

a n

um

ber

decr

ease

d by

eig

ht

is a

t m

ost

ten

mor

e th

an t

wic

e th

e n

um

ber.

5

n -

8 ≤

2n

+ 1

0; {

n �

n ≤

6}

19. S

even

mor

e th

an f

ive

sixt

hs

of a

nu

mbe

r is

mor

e th

an n

egat

ive

thre

e.

5 −

6 n

+ 7

> -

3;

{

n �

n >

-12

} 20

. Fou

r ti

mes

th

e su

m o

f a

nu

mbe

r an

d tw

o in

crea

sed

by t

hre

e is

at

leas

t tw

enty

-sev

en.

4

(n +

2)

+ 3

≥ 2

7; {

n �

n ≥

4}

5-3

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

1912

/21/

10

4:37

PM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

20

Gle

ncoe

Alg

ebra

1

Prac

tice

So

lvin

g M

ult

i-S

tep

In

eq

ualiti

es

Ju

stif

y ea

ch i

nd

icat

ed s

tep

. 1

. x

> 5x

- 1

2 −

8

8x >

(8)

5x -

12

−

8

a.

?

8x >

5x

- 1

2

8x -

5x

> 5

x -

12

- 5

x b

. ?

3x >

-12

3x

−

3 >

-12

−

3

c.

?

x >

-4

a. M

ult

iply

eac

h s

ide

by 8

. b

. Su

btr

act

5x f

rom

eac

h s

ide.

c. D

ivid

e ea

ch s

ide

by 3

.

2.

2(2h

+ 2

) <

2(3

h +

5)

- 1

24h

+ 4

< 6

h +

10

- 1

2 a.

?

4h +

4 <

6h

- 2

4h +

4 -

6h

< 6

h -

2 -

6h

b

. ?

-2h

+ 4

< -

2-

2h +

4 -

4 <

-2

- 4

c.

?

-2h

< -

6

-2h

−

-2

> -

6 −

-

2 d

. ?

h >

3 a

. Dis

trib

uti

ve P

rop

erty

b. S

ub

trac

t 6h

fro

m e

ach

sid

e. c

. Su

btr

act

4 fr

om

eac

h s

ide.

d. D

ivid

e ea

ch s

ide

by -

2 an

d

chan

ge

< t

o >

.

Sol

ve e

ach

in

equ

alit

y. C

hec

k y

our

solu

tion

.

3. -

5 -

t −

6 ≥

-9

4. 4

u -

6 ≥

6u

- 2

0 5.

13

> 2 −

3 a

- 1

{

t � t

≤ 2

4}

{u

� u

≤ 7

} {

a �

a <

21}

6.

w +

3

−

2

< -

8 {w

� w

< -

19}

7. 3f

- 1

0 −

5

>

7 {

f � f

> 1

5}

8. h

≤

6h +

3

−

5 {

h �

h ≥

-3}

9.

3(z

+ 1

) +

11

< -

2(z

+ 1

3) {

z �

z <

-8}

10. 3

r +

2(4

r +

2)

≤ 2

(6r

+ 1

) {r

� r

≥ 2

} 11

. 5n

- 3

(n -

6)

≥ 0

{n

� n

≥ -

9}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k y

our

solu

tion

.

12. A

nu

mbe

r is

les

s th

an o

ne

fou

rth

th

e su

m o

f th

ree

tim

es t

he

nu

mbe

r an

d fo

ur.

n

< 3n

+ 4

−

4 ;

{n �

n <

4}

13. T

wo

tim

es t

he

sum

of

a n

um

ber

and

fou

r is

no

mor

e th

an t

hre

e ti

mes

th

e su

m o

f th

e n

um

ber

and

seve

n d

ecre

ased

by

fou

r.

14. G

EOM

ETRY

Th

e ar

ea o

f a

tria

ngu

lar

gard

en c

an b

e n

o m

ore

than

120

squ

are

feet

. Th

e ba

se o

f th

e tr

ian

gle

is 1

6 fe

et. W

hat

is

the

hei

ght

of t

he

tria

ngl

e?

15. M

USI

C P

RA

CTI

CE

Nab

uko

pra

ctic

es t

he

viol

in a

t le

ast

12 h

ours

per

wee

k. S

he

prac

tice

s fo

r th

ree

fou

rth

s of

an

hou

r ea

ch s

essi

on. I

f N

abu

ko h

as a

lrea

dy p

ract

iced

3

hou

rs i

n o

ne

wee

k, h

ow m

any

sess

ion

s re

mai

n t

o m

eet

or e

xcee

d h

er w

eekl

y pr

acti

ce g

oal?

5-3

12–1

3. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

2(n

+ 4

) ≤

3(n

+ 7

) -

4;

{n �

n ≥

-9}

no

mo

re t

han

15

ft

at le

ast

12 s

essi

on

s

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2012

/21/

10

4:37

PM


Lesson 5-3


NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

21

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

So

lvin

g M

ult

i-S

tep

In

eq

ualiti

es

1.B

EAC

HC

OM

BIN

G J

ay h

as l

ost

his

m

oth

er’s

fav

orit

e n

eckl

ace,

so

he

wil

l re

nt

a m

etal

det

ecto

r to

try

to

fin

d it

. A

ren

tal

com

pan

y ch

arge

s a

one-

tim

e re

nta

l fe

e of

$15

plu

s $2

per

hou

r to

ren

t a

met

al d

etec

tor.

Jay

has

on

ly $

35 t

o sp

end.

Wh

at i

s th

e m

axim

um

am

oun

t of

ti

me

he

can

ren

t th

e m

etal

det

ecto

r?

10 h

ou

rs

2.A

GES

Bob

by, B

illy

, an

d B

arry

Sm

ith

are

ea

ch o

ne

year

apa

rt i

n a

ge. T

he

sum

of

thei

r ag

es i

s gr

eate

r th

an t

he

age

of t

hei

r fa

ther

, wh

o is

60.

How

old

can

th

e ol

dest

br

oth

er c

an b

e?

no

yo

un

ger

th

an 2

2

3.TA

XI

FAR

E Ja

mal

wor

ks i

n a

cit

y an

d so

met

imes

tak

es a

tax

i to

wor

k. T

he

taxi

cabs

ch

arge

$1.

50 f

or t

he

firs

t 1 − 5

mil

e

and

$0.2

5 fo

r ea

ch a

ddit

ion

al 1 − 5

mil

e.

Jam

al h

as o

nly

$3.

75 i

n h

is p

ocke

t. W

hat

is

th

e m

axim

um

dis

tan

ce h

e ca

n t

rave

l by

tax

i if

he

does

not

tip

th

e dr

iver

?

2 m

i

4.PL

AY

GR

OU

ND

Th

e pe

rim

eter

of

a re

ctan

gula

r pl

aygr

oun

d m

ust

be

no

grea

ter

than

120

met

ers,

bec

ause

th

at i

s th

e to

tal

len

gth

of

the

mat

eria

ls

avai

labl

e fo

r th

e bo

rder

. Th

e w

idth

of

the

play

grou

nd

can

not

exc

eed

22 m

eter

s.

Wh

at a

re t

he

poss

ible

len

gth

s of

th

e pl

aygr

oun

d?

less

th

an o

r eq

ual

to

38

met

ers

5. M

EDIC

INE

Cla

rk’s

Ru

le i

s a

form

ula

u

sed

to d

eter

min

e pe

diat

ric

dosa

ges

of

over

-th

e-co

un

ter

med

icin

es.

w

eigh

t of

ch

ild

( lb)

−

15

0

× a

dult

dos

e =

chi

ld

dose

a. I

f an

adu

lt d

ose

of a

ceta

min

oph

en i

s 10

00 m

illi

gram

s an

d a

chil

d w

eigh

s n

o m

ore

than

90

pou

nds

, wh

at i

s th

e re

com

men

ded

chil

d’s

dose

?

x ≤

60

0; n

o m

ore

th

an 6

00

mg

b.

Th

is l

abel

app

ears

on

a c

hil

d’s

cold

m

edic

ine.

Wh

at i

s th

e ad

ult

min

imu

m

dosa

ge i

n m

illi

lite

rs?

Wei

gh

t (l

b)

Ag

e (y

r)D

ose

un

de

r 4

8u

nd

er

6ca

ll a

do

cto

r

48

-95

6-1

12

tsp

or

10

mL

15

.79

mL

c. W

hat

is

the

max

imu

m a

dult

dos

age

in

mil

lili

ters

?

31.2

5 m

L

5-3

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2112

/21/

10

4:37

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

22

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

Carl

os M

on

tezu

ma

Du

rin

g h

is l

ifet

ime,

Car

los

Mon

tezu

ma

(186

6–19

23)

was

on

e of

th

e m

ost

infl

uen

tial

Nat

ive

Am

eric

ans

in t

he

Un

ited

Sta

tes.

He

was

re

cogn

ized

as

a pr

omin

ent

phys

icia

n a

nd

was

als

o a

pass

ion

ate

advo

cate

of

th

e ri

ghts

of

Nat

ive

Am

eric

an p

eopl

es. T

he

exer

cise

s th

at f

ollo

w w

ill

hel

p yo

u l

earn

som

e in

tere

stin

g fa

cts

abou

t D

r. M

onte

zum

a’s

life

.

Sol

ve e

ach

in

equ

alit

y. T

he

wor

d o

r p

hra

se n

ext

to t

he

equ

ival

ent

ineq

ual

ity

wil

l co

mp

lete

th

e st

atem

ent

corr

ectl

y.

1. -

2k >

10

2. 5

≥ r

- 9

Mon

tezu

ma

was

bor

n i

n t

he

stat

e

He

was

a N

ativ

e A

mer

ican

of

the

of

? .

Yav

apai

s, w

ho

are

a ?

peo

ple.

a. k

< -

5 A

rizo

na

a. r

≤ -

4 N

avaj

o

b.

k >

-5

Mon

tan

a b

. r

≥ -

4 M

ohaw

k

c. k

> 1

2 U

tah

c.

r ≤

14

Moh

ave-

Apa

che

3. -

y ≤

-9

4. -

3 +

q >

12

M

onte

zum

a re

ceiv

ed a

med

ical

A

s a

phys

icia

n, M

onte

zum

a’s

fiel

d of

de

gree

fro

m

? i

n 1

889.

s

peci

aliz

atio

n w

as

? .

a. y

≥ 9

C

hic

ago

Med

ical

Col

lege

a.

q >

-4

hea

rt s

urg

ery

b.

y ≥

-9

Har

vard

Med

ical

Sch

ool

b.

q >

15

inte

rnal

med

icin

e

c. y

≤ 9

Jo

hn

s H

opki

ns

Un

iver

sity

c.

q <

-15

re

spir

ator

y di

seas

es

5. 5

+ 4

x -

14

≤ x

6.

