Unit 2 Algebra Concepts HS - Triumph Learning · High School Algebra I, Book . 1. Unit 1 . Number...

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Buckle Down High School Algebra I, Book 1

Unit 1 Number Sense

Lesson 1: Real Numbers Lesson 2: Computation with Real Numbers Lesson 3: Comparing and Ordering Real Numbers Lesson 4: Estimation and Problem Solving

Unit 2 Algebra Concepts

Lesson 5: Expressions Lesson 6: Solving Linear Equations Lesson 7: Using Different Forms of Linear Equations Lesson 8: Solving Linear Inequalities Lesson 9: Systems of Linear Equations and Inequalities Lesson 10: Functions

Go to www.BuckleDown.com to review our complete line of State Test materials for Grades 2–12 READING • WRITING • MATHEMATICS • SCIENCE • ALGEBRA I • BIOLOGY

The quadratic formula, shown on the cover, is used to solve quadratic equations. These equations have many real-world uses. They can be used to solve problems with areas, model business profit, and answer questions in Physics involving projectile motion.

2ND EDITION

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

ISBN 0-7836-5041-85 1 5 9 5

HS A

LGE

BR

A 1, B

OO

K 1

Student Set US05080S2Includes: Student Workbook, Form A Practice Test, Form B Practice Test

Individual Products:Student Workbook US05080W2Form A Practice Test US05080A2Form B Practice Test US05080B2

HSAlgebra 1Book 1

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.TABLE OF CONTENTS

iii

Introduction ..................................................................................... 1

Test-Taking Tips .................................................................. 2

Unit 1 – Number Sense ................................................................... 3

Lesson 1: Real Numbers ..................................................... 4Skills: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.9

Lesson 2: Computation with Real Numbers .................... 20Skills: 1.8, 1.10

Lesson 3: Comparing and Ordering Real Numbers ........ 33Skills: 1.11, 1.12

Lesson 4: Estimation and Problem Solving ..................... 41Skills: 1.7, 1.13, 1.14

Unit 2 – Algebra Concepts ........................................................... 53

Lesson 5: Expressions ....................................................... 54Skills: 2.1, 2.2, 2.3

Lesson 6: Solving Linear Equations ................................. 66Skills: 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.13, 2.14, 2.20

Lesson 7: Using Different Forms of Linear Equations.... 87Skills: 2.12, 2.15, 2.16, 2.17, 2.18, 2.19

Lesson 8: Solving Linear Inequalities ............................ 107Skills: 2.4, 2.5, 2.11

Lesson 9: Systems of Linear Equations and Inequalities ....................................................... 115Skills: 2.21, 2.22, 2.23, 2.24

Lesson 10: Functions ....................................................... 141 Skills: 2.25, 2.26, 2.27, 2.28, 2.29, 2.30

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Table of Contents

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iv

Sample code: N.B.2

Strand Objective Standard

Strands: N � Number and Operations A � Algebra G � Geometry M � Measurement D � Data Analysis and Probability

To the Teacher:

“Skills” codes are listed for each lesson in the table of contents and for each page in the shaded gray bars that run across the tops of the pages in the workbook (see example to the right). These codes indicate which Skills are covered on a given page.

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Unit 1 – Number Sense

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Lesson 1: Real NumbersIn this lesson, you will classify real numbers as members of subsets. You will also distinguish between finite and infinite sets of numbers. You will use exponential and scientific notation. You will also find square roots and the absolute value of real numbers.

The Subsets of the Real Number SystemThe numbers we use everyday are part of the real number system. The real number system is made up of the following subsets:

Natural numbers consist of the counting numbers.

{1, 2, 3, 4, 5, 6, 7, 8, . . .}

Whole numbers consist of the counting numbers and zero.

{0, 1, 2, 3, 4, 5, 6, 7, . . .}

Integers consist of the natural numbers, their opposites, and zero.

