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Daily AffirmationsLet's stand for ourTodayisagoodday!

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Reviewp34ML

EO2Opp#1M/T,Sep2526

IntroductiontoSimilarity

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ParallelLinesCutbyaTransversal

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ParallelLinesCutbya

Transversal(foldable)

OYOp40#1,p43#9

Properties,Postulates,&Theorems

Inv121p30#13,EQ

Sub:Ms.Lawlor

CalculatingScaleFactors

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clarifyingquestions!

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ANSWERS

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EssentialQuestions

LanguageObjective

Activities#13,STMCYU,EQ

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Mathematicianswillbeableto......justifywrittenSTATEMENTSusingREASONSbasedondefinitions,postulates,andtheoremstoprovegeometricclaims.

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Youarelearning:Somemathematicalfactsabouthowtheanglesinthisdiagramarerelated.Someterminologyaboutwhattocallvariouscomponentsofthisdiagram.Howtoprovecertainstatementsrelatedtothisdiagram.

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ANSWERS

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Attachments

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EO2 Inequalities in 1 Variable Name __________________

Unit 2 Lesson 1 With calculator Date: _______________ Period ____

Warm-Up

Match the following number line graphs with the correct symbolic inequality solutions.

1) A)

2) B)

3) C)

4) D)

5) E)

6) F)

Reminders:

7) When you have an open dot (non-filled in circle) you are using the _____ or _____ sign.

8) When you have a closed dot (filled in circle) you are using the _____ or _____ sign.

9) When reading an inequality, like x > 2, it is explaining the answer to some rule of possible x-values that can be plugged in and solve to be true.

a. Therefore, some x-values that do NOT work in the example x > 2 are: __________________________

b. And some x-values that do work in the example x > 2 are: _______________________

c. Can x = 2 here? Why or why not?

Interval Notation: Mathematicians (like yourself) can used interval notation as a kind of shorthand to also describe the solutions like the ones above. Interval notation uses parentheses () and brackets [], like symbolic inequalities and closed and open circles on a number line. Examine the following rules and examples below.

Rules:

When using greater than (>) and less than ( 2

Number Line:

Interval Notation: (2, )

When thinking about Interval Notation, think of it as the range in which all the solutions falls. The parentheses before 2 is telling us that the solutions start as any number larger than 2. The infinity symbol,, is telling us that the solution goes on forever for any number greater than 2. ALL infinity symbols, - or , must have a parentheses next to it since we can never actually reach it.

Example 2: The 3 ways to represent solutions of all x-values less than or equal to -1 or all x-values greater than 2.

Symbolically:

Number Line:

Interval Notation: (, 1] (2, )

The symbol, , means the union of the two intervals. So, the numbers that are in one interval (negative infinity to -1 included) or the numbers in the other interval (any number greater than 2) are the solutions.

Practice writing interval notation from a number line graph: Write the following solutions in interval notation.

1) 2)

3) 4)

Practice writing interval notation from symbolic notation: Write the following solutions in interval notation.

6) 7)

8) 9)

10)

Practice writing solutions symbolically AND on a number line graph when given interval notation:

11) Solution: (, 5]Symbolically:

Number Line:

12) Solution: [, 1)Symbolically:

Number Line:

13) Solution: (, 2) [5, ) Symbolically:

Number Line:

Name: ___________________________ Date: _________________ Period: ____ Page 109

Unit 2 Lesson 1 Investigation 1 Getting the Picture Activities #1-2, STM, CYU, EQ

Essential Questions

Essential Question(s)

Language Objective(s): Mathematicians will be able to

On page 107 of your book:Unit 2: Inequalities and Linear Programing

For some people, athletes and astronauts in particular, selection of a good diet is a carefully planned scientific process. Each person wants maximum performance for minimum cost. The search for an optimum solution is usually constrained by available resources and outcome requirements.

Lesson 1: Inequalities in One Variable

Use numeric and graphic estimation methods and algebraic reasoning to solve problems that involve linear and quadratic inequalities in one variable.