7 -

t <

7 +

t

F

or m

uch

of

his

car

eer,

he

mai

nta

ined

I

n a

ddit

ion

to

mai

nta

inin

g h

is m

edic

ala

med

ical

pra

ctic

e in

?

. p

ract

ice,

he

was

als

o a(

n)

?.

a. x

≤ 9

N

ew Y

ork

Cit

y a.

t >

7

dire

ctor

of

a bl

ood

ban

k

b.

x ≤

3

Ch

icag

o b

. t

> 0

in

stru

ctor

at

a m

edic

al c

olle

ge

c. x

≥ -

9 B

osto

n

c. t

< -

7 le

gal

cou

nse

l to

ph

ysic

ian

s

7. 3

a +

8 ≥

4a

- 1

0 8.

6n

> 8

n -

12

M

onte

zum

a fo

un

ded,

wro

te, a

nd

Mon

tezu

ma

test

ifie

d be

fore

a

edit

ed

?, a

mon

thly

new

slet

ter

com

mit

tee

of t

he

Un

ited

Sta

tes

that

add

ress

ed N

ativ

e A

mer

ican

C

ongr

ess

con

cern

ing

his

wor

k in

co

nce

rns.

t

reat

ing

?.

a. a

≤ -

2 Ya

vapa

i a.

n <

6

appe

ndi

citi

s

b.

a ≥

18

Apa

che

b.

n >

-6

asth

ma

c. a

≤ 1

8 W

assa

ja

c. n

> -

10

hea

rt a

ttac

ks

5-3

013_

022_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2212

/21/

10

4:37

PM

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

NA

ME

DAT

E

P

ER

IOD

Lesson 5-4

Stud

y G

uide

and

Inte

rven

tion

So

lvin

g C

om

po

un

d I

neq

ualiti

es

Ineq

ual

itie

s C

on

tain

ing

an

d A

com

pou

nd

ineq

ual

ity

con

tain

ing

and

is

tru

e on

ly

if b

oth

in

equ

alit

ies

are

tru

e. T

he

grap

h o

f a

com

pou

nd

ineq

ual

ity

con

tain

ing

and

is

the

inte

rsec

tion

of

the

grap

hs

of t

he

two

ineq

ual

itie

s. E

very

sol

uti

on o

f th

e co

mpo

un

d in

equ

alit

y m

ust

be

a so

luti

on o

f bo

th i

neq

ual

itie

s.

G

rap

h t

he

solu

tion

se

t of

x <

2 a

nd

x ≥

-1.

G

raph x

< 2

.

G

raph x

≥ -

1.

F

ind t

he inte

rsection.

Th

e so

luti

on s

et i

s {x

� -

1 ≤

x <

2}.

S

olve

-1

< x

+ 2

< 3

. Th

en

grap

h t

he

solu

tion

set

.-

1 <

x +

2

and

x +

2 <

3-

1 -

2 <

x +

2 -

2

x

+ 2

- 2

< 3

- 2

-3

< x

x <

1

G

raph x

> -

3.

G

raph x

< 1

.

F

ind t

he inte

rsection.

Th

e so

luti

on s

et i

s {x

� -

3 <

x <

1}.

Exer

cise

sG

rap

h t

he

solu

tion

set

of

each

com

pou

nd

in

equ

alit

y.

1. b

> -

1 an

d b

≤ 3

2.

2 ≥

q ≥

-5

3. x

> -

3 an

d x

≤ 4

4. -

2 ≤

p <

4

5. -

3 <

d a

nd

d<

2

6. -

1 ≤

p ≤

3

Sol

ve e

ach

com

pou

nd

in

equ

alit

y. T

hen

gra

ph

th

e so

luti

on s

et.

7. 4

< w

+ 3

≤ 5

8.

-3

≤ p

- 5

< 2

{

w �

1 <

w ≤

2}

{p �

2 ≤

p <

7}

9. -

4 <

x +

2 ≤

-2

10. y

- 1

< 2

an

d y

+ 2

≥ 1

{

x �

-6

< x

≤ -

4}

{y �

-1

≤ y

< 3

}

11. n

- 2

> -

3 an

d n

+ 4

< 6

12

. d -

3 <

6d

+ 1

2 <

2d

+ 3

2

{

n �

-1

< n

< 2

} {d

� -

3 <

d <

5}

-2

-1

-3

01

23

-3

-2

-1

01

23

-3

-2

-1

01

23

-3

-2

-1

01

23

45

-3

-4

-2

-1

01

23

4

-3

-4

-2

-1

01

23

4-

7-

6-

5-

4-

3-

2-

10

1

01

23

45

67

8-

3-

4-

2-

10

12

34

-3

-4

-2

-1

01

23

4-

3-

4-

2-

10

12

34

-3

-2

-1

01

23

45

-4

-3

-2

-1

01

23

4-

4-

3-

6-

5-

2-

10

12

-4

-3

-2

-1

01

23

4

-2

-1

-4

-3

01

2

-3

-4

-2

-1

01

2

-3

-4

-2

-1

01

2

5-4

Cha

pte

r 5

23

Gle

ncoe

Alg

ebra

1

Exam

ple

1Ex

amp

le 2

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2312

/21/

10

4:37

PM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

24

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

So

lvin

g C

om

po

un

d I

neq

ualiti

es

Ineq

ual

itie

s C

on

tain

ing

or

A c

ompo

un

d in

equ

alit

y co

nta

inin

g or

is

tru

e if

on

e or

bo

th o

f th

e in

equ

alit

ies

are

tru

e. T

he

grap

h o

f a

com

pou

nd

ineq

ual

ity

con

tain

ing

or i

s th

e u

nio

n o

f th

e gr

aph

s of

th

e tw

o in

equ

alit

ies.

Th

e u

nio

n c

an b

e fo

un

d by

gra

phin

g bo

th

ineq

ual

itie

s on

th

e sa

me

nu

mbe

r li

ne.

A s

olu

tion

of

the

com

pou

nd

ineq

ual

ity

is a

sol

uti

on o

f ei

ther

in

equ

alit

y, n

ot n

eces

sari

ly b

oth

.

S

olve

2a

+ 1

< 1

1 or

a >

3a

+ 2

. Th

en g

rap

h t

he

solu

tion

set

.

2a

+ 1

< 1

1 or

a

> 3

a +

2 2

a +

1 -

1 <

11

- 1

a -

3a

> 3

a -

3a

+ 2

2a

< 1

0

-2a

> 2

2a

−

2

< 10

−

2

-

2a

−

-2

<

2 −

-2

a

< 5

a <

-1

-2

-1

01

23

45

6

Gra

ph a

< 5

.

-2

-1

01

23

45

6

Gra

ph a

< -

1.

-2

-1

01

23

45

6

Fin

d t

he u

nio

n.

Th

e so

luti

on s

et i

s {a

� a

< 5

}.

Exer

cise

sG

rap

h t

he

solu

tion

set

of

each

com

pou

nd

in

equ

alit

y. 1

. b >

2 o

r b

≤ -

3 2.

3 ≥

q o

r q

≤ 1

3.

y ≤

-4

or y

> 0

4. 4

≤ p

or

p <

8

5. -

3 <

d o

r d

< 2

6.

-2

≤ x

or

3 ≤

x

Sol

ve e

ach

com

pou

nd

in

equ

alit

y. T

hen

gra

ph

th

e so

luti

on s

et.

7. 3

< 3

w o

r 3w

≥ 9

8.

-3p

+ 1

≤-

11 o

r p

< 2

{

w �

1 <

w}

{p

� p

≥ 4

or

p <

2}

9. 2

x +

4 ≤

6 o

r x

≥ 2

x -

4

10. 2

y +

2 <

12

or y

- 3

≥ 2

y

{

x �

x ≤

4}

{y

� y

< 5

}

11.

1 −

2 n

> -

2 or

2n

- 2

< 6

+ n

12

. 3a

+ 2

≥ 5

or

7 +

3a

< 2

a +

6

{

n �

n is

a r

eal n

um

ber

}

{a �

a <

-1

or

a ≥

1}

0

-1

-2

-3

-4

12

34

0-

1-

2-

3-

41

23

4

01

23

45

67

8-

2-

10

12

34

56

01

23

45

67

8-

3-

4-

2-

10

12

34

-3

-4

-2

-1

01

23

40

-1

-2

-3

-4

12

34

0-

1-

21

23

45

6

-3

-4

-5

-2

-1

01

23

-3

-4

-2

-1

01

23

4-

3-

4-

2-

10

12

34

5-4

Exam

ple

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2412

/21/

10

4:37

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-4

Cha

pte

r 5

25

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceS

olv

ing

Co

mp

ou

nd

In

eq

ualiti

es

Gra

ph

th

e so

luti

on s

et o

f ea

ch c

omp

oun

d i

neq

ual

ity.

1. b

> 3

or

b ≤

0

2. z

≤ 3

an

d z

≥ -

2

3. k

> 1

an

d k

> 5

4.

y <

-1

or y

≥ 1

Wri

te a

com

pou

nd

in

equ

alit

y fo

r ea

ch g

rap

h.

5.

-2

-1

-4

-3

01

23

4

6.

-

3 <

x ≤

3

1

≤ x

≤ 4

7.

8.

x

< -

2 o

r x ≥

1

x <

-1

or

x >

2

Sol

ve e

ach

com

pou

nd

in

equ

alit

y. T

hen

gra

ph

th

e so

luti

on s

et.

9. m

+ 3

≥ 5

an

d m

+ 3

< 7

10

. y -

5 <

-4

or y

- 5

≥ 1

{

m �

2 ≤

m <

4}

{y

� y

< 1

or

y ≥

6}

11. 4

< f

+ 6

an

d f

+ 6

< 5

12

. w +

3 ≤

0 o

r w

+ 7

≥ 9

{

f �

-2

< f

< -

1}

{w

� w

≤ -

3 o

r w

≥ 2

}

13. -

6 

1

{

b �

-2

 3

}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m.

Ch

eck

you

r so

luti

on.

15-

17. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

15. A

nu

mbe

r pl

us

one

is g

reat

er t

han

neg

ativ

e fi

ve a

nd

less

th

an t

hre

e.

-5

< n

+ 1

< 3

; {n

� -

6 <

n <

2}

16. A

nu

mbe

r de

crea

sed

by t

wo

is a

t m

ost

fou

r or

at

leas

t n

ine.

n

- 2

≤ 4

or

n -

2 ≥

9;

{n �

n ≤

6 o

r n

≥ 1

1}

17. T

he

sum

of

a n

um

ber

and

thre

e is

no

mor

e th

an e

igh

t or

is

mor

e th

an t

wel

ve.

n

+ 3

≤ 8

or

n +

3 >

12;

{n

� n

≤ 5

or

n >

9}

-3

-4

-2

-1

01

23

4-

2-

10

12

34

56

-3

-4

-2

-1

01

23

4-

4-

3-

2-

10

12

34

-2

-1

01

23

45

6-

2-

10

12

34

56

-4

-3

-2

-1

01

23

4

-3

-4

-2

-1

01

23

4

-4

-3

-2

-1

01

23

4-

3-

4-

2-

10

12

34

-2

-1

01

23

45

6

-4

-3

-2

-1

01

23

4

5-4 0

12

34

56

78

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2512

/21/

10

4:37

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

26

Gle

ncoe

Alg

ebra

1

Prac

tice

So

lvin

g C

om

po

un

d I

neq

ualiti

es

Gra

ph

th

e so

luti

on s

et o

f ea

ch c

omp

oun

d i

neq

ual

ity.