{. . . , 4, 3, 2, 1, 0, 1, 2, 3, 4, . . .}

Rational numbers consist of the numbers that can be expressed as a fraction a

b ,

where a and b are both integers and b 0. This set includes the integers, terminating

decimals, and repeating decimals. Here are a few examples of rational numbers:

5

1 5 17

3 5 2

3 7

10 0.7 2

9 0. 2

Irrational numbers consist of the numbers that cannot be expressed as a fraction of integers. In decimal form, they are the numbers that go on forever without a repeating pattern. Here are a few examples of irrational numbers:

8 2.8284. . . p 3.1415. . . 18.2643. . .

The following tree diagram shows the real number system and its subsets.

Real Numbers

Rational Numbers Irrational Numbers

Nonterminatingand Nonrepeating

Decimals

Integers RepeatingDecimals

Whole Numbers NegativeIntegers

TerminatingDecimals

NaturalNumbers

Zero

Skills: 1.1

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Lesson 1: Real Numbers©

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PracticeDirections: For Numbers 1 through 12, classify each number as natural, whole, integer, rational, or irrational. Write all names that apply.

1. 5 ____________________________________________________________________

2. 7 ___________________________________________________________________

3. 2 3

5 ____________________________________________________________________

4. 13 4

7 _________________________________________________________________

5. 0 _____________________________________________________________________

6. 3.87655 _______________________________________________________________

7. 11 __________________________________________________________________

8. 14 __________________________________________________________________

9. 1 _____________________________________________________________________

10. 4 1

2 ____________________________________________________________________

11. 5.1316. . . _____________________________________________________________

12. 3 2

3 __________________________________________________________________

13. Which number is rational?

A. 15

B. 23.7

C. 4

D. 23.542718. . .

14. Which number is an integer?

A. 43

87

B. 2

7

C. 0

D. p

Skills: 1.1

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Finite and Infinite SetsSets of numbers are classified as finite or infinite depending on the number of elements in the set. Empty braces { } mean a set contains no elements. The three dots (. . .) mean and so on and are used to show that the pattern continues.

A finite set has a countable number of elements. For example, the following set is finite: {1, 3, 5, 7, 9}. Notice the braces { } are used to indicate a set. The elements, or members, of the set are listed within the braces. Sometimes finite sets can be given without listing their elements.

Examples

D {days of the week}

E {positive even numbers less than 50}

P {polygons that have less than 6 sides}

The empty set is a finite set. As the name implies, it is the set that contains no elements. The months of the year with 35 days is an example of the empty set.

An infinite set has an uncountable number of elements. For example, the

following sets are infinite: {. . . , 1, 0, 1, . . .}, {. . . , 6, 7, 8, . . .}, {. . . , 0.05, 0.04,

0.03, . . .}, and . . . , 1

2 , 1

3 , 1

4 , . . . . As with the finite sets, infinite sets can be given

without listing their elements.

Examples

I {integers}

T {multiples of three}

R {real numbers between one and two}

TIP: It is important to recognize how the three-dot notation is used. It may be used in either an infinite set {1, 2, 3, 4, . . .}, meaning that the set continues forever, or a finite set {1, 2, 3, 4, . . . , 20}, meaning all the numbers in between.

Skills: 1.9

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PracticeDirections: For Numbers 1 through 10, classify each set of numbers as finite or infinite.

1. {2, 4, 6, 8, 10} _________________________

2. {even numbers} _________________________

3. {even integers between 5,000 and 5,000} _________________________

4. {integers less than 6} _________________________

5. {whole numbers less than 6} _________________________

6. {6, 5, 4, . . . , 8} _________________________

7. {days of the year in which the sun shines} _________________________

8. {countries of the world} _________________________

9. {20, 10, 0, 10, . . .} _________________________

10. {0} _________________________

11. Which set of numbers is finite?

A. {factors of 45}

B. {odd numbers}

C. {multiples of 3}

D. {negative integers}

12. Which set of numbers is infinite?

A. {whole numbers less than 19}

B. {positive integers less than 19}

C. {natural numbers less than 19}

D. {negative integers less than 19}

Skills: 1.9

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Parts of a WholeYou can use fractions, decimals, or percents to represent parts of a whole.