Page 108: Lesson 1 Inequalities in One Variable.

In previous courses, you learned how to solve a variety of problems by representing and reasoning about them with algebraic equations and inequalities. For example, suppose that the plans for a fundraising raffle show that profit P will depend on ticket price x according to the function. A graph of profit as a function of ticket price is shown here.

Page 109: Think About This Situation: Questions important to the fundraising group can be answered by solving inequalities involving the profit function.

a) What would you learn from solutions of the following inequalities?

i.

ii.

iii.

iv.

b) How could you use the graph to estimate solutions of the inequalities in Part a?

c) In what ways could you record solutions of the inequalities in words, symbols, or diagrams?

Page 109-110: In this lesson, you will learn how to use graphical reasoning and algebraic methods to solve inequalities in one and two variables. You will also learn how to represent the solutions symbolically and graphically and how to interpret them in the contexts of the questions that they help to answer.

Investigation 1: Getting the Picture

You learned in earlier work with inequalities that solutions can be found by first solving related equations. For example, in the raffle fundraiser situation, the solutions of the equation

are approximately $1.23 and $5.44. The reasonableness of these solutions can be seen by scanning the graph of the profit function and the constant function y = 2,500. (Hint: Graph -2500+5000x-750x2 in Y1, and Graph 2500 in Y2 of your graphing calculator. Push 2nd, Push Trace, pick 5: Intersect to find the intersection points).

The solution of the inequality are at values of x between $1.23 and $5.44. Those solutions to the inequality can be represented using symbols or a number line graph.

How would you write the solutions to the inequality in interval notation?

Similarly, the solutions of the inequality are all the values of x that are either less than $1.23 or greater than $5.44. Those solutions can also be represented using symbols or a number line graph.

How would you write the solutions to the inequality in interval notation?

As you work on problems of this investigation, look for answers to these questions:

How can you solve inequalities in one variable?

How can you record the solutions in symbolic (inequality), graphic, and interval notation form?

#1) on page 110-111

The next graph shows the height of the main support cable on a suspension bridge. The function defining the curve is . Where x is horizontal distance (in feet) from the left end of the bridge and h(x) is the height (in feet) of the cable above the bridge surface.

a. Where is the bridge cable less than 40 feet above the bridge surface?

Write the equation or inequality that could be used to solve.

Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).

Express Solution Symbolically

Express Solution Graphically

Express Solution in Interval Notation

b. Where is the bridge cable at least 60 feet above the bridge surface?

Write the equation or inequality that could be used to solve.

Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).

Express Solution Symbolically

Express Solution Graphically

Express Solution in Interval Notation

c. How far is the cable above the bridge surface at a point 45 feet from the left end?

Write the equation or inequality that could be used to solve.

Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).

Express Solution Symbolically

Express Solution Graphically

Express Solution in Interval Notation

d. Where is the cable 80 feet above the bridge surface?

Write the equation or inequality that could be used to solve.

Mark/circle the part or parts of the graph that are solutions. (Use your graphing calculator to help you find more exact answers when writing solutions).

Express Solution Symbolically

Express Solution Graphically

Express Solution in Interval Notation

#2) on page 111

The graph below shows the height of a bungee jumpers head above the ground at various times during her ride on the elastic bungee cord. Suppose that h(t) gives height in feet as a function of time in seconds

For each Part a-d:

Write a question about the bungee jump that can be answered by the indicated mathematical operation.

Use the graph to estimate the answer.

Express your answer (where appropriate) with a number line graph and interval notation.

a) Evaluate h(2)

b) Solve h(t) = 10

c) Solve h(t) 10

d) Solve h(t) < 10.

Summarize the Math p111

a.

b.

c.

Check Your Understanding p112

a.

bi.

ii.

iii.

c.

Now answer the Essential Questions.

How can you solve inequalities in one variable?

How can you record the solutions in symbolic, graphic, and interval form?

SMART Notebook

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SMART Notebook

SMART Notebook

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