1. -

4 ≤

n ≤

1

2. x

> 0

or

x <

3

3. g

< -

3 or

g ≥

4

4. -

4 ≤

p ≤

4

Wri

te a

com

pou

nd

in

equ

alit

y fo

r ea

ch g

rap

h.

5.

6.

x

≤ -

3 o

r x ≥

3

x <

2 o

r x ≥

3

7.

8.

0

≤ x

< 5

-5

< x

< 0

Sol

ve e

ach

com

pou

nd

in

equ

alit

y. T

hen

gra

ph

th

e so

luti

on s

et.

9. k

- 3

< -

7 or

k +

5 ≥

8

10. -

n <

2 o

r 2n

- 3

> 5

{

k �

k <

-4

or

k ≥

3}

{n

� n

> -

2}

11. 5

< 3

h +

2 ≤

11

12. 2

c -

4 >

-6

and

3c +

1 <

13

{

h �

1 <

h ≤

3}

{c

� -

1 <

c <

4}

Def

ine

a va

riab

le, w

rite

an

in

equ

alit

y, a

nd

sol

ve e

ach

pro

ble

m. C

hec

k

you

r so

luti

on.

13-

14. S

amp

le a

nsw

er:

Let

n =

th

e n

um

ber

.

13. T

wo

tim

es a

nu

mbe

r pl

us

one

is g

reat

er t

han

fiv

e an

d le

ss t

han

sev

en.

5

< 2

n +

1 <

7;

{n �

2 <

n <

3}

14. A

nu

mbe

r m

inu

s on

e is

at

mos

t n

ine,

or

two

tim

es t

he

nu

mbe

r is

at

leas

t tw

enty

-fou

r.

n -

1 ≤

9 o

r 2n

≥ 2

4; {

n �

n ≤

10

or

n ≥

12}

15. M

ETEO

RO

LOG

Y S

tron

g w

inds

cal

led

the

prev

aili

ng

wes

terl

ies

blow

fro

m w

est

to e

ast

in a

bel

t fr

om 4

0° t

o 60

° la

titu

de i

n b

oth

th

e N

orth

ern

an

d S

outh

ern

Hem

isph

eres

.

a

. Wri

te a

n i

neq

ual

ity

to r

epre

sen

t th

e la

titu

de o

f th

e pr

evai

lin

g w

este

rlie

s.

{w

� 4

0 ≤

w ≤

60}

b

. Wri

te a

n i

neq

ual

ity

to r

epre

sen

t th

e la

titu

des

wh

ere

the

prev

aili

ng

wes

terl

ies

are

n

ot l

ocat

ed.

16. N

UTR

ITIO

N A

coo

kie

con

tain

s 9

gram

s of

fat

. If

you

eat

no

few

er t

han

4 a

nd

no

mor

e th

an 7

coo

kies

, how

man

y gr

ams

of f

at w

ill

you

con

sum

e?

-3

-4

-2

-1

01

23

4

-2

-1

-4

-3

-6

-5

01

2

-2

-1

01

23

45

6-

2-

1-

4-

30

12

34

-4

-3

-2

-1

01

23

4-

3-

4-

2-

10

12

34

-2

-3

-4

-5

-6

-1

01

2-

2-

10

12

34

56

-2

-1

01

23

45

6-

4-

3-

2-

10

12

34

-2

-1

-4

-3

01

23

4

0-

1-

2-

3-

41

23

4

5-4

{w �

w <

40

or

w >

60}

bet

wee

n 3

6 g

an

d 6

3 g

incl

usi

ve

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2612

/23/

10

3:25

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-4

Cha

pte

r 5

27

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

So

lvin

g C

om

po

un

d I

neq

ualiti

es

1. W

EATH

ER K

en s

aw t

his

gra

ph i

n t

he

new

spap

er w

eath

er f

orec

ast.

It

show

s th

e pr

edic

ted

tem

pera

ture

ran

ge f

or t

he

foll

owin

g da

y. W

rite

an

in

equ

alit

y to

re

pres

ent

the

nu

mbe

r li

ne

grap

h.

2. P

OO

LS T

he

pH o

f a

pers

on’s

eye

s is

7.2

. T

her

efor

e, t

he

idea

l pH

for

th

e w

ater

in

a

swim

min

g po

ol i

s be

twee

n 7

.0 a

nd

7.6.

W

rite

a c

ompo

un

d in

equ

alit

y to

re

pres

ent

pH l

evel

s th

at c

ould

cau

se

phys

ical

dis

com

fort

to

a pe

rson

’s e

yes.

x

≤ 7

.0 o

r x

≥7.

6

3. S

TOR

E SI

GN

S In

Ran

dy’s

tow

n,

stre

et-s

ide

sign

s th

emse

lves

mu

st b

e ex

actl

y 8

feet

hig

h. W

hen

mou

nte

d on

po

les,

th

e si

gns

mu

st b

e sh

orte

r th

an

20 f

eet

or t

alle

r th

an 3

5 fe

et s

o th

at

they

do

not

in

terf

ere

wit

h t

he

pow

er

and

phon

e li

nes

. Wri

te a

com

pou

nd

ineq

ual

ity

to r

epre

sen

t th

e po

ssib

le

hei

ght

of t

he

pole

s.

4. H

EALT

H T

he

hu

man

hea

rt c

ircu

late

s fr

om 7

70,0

00 t

o 1,

600,

000

gall

ons

of

bloo

d th

rou

gh a

per

son

’s b

ody

ever

y ye

ar.

How

man

y ga

llon

s of

blo

od d

oes

the

hea

rt c

ircu

late

th

rou

gh t

he

body

in

on

e da

y?

5. H

EALT

H B

ody

mas

s in

dex

(BM

I) i

s a

mea

sure

of

wei

ght

stat

us.

Th

e B

MI

of a

pe

rson

ove

r 20

yea

rs o

ld i

s ca

lcu

late

d u

sin

g th

e fo

llow

ing

form

ula

.

B

MI

= 7

03 ×

w

eigh

t in

pou

nds

−

−

(

hei

ght

in i

nch

es)2

T

he

tabl

e be

low

sh

ows

the

mea

nin

g of

di

ffer

ent

BM

I m

easu

res.

So

urce

: C

ente

rs f

or D

isea

se C

ontr

ol

a. W

rite

a c

ompo

un

d in

equ

alit

y to

re

pres

ent

the

nor

mal

BM

I ra

nge

.

18.5

≤ x

< 2

5

b.

Wri

te a

com

pou

nd

ineq

ual

ity

to

repr

esen

t an

adu

lt w

eigh

t th

at i

s w

ith

in t

he

hea

lth

y B

MI

ran

ge f

or a

pe

rson

6 f

eet

tall

.

136.

4 ≤

w ≤

183

.6

62°

64°

66°

68°

70°

60°

58°

56°

54°

52°

50°F

5-4

bet

wee

n 2

110

and

438

4 g

al

BM

IW

eig

ht

Sta

tus

less t

ha

n 1

8.5

un

de

rwe

igh

t

18

.5 –

24

.9n

orm

al

25

– 2

9.9

ove

rwe

igh

t

mo

re t

ha

n 3

0o

be

se

54 ≤

x ≤

68

x <

12

or

x >

27

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2712

/21/

10

4:37

PM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

28

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

So

me P

rop

ert

ies o

f In

eq

ualiti

es

Th

e tw

o ex

pres

sion

s on

eit

her

sid

e of

an

in

equ

alit

y sy

mbo

l ar

e so

met

imes

cal

led

the

firs

t an

d se

con

d m

embe

rs o

f th

e in

equ

alit

y.

If t

he

ineq

ual

ity

sym

bols

of

two

ineq

ual

itie

s po

int

in t

he

sam

e di

rect

ion

, th

e in

equ

alit

ies

hav

e th

e sa

me

sen

se. F

or e

xam

ple,

a 

d h

ave

oppo

site

sen

ses.

In t

he

prob

lem

s on

th

is p

age,

you

wil

l ex

plor

e so

me

prop

erti

es

of i

neq

ual

itie

s.

Th

ree

of t

he

fou

r st

atem

ents

bel

ow a

re t

rue

for

all

nu

mb

ers

a a

nd

b (

or a

, b, c

, an

d d

). W

rite

eac

h s

tate

men

t in

alg

ebra

ic

form

. If

the

stat

emen

t is

tru

e fo

r al

l n

um

ber

s, p

rove

it.

If

it i

s n

ot t

rue,

giv

e an

exa

mp

le t

o sh

ow t

hat

it

is f

alse

.

1. G

iven

an

in

equ

alit

y, a

new

an

d eq

uiv

alen

t in

equ

alit

y ca

n b

e cr

eate

d by

in

terc

han

gin

g th

e m

embe

rs a

nd

reve

rsin

g th

e se

nse

.

I

f a >

b, t

hen

b <

a.

a

> b

, a -

b >

0, -

b >

-a, (

-1)

(-b

) <

(-

1)(-

a),

b <

a

2. G

iven

an

in

equ

alit

y, a

new

an

d eq

uiv

alen

t in

equ

alit

y ca

n b

e cr

eate

d by

ch

angi

ng

the

sign

s of

bot

h t

erm

s an

d re

vers

ing

the

sen

se.

I

f a >

b, t

hen

-a <

-b

.

a >

b, a

- b

> 0

, -b

> -

a, -

a <

-b

3. G

iven

tw

o in

equ

alit

ies

wit

h t

he

sam

e se

nse

, th

e su

m o

f th

e co

rres

pon

din

g m

embe

rs a

re m

embe

rs o

f an

equ

ival

ent

ineq

ual

ity

wit

h t

he

sam

e se

nse

.

I

f a >

b a

nd

c >

d, t

hen

a +

c >

b +

d.

a

> b

an

d c

> d

, so

(a -

b)

and

(c -

d)

are

po

siti

ve n

um

ber

s,

so t

he

sum

(a -

b)

+ (

c -

d)

is a

lso

po

siti

ve.

(a -

b)

+ (c

- d

) >

0, s

o a

+ c

> b

+ d

.

4. G

iven

tw

o in

equ

alit

ies

wit

h t

he

sam

e se

nse

, th

e di

ffer

ence

of

the

corr

espo

ndi

ng

mem

bers

are

mem

bers

of

an e

quiv

alen

t in

equ

alit

y w

ith

th

e sa

me

sen

se.