FractionsA fraction is a rational number whose numerator (top number) shows the number of parts of the whole or set you have. The denominator (bottom number) shows the number of parts the whole or set is divided into.

numerator ’4

7 (the number of parts you have)

denominator ’

(the number of parts the whole is divided into)

Example

Four of ten pieces of candy in a pack are cherry. The rest are orange.

What fraction of the pieces of candy are cherry?

4

10

or 2

5

What fraction of the pieces of candy are orange?

6

10

or 3

5

Example

Joanna tags birds for an animal biologist. One day she tagged six cardinals, five robins, three blue jays, and eight orioles.

What fraction of the birds Joanna tagged that day were cardinals?

6

22

or 3

11

What fraction of the birds Joanna tagged that day were orioles?

8

22

or 4

11

A fraction is in simplest form when there is no number (other than 1) that evenly divides both the numerator and the denominator.

Skills: 1.2

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.Lesson 1: Real Numbers

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DecimalsA decimal is a number in which each place to the left or right of the decimal point

represents a power of 10. Each decimal place is based on a fraction out of

10, 100, 1,000, etc. For example, the value of the 7 in the table below is 7

10

, the value

of the 1 is 1

100

, the value of the 9 is 9

1,000

.

Fifty-two and seven thousand one hundred ninety-three ten-thousandths is shown in the following place-value table.

Example

Write each of these numbers as a decimal.

fifty-one and four hundred sixty-seven thousandths: 51.467

two hundred one and ninety-three ten thousandths: 201.0093

seven thousand nineteen and eight hundredths: 7,019.08

PercentsA percent (%) is a way of comparing a number to 100. Percent means ‘‘out of one hundred.’’ In the following figure, 36% of the squares are shaded.

36% is the same as 36 out of 100 or, in simplest form, 9 out of 25.

Example

Write each percent as a number out of 100. Then write it in simplest form.

55%: 55 out of 100 or 11 out of 20

84%: 84 out of 100 or 21 out of 25

71%: 71 out of 100

Skills: 1.2

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PracticeDirections: Use the following information to answer Numbers 1 through 5. Write your answers in simplest form.

Juanita just got her report card. She received two A’s, five B’s, four C’s, one D, and no F’s.

1. What fraction of Juanita’s grades were A’s? ____________________

2. What fraction of Juanita’s grades were B’s? ____________________

3. What fraction of Juanita’s grades were C’s? ____________________

4. What fraction of Juanita’s grades were D’s? ____________________

5. What fraction of Juanita’s grades were F’s? ____________________

Directions: For Numbers 6 through 8, write each number as a decimal.

6. five hundred seventy-eight ten thousandths ____________________

7. nine hundred two and forty-three hundredths ____________________

8. twenty-one and sixteen thousandths ____________________

Directions: For Numbers 9 through 11, write each percent as a number out of 100. Then write it in simplest form.

9. 94% _____________________________________________________________________

10. 30% _____________________________________________________________________

11. 8% ______________________________________________________________________

Skills: 1.2

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Exponential NotationAn exponent shows how many times to multiply a base number by itself. A negative exponent tells you how many times to multiply the reciprocal of a base number by itself.

For example, in 74, the base number is 7 and the exponent is 4.

exponent

‘ 74 7 • 7 • 7 • 7 2,401 “base

Examples

nine to the zero power: 90 1

four to the first power: 41 4

d to the second power: d2 d • d

h to the fifth power: h5 h • h • h • h • h

two thirds to the fourth power: 2

3

4 2

3 • 2

3 • 2

3 • 2

3

16

81

eight to the negative first power: 81 1

8

w to the negative second power: w2 1

w

• 1

w

1

w2

three fourths to the negative third power: 3

4 3

4

3 • 4

3 • 4

3

64

27

TIP: Any base number with 1 as an exponent equals that base number. A base number with 0 as an exponent equals 1. The exception is 00, which has no meaning.

Skills: 1.3

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PracticeDirections: For Numbers 2 through 5, write each expression two other ways, as shown in Number 1.