I

f a >

b a

nd

c >

d, t

hen

a -

c >

b -

d. T

he

stat

emen

t is

fal

se. 5

> 4

an

d 3

> 2

, bu

t 5

- 3

≯ 4

- 2

.

5-4

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2812

/21/

10

4:37

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-5

Cha

pte

r 5

29

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

In

eq

ualiti

es I

nvo

lvin

g A

bso

lute

Valu

e

Ineq

ual

itie

s In

volv

ing

Ab

solu

te V

alu

e (<

) W

hen

so

lvin

g in

equ

alit

ies

that

in

volv

e ab

solu

te v

alu

e, t

her

e ar

e tw

o ca

ses

to c

onsi

der

for

ineq

ual

itie

s in

volv

ing

< (

or ≤

).

Rem

embe

r th

at i

neq

ual

itie

s w

ith

an

d a

re r

elat

ed t

o in

ters

ecti

ons.

S

olve

|3a

+ 4

| <

10.

Th

en g

rap

h t

he

solu

tion

set

.

Wri

te �

3a +

4 �

< 1

0 as

3a

+ 4

< 1

0 an

d 3a

+ 4

> -

10.

3a

+ 4

< 1

0 an

d 3a

+ 4

> -

10 3

a +

4 -

4 <

10

- 4

3a +

4 -

4 >

-10

- 4

3a

< 6

3a >

-14

3a

−

3

< 6 −

3

3a

−

3

> -

14

−

3

a

< 2

a >

-4

2 −

3

Th

e so

luti

on s

et i

s {a

� -

4 2 −

3 <

a <

2} .

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Th

en g

rap

h t

he

solu

tion

set

.

1. �

y � <

3

2. �

x -

4 �

< 4

3.

� y

+ 3

� ≤

2

{

y �

-3

< y

< 3

} {

x �

0 <

x <

8}

{y

� -

5 ≤

y ≤

-1}

-3

-4

-2

-1

01

23

4

0

12

34

56

78

-8

-7

-6

-5

-4

-3

-2

-1

0

4. �

b +

2 � ≤

3

5. �

w -

2 �

≤ 5

6.

� t

+ 2

� ≤

4

{b �

-5

≤b

≤ 1

} {

w �

-3

≤w

≤ 7

} {

t �

-6

≤t

≤ 2

}

-6

-5

-4

-3

-2

-1

01

2

-

8-

6-

4-

20

24

68

-8

-6

-4

-2

02

46

8

7. �

2x

� ≤ 8

8.

� 5y

- 2

� ≤

7

9. �

p -

0.2

� <

0.5

{

x �

-4

≤ x

≤ 4

} {y

� -

1 ≤

y ≤

1 4 −

5 }

{p

� -

0.3

 -

n a

nd

x <

n.

Now

gra

ph t

he

solu

tion

set

.

-2

-1

-4

-5

-3

01

23

5-5

Exam

ple

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

2912

/21/

10

5:18

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

30

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Ineq

ualiti

es I

nvo

lvin

g A

bso

lute

Valu

e

Solv

e A

bso

lute

Val

ue

Ineq

ual

itie

s (>

) W

hen

sol

vin

g in

equ

alit

ies

that

in

volv

e ab

solu

te v

alu

e, t

her

e ar

e tw

o ca

ses

to c

onsi

der

for

ineq

ual

itie

s in

volv

ing

> (

or ≥

). R

emem

ber

that

in

equ

alit

ies

wit

h o

r ar

e re

late

d to

un

ion

s.

S

olve

� 2

b +

9 �

> 5

. Th

en g

rap

h t

he

solu

tion

set

.

Wri

te �

2b +

9 �

> 5

as

� 2b

+ 9

� >

5 o

r � 2b

+ 9

� <

-5.

2b +

9 >

5

or

2b +

9 <

-5

2b +

9 -

9 >

5 -

9

2b

+ 9

- 9

< -

5 -

9

2b >

-4

2b

< -

14

2b

−

2 >

-4

−

2 2b

−

2

< -

14

−

2

b >

-2

b

< -

7

Th

e so

luti

on s

et i

s {b

� b

> -

2 or

b <

-7 }

.

Exer

cise

sS

olve

eac

h i

neq

ual

ity.

Th

en g

rap

h t

he

solu

tion

set

.

1. �

c -

2 �

> 6

2.

� x

- 3

� >

0

3. �

3f +

10

� ≥

4

{

c �

c <

-4

or

c >

8}

{x

� x

> 3

or

x <

3}

{f �

f ≤

-4

2 −

3 o

r f

≥ -

2 }

20

-2

-4

-6

46

810

0-

1-

2-

3-

41

23

4

-

6-

5-

4-

3-

2-

10

12

4. �

x � ≥

2

5. �

x � ≥

3

6. �

2x +

1 � ≥

-2

{

x �

x ≤

-2

or

x ≥

2}

{x

� x

≤ -

3 o

r x ≥

3}

{x

� x

is a

rea

l num

ber}

-4

-3

-2

-1

01

23

4

0

-1

-2

-3

-4

12

34

0-

1-

2-

3-

41

23

4

7. �

2d

- 1

� ≥

4

8. �

3 -

(x

- 1

) � ≥

8

9. �

3r +

2 �

> -

5

{d

� d

≤ -

1 1 −

2 o

r d

≥ 2

1 −

2 }

{x

� x

≤ -

4 o

r x ≥

12}

{

r � r

is a

rea

l num

ber}

-4

-3

-2

-1

01

23

4

-

4-

20

24

68

1012

0-

1-

2-

3-

41

23

4

5-5

-9

-8

-7

-6

-5

-4

-3

-2

0-

1

Now

gra

ph t

he

solu

tion

set

.

Exam

ple

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3012

/21/

10

4:38

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-5

Cha

pte

r 5

31

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceIn

eq

ualiti

es I

nvo

lvin

g A

bso

lute

Valu

e

Mat

ch e

ach

op

en s

ente

nce

wit

h t

he

grap

h o

f it

s so

luti

on s

et.

1.

� x

� >

2 b

a.

-

2-

3-

4-

5-

10

12

34

5

2.

� x

- 2

� ≤

3 c

b

. -

2-

3-

4-

5-

10

12

34

5

3.

� x

+ 1

� <

4 a

c.

-

2-

3-

4-

10

12

34

56

Exp

ress

eac

h s

tate

men

t u

sin

g an

in

equ

alit

y in

volv

ing

abso

lute

val

ue.

4. T

he

wea

ther

man

pre

dict

ed t

hat

th

e te

mpe

ratu

re w

ould

be

wit

hin

3°

of 5

2°F.

� x -

52

� ≤

3

5. S

eren

a w

ill

mak

e th

e B

tea

m i

f sh

e sc

ores

wit

hin

8 p

oin

ts o

f th

e te

am a

vera

ge o

f 92

.

� x -

92

� ≤

8

6. T

he

dan

ce c

omm

itte

e ex

pect

s at

ten

dan

ce t

o n

um

ber

wit

hin

25

of l

ast

year

’s 8

7 st

ude

nts

. � x -

87

� ≤

25

Sol

ve e

ach

in

equ

alit

y. T

hen

gra

ph

th

e so

luti

on s

et.

7. �

x +

1 �

< 0

8.

� c

- 3

� <

1

9. �

n +

2 �

≥ 1

10

. � t

+ 6

� >

4

11. �

w -

2 �

< 2

12

. � k

- 5

� ≤

4

-8

-7

-10

-9

-6

-5

-4

-3

-2

-1

0-

2-

1-

4-

5-

6-

30

12

34

-3

-1

01

23

45

6-

27

-3

-4

-5

-6

-2

-1

01

23

4

5-5

-3

-4

-2

-1

01

23

45

60

12

34

56

78

910

{c �

2 <

c <

4}

{n �

n ≥

-1

or

n ≤

- 3

}{t

� t

< -

10 o

r t

> -

2}

no s

olu

tion

{w �

0 <

w <

4}

{k �

1 ≤

k ≤

9}

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3112

/22/

10

5:33

PM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

32

Gle

ncoe

Alg

ebra

1

Prac

tice

Ineq

ualiti

es I

nvo

lvin

g A

bso

lute

Valu

e

Mat

ch e

ach

op

en s

ente

nce

wit

h t

he

grap

h o

f it

s so

luti

on s

et.

1. �

x -

3 �

≥ 1

a

a.

2. �

2x

+ 1

� <

5 c

b

.

3. �

5 -

x �

≥ 3

b

c.

Exp

ress

eac

h s

tate

men

t u

sin

g an

in

equ

alit

y in

volv

ing

abso

lute

val

ue.

4. T

he

hei

ght

of t

he

plan

t m

ust

be

wit

hin

2 i

nch

es o

f th

e st

anda

rd 1

3-in

ch s

how

siz

e.

�

h -

13

� ≤

2

5. T

he

maj

orit

y of

gra

des

in S

ean

’s E

ngl

ish

cla

ss a

re w

ith

in 4

poi

nts

of

85.

�

g -

85

� ≤

4

Sol

ve e

ach

in

equ

alit

y. T

hen

gra

ph

th

e so

luti

on s

et.

6. |

2z -

9|

≤ 1

{z

� 4

≤ z

≤ 5

} 7.

|3

- 2

r| >

7 {

r � r

< -

2 o

r r

> 5

}

8. |

3t +

6|

< 9

{t

� -

5 <

t <

1}

9. |

2g -

5|

≥ 9

{g

� g

≤ -

2 o

r g

≥ 7

}

Wri

te a

n o

pen

sen

ten

ce i

nvo

lvin

g ab

solu

te v

alu

e fo

r ea

ch g

rap

h.

10.

11.

�

x -

6 � <

5

�

x +

4 � >

2 12

. 13

.

�

x +

3 �

≥ 4

� x

- 2

� ≤

4

14. R

ESTA

UR

AN

TS T

he

men

u a

t Je

ann

e’s

favo

rite

res

tau

ran

t st

ates

th

at t

he

roas

ted

chic

ken

wit

h v

eget

able

s en

tree

typ

ical

ly c

onta

ins

480

Cal

orie

s. B

ased

on

th

e si

ze o

f th

e ch

icke

n, t

he

actu

al n

um

ber

of C

alor

ies

in t

he

entr

ee c

an v

ary

by a

s m

any

as 4

0 C

alor

ies

from

th

is a

mou

nt.

a. W

rite

an

abs

olu

te v

alu

e in

equ

alit

y to

rep

rese

nt

the

situ

atio

n.

� x -

480

� ≤

40

b.

Wh

at i

s th

e ra

nge

of

the

nu

mbe

r of

Cal

orie

s in

th

e ch

icke

n e

ntr

ee?

440

≤ x

≤ 5

20

-2

-3

-1

01

23

45

67

-2

-3

-4

-5

-6

-7

-8

-1

01

2

12

34

56

78

910

11

-2

-1

01

23

45

67

8-

5-

4-

3-

2-

10

12

34

5

-5

-4

-3

-2

-1

01

23

45

-5

-4

-3

-2

-1

01

23

45

-2

-3

-4

-5

-1

01

23

45

-2

-1

01

23

45

67

8

-2

-3

-4

-5

-1

01

23

45

5-5

-2

-3

-4

-5

-6

-7

-8

-1

01

2

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3212

/21/

10

4:38

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-5

Cha

pte

r 5

33

Gle

ncoe

Alg

ebra

1

Wor

d Pr

oble

m P

ract

ice

Ineq

ualiti

es I

nvo

lvin

g A

bso

lute

Valu

e

1. S

PEED

OM

ETER

S T

he

gove

rnm

ent

requ

ires

spe

edom

eter

s on

car

s so

ld i

n

the

Un

ited

Sta

tes

to b

e ac

cura

te w

ith

in

±2.

5% o

f th

e ac

tual

spe

ed o

f th

e ca

r. If

yo

ur

spee

dom

eter

rea

ds 6

0 m

iles

per

h

our

wh

ile

you

are

dri

vin

g on

a h

igh

way

, w

hat

is

the

ran

ge o

f po

ssib

le a

ctu

al

spee

ds a

t w

hic

h y

our

car

cou

ld b

e tr

avel

ing?

{x|

58.5

≤x

≤ 6

1.5}

2. B

AK

ING

Pet

e is

mak

ing

mu

ffin

s fo

r a

bake

sal

e. B

efor

e h

e st

arts

bak

ing,

he

goes

on

lin

e to

res

earc

h d

iffe

ren

t m

uff

in

reci

pes.

Th

e re

cipe

s th

at h

e fi

nds

all

sp

ecif

y ba

kin

g te

mpe

ratu

res

betw

een

35

0°F

an

d 40

0°F,

in

clu

sive

. Wri

te a

n

abso

lute

val

ue i

nequ

alit

y to

rep

rese

nt t

he

poss

ible

tem

pera

ture

s t

call

ed f

or i

n t

he

mu

ffin

rec

ipes

Pet

e is

res

earc

hin

g.

|t-

375

| ≤ 2

5

3. A

RC

HER

Y I

n a

n O

lym

pic

arch

ery

even

t,

the

cen

ter

of t

he

targ

et i

s se

t ex

actl

y 13

0 ce

nti

met

ers

off

the

grou

nd.

To

get

the

hig

hes

t sc

ore

of t

en p

oin

ts, a

n a

rch

er

mu

st s

hoo

t an

arr

ow n

o fu

rth

er t

han

3.

05 c

enti

met

ers

from

th

e ex

act

cen

ter

of

the

targ

et.

a. W

rite

an

abs

olu

te v

alu

e in

equ

alit

y to

re

pres

ent

the

poss

ible

dis

tan

ces

dfr

om t

he

grou

nd

an a

rch

er c

an h

it t

he

targ

et a

nd

stil

l sc

ore

ten

poi

nts

.

|d-

130

|≤

3.0

5

b.

Gra

ph t

he

solu

tion

set

of

the

ineq

ual

ity

you

wro

te i

n p

art

a.

124

126

128

130

132

134

4. C

ATS

Du

rin

g a

rece

nt

visi

t to

th

e ve

teri

nar

ian

’s o

ffic

e, M

rs. V

an A

llen

was

in

form

ed t

hat

a h

ealt

hy

wei

ght

for

her

ca

t is

app

roxi

mat

ely

10 p

oun

ds, p

lus

or

min

us

one

pou

nd.

Wri

te a

n a

bsol

ute

va

lue

ineq

ual

ity

that

rep

rese

nts

u

nh

ealt

hy

wei

ghts

w f

or h

er c

at.

⎪ w-

10⎥

≥ 1

5. S

TATI

STIC

S T

he

mos

t fa

mil

iar

stat

isti

cal

mea

sure

is

the

arit

hm

etic

m

ean

, or

aver

age.

A s

econ

d im

port

ant

stat

isti

cal

mea

sure

is

the

stan

dard

de

viat

ion

, wh

ich

is

a m

easu

re o

f h

ow f

ar

the

indi

vidu

al s

core

s de

viat

e fr

om t

he

mea

n. F

or e

xam

ple,

in

a r

ecen

t ye

ar t

he

mea

n s

core

on

th

e m

ath

emat

ics

sect

ion

of

th

e S

AT

tes

t w

as 5

15 a

nd

the

stan

dard

de

viat

ion

was

114

. Th

is m

ean

s th

at

peop

le w

ith

in o

ne

devi

atio

n o

f th

e m

ean

h

ave

SA

T m

ath

sco

res

that

are

no

mor

e th

an 1

14 p

oin

ts h

igh

er o

r 11

4 po

ints

lo

wer

th

an t

he

mea

n.

a. W

rite

an

abs

olu

te v

alu

e in

equ

alit

y to

fin

d th

e ra

nge

of

SA

T m

ath

emat

ics

test

sco

res

wit

hin

on

e st

anda

rd

devi

atio

n o

f th

e m

ean

.

|x-

515

| ≤ 1

14

b.

Wh

at i

s th

e ra

nge

of

SA

T

mat

hem

atic

s te

st s

core

s ±

2 st

anda

rd

devi

atio

n f

rom

th

e m

ean

?

287

to 7

43

5-5

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3312

/21/

10

5:19

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF 2nd



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

34

G

lenc

oe A

lgeb

ra 1

Enri

chm

ent

Pre

cis

ion

of

Measu

rem

en

tT

he

prec

isio

n o

f a

mea

sure

men

t de

pen

ds b

oth

on

you

r ac

cura

cy i

n

mea

suri

ng

and

the

nu

mbe

r of

div

isio

ns

on t

he

rule

r yo

u u

se. S

upp

ose

you

mea

sure

d a

len

gth

of

woo

d to

th

e n

eare

st o

ne-

eigh

th o

f an

in

ch

and

got

a le

ngt

h o

f 6

5 −

8 i

nch

es.

Th

e dr

awin

g sh

ows

that

th

e ac

tual

mea

sure

men

t li

es s

omew

her

e

betw

een

6 9 −

16

an

d 6

11

−

16 i

nch

es. T

his

mea

sure

men

t ca

n b

e w

ritt

en u

sin

g

the

sym

bol

±, w

hic

h i

s re

ad p

lus

or m

inu

s. I

t ca

n a

lso

be w

ritt

en a

s a

com

pou

nd

ineq

ual

ity.

6 5 −

8 ±

1 −

16

in

.

6

9 −

16

in

. ≤ m

≤ 6

11

−

16 i

n.

In t

his

exa

mpl

e,

1 −

16

in

ch i

s th

e ab

solu

te e

rror

. Th

e ab

solu

te e

rror

is

one-

hal

f th

e sm

alle

st u

nit

use

d in

a m

easu

rem

ent.

Wri

te e

ach

mea

sure

men

t as

a c

omp

oun

d i

neq

ual

ity.

Use

th

e va

riab

le m

.

1. 5

1 −

2 ±

1 −

4 i

n.

2. 3

.78

± 0

.005

kg

3. 7

.11

± 0

.005

g

5

1

−

4 i

n.

≤ m

≤ 5

3

−

4 i

n.

3.7

75 k

g ≤

m ≤

7

.105 g

≤ m

≤ 7

.115 g

3

.785 k

g

4. 1

6 ±

1 −

2 y

d 5.

22

± 0

.5 c

m

6.

9 −

16

±

1 −

32

in

.

1

5 1

−

2 y

d ≤

m ≤

16 1

−

2 y

d

21.

5 c

m ≤

m ≤

17

−

32 i

n.

≤ m

≤ 1

9

−

32 i

n.

2

2.5

cm

For

eac

h m

easu

rem

ent,

giv

e th

e sm

alle

st u

nit

use

d a

nd

th

e ab

solu

te e

rror

.

7. 9

.5 i

n. ≤

m ≤

10.

5 in

. 8.

4 1 −

4 i

n. ≤

m ≤

4 3 −

4 i

n.

1

in

., 0

.5 i

n.

1

−

2 i

n., 1

−

4 i

n.

9. 2

3 1 −

2 cm

≤ m

≤ 2

4 1 −

2 c

m

10. 7

.135

mm

≤ m

≤ 7

.145

mm

1

cm

, 1

−

2 c

m

0.0

1 m

m, 0.0

05 m

m

56

78

65 – 8

5-5

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3412

/24/

10

7:41

AM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-5

Cha

pte

r 5

35

G

lenc

oe A

lgeb

ra 1

Gra

phin

g Ca

lcul

ator

Act

ivit

yA

bso

lute

Valu

e I

neq

ualiti

es

Th

e T

ES

T m

enu

can

be

use

d to

sol

ve a

nd

grap

h a

bsol

ute

val

ue

ineq

ual

itie

s by

usi

ng

the

equ

ival

ent

com

pou

nd

ineq

ual

itie

s re

late

d to

abs

olu

te v

alu

e.

Exer

cise

sG

rap

h a

nd

sol

ve e

ach

in

equ

alit

y.

1. |x

+ 3

| ≥

2

2. |

2x +

6|

≤ 4

3

. ⎪ 2

- 4

x −

5

⎥ >

2

4. |

x +

8|

< -

3

G

rap

h a

nd

sol

ve e

ach

in

equ

alit

y.

a.

| x +

4|

≥ 8

En

ter

the

ineq

ual

ity

into

Y1.

Th

en e

nte

r th

e eq

uiv

alen

t co

mpo

un

d in

equ

alit

y in

to Y

2 an

d gr

aph

to

view

th

e re

sult

s. B

e su

re t

o ch

oose

ap

prop

riat

e se

ttin

gs f

or t

he

view

win

dow

.K

eyst

roke

s:

MA

TH

EN

TE

R

+ 4

)

2nd

[T

ES

T]

4 8

EN

TE

R

+ 4

2nd

[T

ES

T]

6 (–

) 8

2nd

[T

ES

T]

2

+ 4

2nd

[T

ES

T]

4 8

EN

TE

R

GR

AP

H.

Use

TR

AC

E t

o co

nfi

rm t

he

solu

tion

. Wh

en y

= 1

th

e st

atem

ent

is

tru

e, a

nd

wh

en y

= 0

th

e st

atem

ent

is f

alse

. Th

us,

th

e so

luti

on i

s x

≤

-12

or

x ≥

4.

b.

⎪ 5x +

2

−

4 ⎥

≤ 7

En

ter

the

ineq

ual

ity

into

Y1

and

the

equ

ival

ent

com

pou

nd

ineq

ual

itie

s in

to Y

2. T

hen

gra

ph t

he

solu

tion

set

.K

eyst

roke

s:

MA

TH

EN

TE

R

( 5

+

2

)

÷ 4

)

2nd

[T

ES

T]

6 7

EN

TE

R

( 5

+

2

)

÷ 4

2nd

[TE

ST

] 4

(–) 7

2nd

[T

ES

T]

EN

TE

R

( 5

+

2

)

÷

4

2nd

[T

ES

T]

6 7

EN

TE

R

GR

AP

H.