1. 123 12 • 12 • 12 1,728

2. (9)2 _______________ _______________

3. 53 _______________ _______________

4. 36 _______________ _______________

5. (2)7 _______________ _______________

Directions: For Numbers 6 through 11, rewrite each expression using exponential notation.

6. 13 • 13 • 13 _____________ 9. b • b • b • b • b • b _____________

7. 16 • 16 • 16 • 16 _____________ 10. r • r • r • r • r _____________

8. (4) • (4) _____________ 11. 7 • 7 • . . . • 7 (n times) ___________

Directions: For Numbers 12 through 17, write the numbers using exponential notation with exponents other than 1. (Some may be written in more than one way.)

12. 121 _______________ 15. 49 _______________

13. 32 _______________ 16. 64 _______________

14. 256 _______________ 17. 243 _______________

Skills: 1.3

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Scientific NotationScientific notation expresses very large and very small numbers more conveniently than standard form.

In scientific notation, numbers are expressed as the product of a number, called the coefficient, that is greater than or equal to 1 but less than 10, and some power of 10.

101

0.1

102

0.01

103

0.001

10 4

0.0001

105

0.00001

and so on . . .

101

10

102

100

103

1,000

104

10,000

105

100,000

and so on . . .

Powers of 10

Positive Negative

Example

Write 2,520,000 in scientific notation.

For numbers greater than or equal to 10, move the decimal point to the left and use positive exponents.

2,520,000 2.52 • 106

Example

Write 0.0029 in scientific notation.

For decimal numbers less than 1, move the decimal point to the right and use negative exponents.

0.0029 2.9 • 103

TIP: To convert a number with positive exponents from scientific notation to

standard form, move the decimal point to the right. For example, 4.9 • 107

4.9000000 49,000,000. To convert a number with negative exponents from

scientific notation to standard form, move the decimal point to the left. For

example, 1.063 • 102 01.063 0.01063.

Skills: 1.4

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PracticeDirections: For Numbers 1 through 6, write each number in scientific notation.

1. 3,500,000,000 __________________ 4. 0.0000000024 __________________

2. 0.0067 __________________ 5. 49,000 __________________

3. 0.941 __________________ 6. 759,000,000 __________________

Directions: For Numbers 7 through 12, write each number in standard form.

7. 4.6 • 105 __________________ 10. 2.1 • 108 __________________

8. 7.308 • 105 __________________ 11. 7.035 • 106 __________________

9. 6.13 • 103 __________________ 12. 8.51 • 104 __________________

Directions: For Numbers 13 through 16, write the given number in scientific notation.

13. After five years, the company that made the first PC with color graphics was worth $500,000,000.

_____________________________

14. In 2004, there were about 6,300,000,000 people in the world.

_____________________________

15. The diameter of a proton is about 0.00000000000002 meters.

_____________________________

16. One inch is 0.0000157828 miles. _____________________________

Skills: 1.4

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Square RootsThe square root of a number x is represented by the symbol x . The is called

the radical sign and x is called the radicand. Finding the square root of a number

is the inverse of squaring, or raising a number to the 2nd power. Therefore, x r

if and only if r2 x. There are two roots of x, r and r, since both r2 x and (r)2

x. Most times, however, x represents the principal, or positive, square root of x.

Example

What is 81 ?

Since 92 81, 81 9.

Example

What is 121 ?

Since 112 121, 121 11.

Unless otherwise stated, it is assumed that the roots should be positive.

Example

What is 25 ? Find the positive and negative roots.

Since 52 25, 25 5.

Since (5)2 25, 25 5.

If x r where r is an integer, then x is called a perfect square. From the example above, 25 is a perfect square. The square root of a non-perfect square results in a non-terminating decimal.

Example

What is 56 ?

Since 72 49 and 82 64, 56 is between 7 and 8.

56 7.4833. . .

56 7.48

Skills: 1.5

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PracticeDirections: For Numbers 1 through 8, find the square root of each number. Round to the nearest tenth, if necessary.