Th

e st

atem

ent

is t

rue

betw

een

-6

and

5.2.

Th

us

the

solu

tion

is

-

6 ≤

x ≤

5.2

.

[-18.8

, 18.8

] scl:

2 b

y [

-3.1

, 3.1

] scl:

1

[-18.8

, 18.8

] scl:

2 b

y [

-3.1

, 3.1

] scl:

1

[-18.8

, 18.8

] scl:

2 b

y [

-3.1

, 3.1

] scl:

1

[-18.8

, 18.8

] scl:

2 b

y [

-3.1

, 3.1

] scl:

1

5-5 x

≤

-5

or

x ≥

-1

-5

≤ x

≤ -

1x

<

-2

or

x >

3

Exam

ple

no

so

luti

on

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3512

/22/

10

3:58

PM


A13_A21_ALG1_A_CRM_C05_AN_661384.indd A16A13_A21_ALG1_A_CRM_C05_AN_661384.indd A16 12/24/10 7:59 AM12/24/10 7:59 AM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

36

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

Gra

ph

ing

In

eq

ualiti

es i

n T

wo

Vari

ab

les

Gra

ph

Lin

ear

Ineq

ual

itie

s T

he

solu

tion

set

of

an i

neq

ual

ity

that

in

volv

es t

wo

vari

able

s is

gra

phed

by

grap

hin

g a

rela

ted

lin

ear

equ

atio

n t

hat

for

ms

a bo

un

dary

of

a h

alf-

pla

ne.

Th

e gr

aph

of

the

orde

red

pair

s th

at m

ake

up

the

solu

tion

set

of

the

ineq

ual

ity

fill

a r

egio

n o

f th

e co

ordi

nat

e pl

ane

on o

ne

side

of

the

hal

f-pl

ane.

G

rap

h y

≤ -

3x -

2.

Gra

ph y

= -

3x -

2.

Sin

ce y

≤ -

3x -

2 i

s th

e sa

me

as y

< -

3x -

2 a

nd

y =

-3x

- 2

, th

e bo

un

dary

is

incl

ude

d in

th

e so

luti

on s

et a

nd

the

grap

h s

hou

ld b

e dr

awn

as

a so

lid

lin

e.S

elec

t a

poin

t in

eac

h h

alf

plan

e an

d te

st i

t. C

hoo

se (

0, 0

) an

d (-

2, -

2).

y ≤

-3x

- 2

y

≤ -

3x -

2 0

≤ -

3(0)

- 2

-

2 ≤

-3(

-2)

- 2

0 ≤

-2

is f

alse

. -

2 ≤

6 -

2-

2 ≤

4 i

s tr

ue.

Th

e h

alf-

plan

e th

at c

onta

ins

(-2,

-2)

con

tain

s th

e so

luti

on. S

had

e th

at h

alf-

plan

e.

Exer

cise

sG

rap

h e

ach

in

equ

alit

y.

1. y

< 4

2.

x ≥

1

3. 3

x ≤

y

4. -

x >

y

5. x

- y

≥ 1

6.

2x

- 3

y ≤

6

7. y

< -

1 −

2 x

- 3

8.

4x

- 3

y <

6

9. 3

x +

6y

≥ 1

2

x

y

O

x

y

O

x

y

O

x

y

Ox

y

O

x

y O

x

y

Ox

y

O

x

y

O

5-6

x

y

O

Exam

ple

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3612

/21/

10

4:38

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-6

Cha

pte

r 5

37

Gle

ncoe

Alg

ebra

1

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Gra

ph

ing

In

eq

ualiti

es i

n T

wo

Vari

ab

les

Solv

e Li

nea

r In

equ

alit

ies

We

can

use

a c

oord

inat

e pl

ane

to s

olve

in

equ

alit

ies

wit

h

one

vari

able

.

U

se a

gra

ph

to

solv

e 2x

+ 2

> -

1 .

Ste

p 1

F

irst

gra

ph t

he

bou

nda

ry, w

hic

h i

s th

e re

late

d fu

nct

ion

. R

epla

ce t

he

ineq

ual

ity

sign

wit

h a

n e

qual

s si

gn, a

nd

get

0 on

a s

ide

by i

tsel

f.

2x

+ 2

> -

1 O

rigin

al in

equalit

y

2x

+ 2

= -

1 C

hange <

to =

.

2x

+ 2

+ 1

= -

1 +

1

Add 1

to e

ach s

ide.

2x +

3 =

0

Sim

plif

y.

G

raph

2x

+ 3

= y

as

a da

shed

lin

e.

Ste

p 2

C

hoo

se (

0, 0

) as

a t

est

poin

t, s

ubs

titu

tin

g th

ese

valu

es

into

th

e or

igin

al i

neq

ual

ity

give

us

3 >

-5.

Ste

p 3

B

ecau

se t

his

sta

tem

ent

is t

rue,

sh

ade

the

hal

f pl

ane

con

tain

ing

the

poin

t (0

, 0).

Not

ice

that

th

e x-

inte

rcep

t of

th

e gr

aph

is

at -

1 1 −

2 .

Bec

ause

th

e h

alf-

plan

e to

th

e ri

ght

of t

he

x-in

terc

ept

is s

had

ed, t

he

solu

tion

is

x >

-1

1 −

2 .

Exer

cise

sU

se a

gra

ph

to

solv

e ea

ch i

neq

ual

ity.

1. x

+ 7

≤ 5

x ≤

-2

2. x

- 2

> 2

x >

4

3. -

x +

1 <

-3

x >

4

4. -

x -

7 ≥

-6

x ≤

-1

5. 3

x -

20

< -

17 x

< 1

6.

-2x

+ 1

1 ≥

15

x ≤

-2

5-6

y

x

y

x

y

xO

y

x

y

x

y

x

y

x

Exam

ple

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3712

/21/

10

4:38

PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

38

Gle

ncoe

Alg

ebra

1

Skill

s Pr

acti

ceG

rap

hin

g I

neq

ualiti

es i

n T

wo

Vari

ab

les

Mat

ch e

ach

in

equ

alit

y to

th

e gr

aph

of

its

solu

tion

.

1. y

- 2

x <

2 b

a.

b

.

2. y

≤ -

3x d

3. 2

y -

x ≥

4 a

4. x

+ y

> 1

c

c.

d

.

Gra

ph

eac

h i

neq

ual

ity.

5. y

< -

1 6.

y ≥

x -

5

7. y

> 3

x

8. y

≤ 2

x +

4

9. y

+ x

> 3

10

. y -

x ≥

1

Use

a g

rap

h t

o so

lve

each

in

equ

alit

y.

11. 1

> 2

x +

5

12. 7

≤ 3

x +

4

13. -

1 −

2 <

- 1 −

2 x

+ 1x

y

Ox

y

Ox

y

O

x

y

O

x

y

O

x

y

O

x

y

Ox

y

O

x

y O

x

y

O

5-6 x <

-2

x ≥

1

x <

3

x

y

O

y

xO

y

x

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3812

/21/

10

4:38

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-6

Cha

pte

r 5

39

Gle

ncoe

Alg

ebra

1

Prac

tice

Gra

ph

ing

In

eq

ualiti

es i

n T

wo

Vari

ab

les

Det

erm

ine

wh

ich

ord

ered

pai

rs a

re p

art

of t

he

solu

tion

set

for

eac

h i

neq

ual

ity.

1. 3

x +

y ≥

6, {

(4, 3

), (-

2, 4

), (-

5, -

3), (

3, -

3)}

{(4,

3),

(3, -

3)}

2. y

≥ x

+ 3

, {(6

, 3),

(-3,

2),

(3, -

2), (

4, 3

)} {

(-3,

2)}

3. 3

x -

2y

< 5

, {(4

, -4)

, (3,

5),

(5, 2

), (-

3, 4

)} {

(3, 5

), (-

3, 4

)}

Gra

ph

eac

h i

neq

ual

ity.

4. 2

y -

x <

-4

5. 2

x -

2y

≥ 8

6.

3y

> 2

x -

3

Use

a g

rap

h t

o so

lve

each

in

equ

alit

y.

7. -

5 ≤

x -

9

8. 6

> 2 −

3 x

+ 5

9.

1 −

2 >

-2

x +

7 −

2

10. M

OV

ING

A m

ovin

g va

n h

as a

n i

nte

rior

hei

ght

of 7

fee

t (8

4 in

ches

). Yo

u h

ave

boxe

s in

12

in

ch a

nd

15 i

nch

hei

ghts

, an

d w

ant

to s

tack

th

em a

s h

igh

as

poss

ible

to

fit.

Wri

te a

n

ineq

ual

ity

that

rep

rese

nts

th

is s

itu

atio

n.

12x +

15y

≤ 8

4

11. B

UD

GET

ING

Sat

chi

fou

nd

a u

sed

book

stor

e th

at s

ells

pre

-ow

ned

DV

Ds

and

CD

s. D

VD

s co

st $

9 ea

ch, a

nd

CD

s co

st $

7 ea

ch. S

atch

i ca

n s

pen

d n

o m

ore

than

$35

.

a. W

rite

an

in

equ

alit

y th

at r

epre

sen

ts t

his

sit

uat

ion

. 9x

+ 7

y ≤

35

b.

Doe

s S

atch

i h

ave

enou

gh m

oney

to

buy

2 D

VD

s an

d 3

CD

s?

No

, th

e p

urc

has

es w

ill b

e $3

9, w

hic

h is

gre

ater

th

an $

35.

x

y

O

x

y

O

x

y

O

5-6

x

y

O

x

y

O

x <

1 1 −

2

x ≥

4x >

1 1 −

2 y

x

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

3912

/21/

10

4:38

PM

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF 2nd



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

40

Gle

ncoe

Alg

ebra

1

1. F

AM

ILY

Tyr

one

said

th

at t

he

ages

of

his

si

blin

gs a

re a

ll p

art

of t

he

solu

tion

set

of

y >

2x,

wh

ere

x is

th

e ag

e of

a s

ibli

ng

and

y is

Tyr

one’

s ag

e. W

hic

h o

f th

e fo

llow

ing

ages

is

poss

ible

for

Tyr

one

and

a si

blin

g?T

yron

e is

23;

Max

ine

is 1

4.

no

Tyr

one

is 1

8; C

amil

le i

s 8.

ye

sT

yron

e is

12;

Fra

nci

s is

4.

yes

Tyr

one

is 1

1; M

arti

n i

s 6.

n

oT

yron

e is

19;

Pau

l is

9.

yes

2. F

AR

MIN

G T

he

aver

age

valu

e of

U.S

. fa

rm c

ropl

and

has

ste

adil

y in

crea

sed

in

rece

nt

year

s. I

n 2

000,

th

e av

erag

e va

lue

was

$14

90 p

er a

cre.

Sin

ce t

hen

, th

e va

lue

has

in

crea

sed

at l

east

an

ave

rage

of

$77

per

acre

per

yea

r. W

rite

an

in

equ

alit

y to

sh

ow l

and

valu

es a

bove

th

e av

erag

e fo

r fa

rmla

nd.

3. S

HIP

PIN

G A

n i

nte

rnat

ion

al s

hip

pin

g co

mpa

ny

has

est

abli

shed

siz

e li

mit

s fo

r pa

ckag

es w

ith

all

th

eir

serv

ices

. Th

e to

tal

of t

he

len

gth

of

the

lon

gest

sid

e an

d th

e gi

rth

(di

stan

ce c

ompl

etel

y ar

oun

d th

e pa

ckag

e at

its

wid

est

poin

t pe

rpen

dicu

lar

to t

he

len

gth

) m

ust

be

less

th

an o

r eq

ual

to

419

cen

tim

eter

s. W

rite

an

d gr

aph

an

in

equ

alit

y th

at r

epre

sen

ts t

his

sit

uat

ion

.

4. F

UN

DR

AIS

ING

Tro

op 2

00 s

old

cide

r an

d do

nu

ts t

o ra

ise

mon

ey f

or c

har

ity.

Th

ey

sold

sm

all

boxe

s of

don

ut

hol

es f

or $

1.25

an

d ci

der

for

$2.5

0 a

gall

on. I

n o

rder

to

cove

r th

eir

expe

nse

s, t

hey

nee

ded

to

rais

e at

lea

st $

100.

Wri

te a

nd

grap

h a

n

ineq

ual

ity

that

rep

rese

nts

th

is s

itu

atio

n.

1.25

d +

2.5

c ≥

10

0

5.IN

CO

ME

In 2

006

the

med

ian

yea

rly

fam

ily

inco

me

was

abo

ut

$48,

200

per

year

. Su

ppos

e th

e av

erag

e an

nu

al r

ate

of c

han

ge s

ince

th

en i

s $1

240

per

year

.

a. W

rite

an

d gr

aph

an

in

equ

alit

y fo

r th

e an

nu

al f

amil

y in

com

es y

th

at

are

less

th

an t

he

med

ian

for

x y

ears

af

ter

2006

.

y <

124

0x +

48,

200

b.

Det

erm

ine

wh

eth

er e

ach

of

the

foll

owin

g po

ints

is

part

of

the

solu

tion

set

. (2

, 51,

000)

n

o

(8, 6

9,20

0)

no

(5, 5

0,00

0)

yes

(10,

61,

000)

n

oO

Girth

Len

gth

100

150 50200

250

300

350

400

450

500

g

150

100

5025

035

020

030

040

0450 50

0�

Cider (gal)

Do

nu

t h

ole

s

2030 10405060708090c

3020

1050

7040

60d

O

1.25

d +

2.5

0c ≥

100

O

Income ($1000)

Year

s si

nce

200

6

46,0

00

48,0

00

44,0

00

50,0

00

52,0

00

54,0

00

56,0

00

58,0

00

y

32

15

78

910

46

x

lengt

h

girth

l + g

≤ 4

19

5-6

Wor

d Pr

oble

m P

ract

ice

Gra

ph

ing

In

eq

ualiti

es i

n T

wo

Vari

ab

les

y >

77x

+ 1

490

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

4012

/21/

10

4:38

PM


NA

ME

DAT

E

P

ER

IOD

Lesson 5-6

Cha

pte

r 5

41

Gle

ncoe

Alg

ebra

1

Enri

chm

ent

Lin

ear

Pro

gra

mm

ing

Lin

ear

prog

ram

min

g ca

n b

e u

sed

to m

axim

ize

or m

inim

ize

cost

s. I

t in

volv

es g

raph

ing

a se

t of

lin

ear

ineq

ual

itie

s an

d u

sin

g th

e re

gion

of

inte

rsec

tion

. You

wil

l u

se l

inea

r pr

ogra

mm

ing

to s

olve

th

e fo

llow

ing

prob

lem

.

L

ayn

e’s

Gif

t S

hop

pe

sell

s at

mos

t 50

0 it

ems

per

wee

k. T

o m

eet

her

cu

stom

ers’

dem

and

s, s

he

sell

s at

lea

st 1

00 s

tuff

ed a

nim

als

and

75

gree

tin

g ca

rds.

If

th

e p

rofi

t fo

r ea

ch s

tuff

ed a

nim

al i

s $2

.50

and

th

e p

rofi

t fo

r ea

ch g

reet

ing

card

is

$1.

00, t

he

equ

atio

n P

(a, g

) =

2.5

0a +

1.0

0g c

an b

e u

sed

to

rep

rese

nt

the

pro

fit.

H

ow m

any

of e

ach

sh

ould

sh

e se

ll t

o m

axim

ize

her

pro

fit?

Wri

te t

he

ineq

ual

itie

s:

G

raph

th

e in

equ

alit

ies:

a +

g ≤

500

a ≥

100

g ≥

75

Fin

d th

e ve

rtic

es o

f th

e tr

ian

gle

form

ed: (

100,

75)

, (10

0, 4

00),

and

(425

, 75)

Su

bsti

tute

th

e va

lues

of

the

vert

ices

in

to t

he

equ

atio

n f

oun

d ab

ove:

2.

50(1

00)

+ 1

(75)

= 3

25

2.

50(1

00)

+ 1

(400

) =

650

2.50

(425

) +

1(7

5) =

113

7.50

T

he

max

imu

m p

rofi

t is

$11

37.5

0.

Exer

cise

s T

he

Sp

irit

Clu

b i

s se

llin

g sh

irts

an

d b

ann

ers.

Th

ey s

ell

at m

ost

400

of t

he

two

item

s. T

o m

eet

the

dem

and

s of

th

e st

ud

ents

, th

ey m

ust

sel

l at

lea

st 5

0 T-

shir

ts a

nd

10

0 b

ann

ers.

Th

e p

rofi

t on

eac

h s

hir

t is

$4.

00 a

nd

th

e p

rofi

t on

eac

h b

ann

er i

s $1

.50,

th

e eq

uat

ion

P(t

, b)

= 4

.00t

+ 1

.50b

can

be

use

d t

o re

pre

sen

t th

e p

rofi

t. H

ow

man

y sh

ould

th

ey s

ell

of e

ach

to

max

imiz

e th

e p

rofi

t?

1. W

rite

th

e in

equ

alit

ies

to r

epre

sen

t th

is s

itu

atio

n.

t +

b ≤

40

0; t

≥ 5

0; b

≥ 1

00

2. G

raph

th

e in

equ

alit

ies

from

Exe

rcis

e 1.

See

gra

ph

at

rig

ht.

3. F

ind

the

vert

ices

of

the

figu

re f

orm

ed.

(50,

10

0), (

50, 3

50),

(30

0, 1

00)

4. W

hat

is

the

max

imu

m p

rofi

t th

e S

piri

t C

lub

can

mak

e?

$135

0

b

tO

100

300

400

500

100

200

200

300

400

500

g

aO

100

-30

0

-40

0

-50

0

-10

0

-20

0

200

300

400

500

5-6

Exam

ple

023_

042_

ALG

1_A

_CR

M_C

05_C

R_6

6138

4.in

dd

4112

/24/

10

11:3

5 A

M

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DAT

E

P

ER

IOD

Cha

pte

r 5

42

Gle

ncoe

Alg

ebra

1

Spre

adsh

eet

Act

ivit

yIn

eq

ualiti

es i

n T

wo

Vari

ab

les

You

can

use

a s

prea

dsh

eet

to d

eter

min

e w

het

her

ord

ered

pai

rs s

atis

fy a

n i

neq

ual

ity.

Exer

cise

sU

se a

sp

read

shee

t to

det

erm

ine

wh

ich

ord

ered

pai

rs a

re p

art

of t

he

solu

tion

set

for

eac

h i

neq

ual

ity.

1. 2

x +

3y

> 1

; {(0

, 3),

(1, -

3), (

-2,

-1)

, (6,

8)}

{(0

, 3),

(6, 8

)}

2. 7

x -

y <

8; {

(1, 2

), (-

3, -

1), (

0, -

10),

(6, 9

)} {

(1, 2

), (-

3, -

1)}

3. y

≥ 3

x; {

(3, 1

), (-

4, 5

), (-

1, 0

), (7

, -1)

, (2,

7)}

{(-

4, 5

), (-

1, 0

), (2

, 7)}

4. y

≤ -

4x; {

(9, 3

), (-

3, 5

), (0

, 0),

(12,

1),

(3, 9

)} {

(-3,

5),

(0, 0

)}

5. y

> 1

2 -

2x;

{(-

3, -

3), (

-1,

9),

(12,

13)

, (-

4, 1

1)}

{(12

, 13)

}

6. y

> 2

+ 6

x; {

(0, -

4), (

-4,

8),

(9, 1

7), (

-2,

18)

, (-

5, -

5)}

{(-

4, 8

), (-

2, 1

8), (

-5,

-5)

}

7. |

x +

1|

≤ y

; {(1

, -8)

, (0,

4),

(5, 1

6), (

-2,

-8)

, (11

, -2)

} {(

0, 4

), (5

, 16)

}

8. |

y -

7|

< x

; {(5

, 8),

(-1,

3),

(2, 1

9), (

-6,

-6)

, (10

, -22

)} {

(5, 8

)}

U

se a

sp

read

shee

t to

det

erm

ine

wh

ich

ord

ered

pai

rs f

rom

th

e se

t {(

2, 3

), (

4, 1

), (

-1,

2),

(0,

7),

(-

8, -

10)}

are

par

t of

th

e so

luti

on s

et f

or 5

x -

2y

> 1

2.

Ste

p 1

U

se c

olu

mn

s A

an

d B

of

the

spre

adsh

eet

for

the

repl

acem

ent

set.

E

nte

r th

e x-

coor

din

ates

in

col

um

n A

an

d th

e y-

coor

din

ates

in

col

um

n B

.

Ste

p 2

C

olu

mn

C c

onta

ins

the

form

ula

for

th

e in

equ

alit

y.

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{w | w ≤ -14}

1.

-15-16-17-18 -14-13 -11-10-12

2.

3.

4.

5.

{n � n ≥ -28}

Sample answer: n = the number;

n - 7 ≥ 15; {n � n ≥ 22}

{m � m > -78}

B

1.

2.

3.

4.

5.

Ø

{d � d ≤ -100}

Sample answer: n = the number; n + 3 < 19 - n;

{ n |n < 8 }

D

{t � t > 6}

1.

2.

3.

4.

5.

B

1.

2.

3.

4.

5.