1. 225 ______________________

2. 441 ______________________

3. 641 ______________________

4. 6,400 ______________________

5. 100 ______________________

6. 576 ______________________

7. 49 ______________________

8. 140 ______________________

9. What is the negative root of 36 ?

A. 8

B. 6

C. 6

D. 8

Directions: For Numbers 10 through 17, find the square root of each number. Round to the nearest tenth, if necessary.

10. 62 ______________________

11. 500 ______________________

12. 324 ______________________

13. 676 ______________________

14. 2,000 ______________________

15. 1,000 ______________________

16. 2,500 ______________________

17. 529 ______________________

18. What is the negative root of 169 ?

A. 13

B. 9

C. 9

D. 13

Skills: 1.5

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Absolute ValueThe absolute value of a real number (n) is the net distance from zero on a number line. It is shown by the symbol un u.

2 1 0 1 2

34

1 units34

1 units

The number 1 3

4 is 1 3

4 units from 0 on a number line, so u 1 3

4 u 1 3

4 . The number 1 3

4

is also 1 3

4 units from 0 on a number line, so u 1 3

4 u 1 3

4 .

Example

What is the absolute value of 0.546?

The number 0.546 is 0.546 units from 0 on a number line, so

u0.546u 0.546.

Example

What is the absolute value of 1

8 ?

The number 1

8 is 1

8 units from 0 on a number line, so u 1

8 u 1

8 .

PracticeDirections: For Numbers 1 through 8, find the absolute value of the given number.

1. u12u _______________

2. u18.5u _______________

3. u23.4u _______________

4. u 2

7 u ______________

5. u 3 4

9 u _______________

6. u 2 1

2 u _______________

7. u0.516u ______________

8. u 3

10

u ______________

Skills: 1.6

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Algebra I Practice

1. Which rational number will convert to a repeating decimal?

A. 1

5

B. 1

4

C. 1

3

D. 1

2

2. Which statement is false?

A. 25 is an irrational number.

B. All integers are rational numbers.

C. All counting numbers are integers.

D. All repeating decimals are rational numbers.

3. The Big Bang Theory proposes that the universe began with an enormous explosion 13 billion years ago. How is this number written in scientific notation?

A. 1.3 • 1010

B. 13 • 1010

C. 1.3 • 109

D. 13 • 109

4. Which expression does not equal 729?

A. 36

B. 94

C. 272

D. 7.29 • 102

5. Pauline’s family consists of her mom, dad, three sisters, and four brothers. What fraction of Pauline’s family is female?

A. 2

5

B. 4

9

C. 1

2

D. 5

9

6. What is the value of the expression below?

196

A. 14

B. 13

C. 12

D. 9

7. Lonnie has 100 CDs. If each CD has 10 songs, which expression shows the total number of songs Lonnie has on CD?

A. 102

B. 103

C. 1002

D. 10010

8. Which number is rational?

A. p

B. 10

C. 4. 3

D. 6.342894. . .

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Lesson 1: Algebra I Practice©

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u45u

A. 45

B. 9

C. 9

D. 45

10. Which rational number will convert to a repeating decimal?

A. 7

12

B. 1

2

C. 2

5

D. 4

19

11. Which is not a way to express 15,000?

A. u(15,000)u

B. 2,250,000

C. 23 • 3 • 54

D. 1.5 • 104

12. How is the following number written in scientific notation?

0.000002039

A. 2.039 • 105

B. 2.039 • 106

C. 2.039 • 105

D. 2.039 • 106


256

A. 18

B. 16

C. 14

D. 12

14. Which of the following is a finite set?

A. {1, 2, 3, 4, . . .}

B. {. . . , 1, 2, 3, 4}

C. {even numbers}

D. {1, 2, . . . , 15, 16}

15. Which of the following is an example of the empty set?

A. {integers less than 0}

B. {counting numbers less than 0}

C. {positive real numbers less than 1}

D. {days of the week that end in “day”}

16. What is the negative root of 300 ? Round to the nearest tenth.

A. 17.3

B. 13.7

C. 13.7

D. 17.3

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Unit 2 Algebra Concepts HS - Triumph Learning · High School Algebra I, Book . 1. Unit 1 . Number...

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