D

{x � 0 < x ≤ 3}

� x � > 1

{x � -1 ≤ x ≤ 3}

{x � x ≥ 2 or ≤ -1}

yes

-4 54321-3 0-1-2

-4 54321-3 0-1-2

-4 54321-3 0-1-2

y

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y

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-2(x -y )≤ 4

y

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x +1 ≤ y13

Chapter 5 Assessment Answer KeyQuiz 1 (Lessons 5-1 and 5-2) Quiz 3 (Lessons 5-4 and 5-5) Mid-Chapter TestPage 45 Page 46 Page 47

Quiz 2 (Lessons 5-3)

Page 45

1.

2.

3.

4.

5.

6.

7.

8.

9.

Sample answer:

a = no. of apples;

6a ≥ 50; at least 8 1 − 3

apples

Sample answer:

� = length; � + 63 ≤ 85;

22 in. or less

C

J

B

F

{t | t < 15.2}

{x | x > 4}

C

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1.

2.

3.

4.

5.

6.

7.

intersection

half-plane

compound inequality

boundary

union

Sample answer: An open

half-plane is the half of a

coordinate grid that

contains all solutions to

an inequality, boundary

included.

Sample answer: Set-

builder notation is a way

of writing a solution set.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. J

C

F

C

G

D

J

A

J

B

H

A

13.

14.

15.

16.

17.

18.

19.

20. G

A

G

B

J

A

G

A

-1B:

Chapter 5 Assessment Answer KeyVocabulary Test Form 1Page 48 Page 49 Page 50


Chapter 5 Assessment Answer KeyForm 2A Form 2BPage 51 Page 52 Page 53 Page 54

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. C

F

D

H

D

G

A

J

A

F

C

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12.

13.

14.

15.

16.

17.

18.

19.

20. G

B

J

D

J

G

A

A

H

{n � -12 ≤ n ≤ 12}B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. C

F

D

H

B

H

A

J

D

F

A 12.

13.

14.

15.

16.

17.

18.

19.

20. F

A

J

J

A

H

B

C

H

{x |-3 ≤ x < 4}B:

Chapter 5 Assessment Answer KeyForm 2CPage 55 Page 56

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

-1-2-3-4 0 1 2 43

-1-2-3-4 0 1 2 43

{x |x ≤ 1 or x ≥ 2}

-1-2-3-4 0 1 2 43

0-1-2-3-4 1 2 43

-1-2-3-4-5-6 0 1 2

{x |x > 13}

131211109 14 15 1716

{x |x < 3}

{c |c ≤ 13}

{x |x is a real number}

{w |-5 < w < 2}

{p |p ≤ 2}

{x |x < 8}

{r |r > -5.5}

{y |y ≤ -5.5}

{t |t ≥ 84}

{

b|b > -1 3 −

5 }

{z |z < -4}

{x | -1 ≤ x ≤ 3}

15.

16.

17.

18.

19.

20.

B:

3x + 0.75y ≤ 20; No, the cost

would be more than $20.00.

y < x - 1

y

xO

y = - x + 2 13

y

xO

2x - 3y ≤ 6

Sample answer:

x = car price;

� 6000 - x �

≤

1500;

{x � 4500 ≤ x ≤ 7500}; from

$4500 to $7500

{x | -2 < x < 1.5}

-3 1 117

Chapter 5 Assessment Answer KeyForm 2DPage 57 Page 58

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

{w| -3 ≤ w ≤ 5}-11 30

-1-2-3-4 0 1 2 43

0-1-2-3-4 1 2 3 4

-1-2-3-4 0 1 2 43

1312111098 14 15 16

{x | x > -11}

{x | x > 2}

{w | w is a real number}

{w | 2 < w < 3}

{w | w > -5}

{t | t ≤ -5.5}

{m | m > 6}

{k | k ≤ 3.5}

{

z | z < -3 1 −

3 }

{h | h < 27}

{k|k ≥ 5}

{y |y ≤ 12}

{z | z ≤ -11 or z ≥ 3}

{x| 1 < x < 4}-1-2-3 0 1 2 4 53

16.

17.

18.

19.

20.

B:

28x + 4y ≤ 75;

no, the cost would be more than $75.00.

Sample answer: x = boat

price; � 4500 - x � ≤ 1300; {x � 3200 ≤ x ≤ 5800}; from

$3200 to $5800

y

xO

y = 3x

2x - y ≤ 1

y

xO

2y - 4x < 8

-7 -5 3 5

Chapter 5 Assessment Answer KeyForm 3Page 59 Page 60

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1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

9.3

3635343332 37 38 4039

Sample answer: s = amount of sales; 32,500 < 0.1s + 25,600 < 41,900; between $69,000 and $163,000

Sample answer: n = small positive even integer; n + n + 2 ≤ 15; 6, 8; 4, 6; 2, 4

Sample answer: j = cost of jeans;

2(19.89) + j ≤ 78; no more than $38.22

Ø

{x | x ≤ -9}

{b | b < 0.5}

{t | t >

9 −

11 }

{w | w ≤ -10.4}

Sample answer: n = the

number; n - 15 > 2n + 8;

{n | n < -23}

Sample answer: n = the number;

{t | t < 36}

{m |m ≥ 9.3}

-

3

−

7 + n ≥ 2; {n | n ≥ 2 3

−

7 }

13.

14.

-6 80

15.

16.

17.

18.

19.

20.

B:

-1-2-3-4-5-6-7-8-9

-1-2 0 1 2 4 5 63

-1-2 0 1 2 4 5 63

y

xO

x + 3y > -12

y

xO

-y = 3x

16x + 12y ≤ 120; 3

{x|x ≤ -1 or x ≥ 4}

{x |-2 < x < 6}

{x |-6 < x ≤ 8}

{n|n > -6}

False; sample answer: x > 3 and y > 1. If xy < 0, x and y cannot both be positive, so x > 3 and y > 1 is false.

{x|x ≤ -2.8 or x ≥ 4}

-2.8 40

Chapter 5 Assessment Answer Key Page 61, Extended-Response Test Scoring Rubric

Score General Description Specifi c Criteria

4 Superior

A correct solution that

is supported by well-

developed, accurate

explanations

• Shows thorough understanding of the concepts of using the properties of inequalities, solving inequalities, solving compound inequalities, solving open sentences involving absolute value, and graphing inequalities in two variables.

• Uses appropriate strategies to solve problems.

• Computations are correct.

• Written explanations are exemplary.

• Graphs are accurate and appropriate.

• Goes beyond requirements of some or all problems.

3 Satisfactory

A generally correct solution,

but may contain minor fl aws

in reasoning or computation

• Shows an understanding of the concepts of using the properties of inequalities, solving inequalities, solving compound inequalities, solving open sentences involving absolute value, and graphing inequalities in two variables.

• Uses appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are effective.

• Graphs are mostly accurate and appropriate.

• Satisfi es all requirements of problems.

2 Nearly Satisfactory

A partially correct

interpretation and/or

solution to the problem

• Shows an understanding of most of the concepts of using the properties of inequalities, solving inequalities, solving compound inequalities, solving open sentences involving absolute value, and graphing inequalities in two variables.

• May not use appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are satisfactory.

• Graphs are mostly accurate.

• Satisfi es the requirements of most of the problems.

1 Nearly Unsatisfactory

A correct solution with no

supporting evidence or

explanation

• Final computation is correct.

• No written explanations or work is shown to substantiate the

fi nal computation.

• Graphs may be accurate but lack detail or explanation.

• Satisfi es minimal requirements of some of the problems.

0 Unsatisfactory

An incorrect solution

indicating no mathematical

understanding of the

concept or task, or no

solution is given

• Shows little or no understanding of most of the concepts of

using the properties of inequalities, solving inequalities, solving compound inequalities, solving open sentences involving absolute value, and graphing inequalities in two variables.

• Does not use appropriate strategies to solve problems.

• Computations are incorrect.

• Written explanations are unsatisfactory.

• Graphs are inaccurate or inappropriate.

• Does not satisfy requirements of problems.

• No answer may be given.

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PDF Pass



Chapter 5 Assessment Answer Key Page 61, Extended-Response Test Sample AnswersIn addition to the scoring rubric found on page A34, the following sample answers may be used as guidance in evaluating extended-response assessment items.

2a. After drawing a graph, students should write an inequality that corresponds with the line they have drawn. The inequality may or may not include equality.

2b. The solution set includes the boundary (line) if the inequality written for part a includes equality. If the inequality written for part a does not include equality, then the student should state that the solution set of the inequality does not include the line.

3. The inequality ab > 2a can be determined to be true or false by considering the value of a. Since b > 2, by the Multiplication Property of Inequality ab > 2a is true if a is a positive number.

4. The solution set for � x - 2 � > 4 is {x � x < -2 or x > 6} . The solution set for -2x < 4 or x > 6 is {x � x > -2} . One includes numbers greater than -2, and the other includes numbers less than -2 or greater than 6. These solution sets are not the same.

5a. w ≤ 90 - 2(20); {w � w ≤ 50}

5b. The formula for the area of a rectangle with 50 substituted for the width can be used to write the compound inequality 2800≤ 50 � ≤ 3200. The possible lengths are found by solving this compound inequality for �. The solution set is {� � 56 ≤ � ≤ 64} .

5c. � 175,000 - x � ≤ 20,000; 155,000 ≤ x ≤ 195,000; The Fraziers are willing to pay from $155,000 to $195,000 for the house.

1. 10n - 7(n + 2) > 5n - 12 Original inequality

10n - 7n - 14 > 5n - 12 Distributive Property

3n - 14 > 5n - 12 Combine like terms.

3n - 14 - 5n > 5n - 12 - 5n Subtract 5n from each side.

-2n - 14 > -12 Simplify.

-2n - 14 + 14 > -12 + 14 Add 14 to each side.

-2n > 2 Simplify.

-2n −

-2 < 2 ÷ (-2) Divide each side by -2. Change > to <.

n < - 1 Simplify.

The solution set is {n�n < -1} .

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Chapter 5 Assessment Answer KeyStandardized Test PracticePage 62 Page 63

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. F G H J

F G H J

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

A B C D

A B C D

18.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

1 19.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

5

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11.

12.

13.

14.

15.

16.

17. A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

A B C D


Chapter 5 Assessment Answer KeyStandardized Test PracticePage 64

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20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30a.

30b.

21

102

y

xO

y = x - 4

a ≤ -15

y = 4x - 4

{(-4, -2), (-2, -1.5),

(0, -1), (2, -0.5), (4, 0)}

{

f |

|

-3 ≤ f ≤

5 −

3 }

$400 ≤ x ≤ $800

$428 ≤ x ≤ $856

x = 10

{a � -2 ≤ a < 5}

-4 -3 -2 -1 0 1 2 3 4

-3 -2 -1 0 1 2 3 4 5

y

xO

Chapter 5 Resource Masters - Commack Schools

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Transcript of Chapter 5 Resource Masters - Commack Schools