chapter 3 solving equations and inequalities - denton · pdf filechapter 3 solving equations...

55a

Mathematical OverviewIn this chapter, students use algebraic expressions as the building blocks for writing equations and inequalities. Arrow diagrams are introduced as a way to represent expressions and then later as a way to model the steps used when solving equations. The order of operations guidelines and the TABLE feature on a calculator, as well as the Properties of Equality and the Properties of Inequality, are all explored in the context of fi nding solutions for real-world problems. Students note that whereas a single number solution is graphed as a point on a number line, the graph of an inequality is an interval on the line. The concept of an inequality is extended to include compound inequalities. Students also transform statements into inequalities and graph the solution sets on number lines.

Lesson Summaries

Lesson 3.1 Activity: Tree Growth

In this Activity, students measure the heights of drawings of a tree in six different years of growth. They then use a proportion to compute the actual height of the tree for each of those six years. Using their tables of data, students write expressions that represent the height of the tree for any year of growth and use them to make predictions about the height of the tree. Students are introduced to arrow diagrams and use them to represent/evaluate expressions. They also review the order of operations guidelines to help them evaluate expressions that contain many mathematical operations.

Lesson 3.2 Investigation: From Expression to Equation

In this Investigation, students build on their work with simple expressions from Lesson 3.1 and construct equations that model a second stage of tree growth. They learn that an equation is a statement that two expressions are equal. Students become aware that solving some problems may require different strategies than they normally use. The process of using equations to model real-world problems that are unlike problems students have solved in the past is the main focus of this lesson.

Lesson 3.3 Using Arrow Diagrams and Tables to Solve Equations

In this lesson, the use of arrow diagrams is expanded to represent two-step equations. Students solve equations by using a completed diagram to tell them which operations to perform and in which order. Students also learn to use the calculator TABLE feature to solve equations. The methods used in this lesson illustrate important algebraic properties, the Properties of Equality, which are used to solve many types of equations.

Lesson 3.4 R.A.P.

In this lesson, students Review And Practice solving problems that require the use of skills and concepts taught in previous math levels. The skills reviewed in this lesson are skills that are needed as a basis for solving problems throughout this course.

Lesson 3.5 Investigation: Solving Multi-Step Equations and 3.6 Investigation: Solving Inequalities

In Lesson 3.5, students explore using expressions and equations to model, evaluate, and solve real-world problems. Emphasis is placed on fi nding and combining like terms and isolating the variable. In Lesson 3.6 students apply this same knowledge to solving and graphing inequalities. The main purpose of the Investigation in this lesson is to provide students with a review of the Properties of Inequality.

Lesson 3.7 Compound Inequalities

In this lesson, students use what they learned inLessons 3.5 and 3.6 to write, solve, and graph compound inequalities. Students learn that the intersection of two graphs consists of all points that are common to both graphs, and the union of two graphs consists of all the points in one or the other or in both. Absolute value inequalities are included in some of the practice exercises.

Chapter 3Solving Equations and Inequalities

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55b

Lesson/Objectives Materials

Chapter 3 Opener: How Can We Care for Our Forests?• recognize that equations can be used to help maintain and

protect our country’s forests.

3.1 Activity: Tree Growth • use arrow diagrams to represent expressions.• evaluate expressions.• write expressions to model real-world situations.

Per pair:• inch ruler

Optional:• TRM table shell for Question 2.• TRM table shell for Exercises 6 and 9.

3.2 Investigation: From Expression to Equation• write an equation that models a real-world situation.

Optional:• TRM table shells for Exercises 1 and 18.• grid paper

3.3 Using Arrow Diagrams and Tables to Solve Equations• use arrow diagrams to solve two-step equations.• use tables to solve equations.

3.4 R.A.P.• solve problems that require previously learned concepts and

skills.

3.5 Investigation: Solving Multi-Step Equations• combine like terms in an expression.• solve equations.• use equations to solve real-world problems.

Optional:• algebra tiles

3.6 Investigation: Solving Inequalities• solve inequalities.• graph inequalities.

3.7 Compound Inequalities• write compound inequalities.• graph compound inequalities on a number line.• solve compound inequalities.

Lesson Guide

Supplement SupportSee the Book Companion Website at www.highschool.bfwpub.com/ModelingwithMathematics and the Teacher’s Resource Materials (TRM)for additional resources.

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9 Day 10

Basic p. 56, 3.1 3.2 3.3 3.4 3.5 3.5 3.6 3.7 project review

Standard p. 56, 3.1 3.2 3.3 3.4 3.5 3.5 3.6 3.7 project review

Block p. 56, 3.1, 3.2

3.2, 3.3 3.3, 3.4 3.5 3.6, 3.7 3.7, project

review

Pacing Guide

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CHAPTER 3

Solving Equations and

Inequalities

55

Chapter 2 Direct VariationMathematical

Model

CONTENTSHow Is Mathematics Related to

Bungee Jumping? 35

Lesson 2.1 Activity: Bungee Jumping 37

Lesson 2.2 Investigation: Proportional Relationships 39

Lesson 2.3 Direct Variation Functions 43

Lesson 2.4 RAP 47

Lesson 2.5 Slope 53

Modeling Project: It’s Only Water Weight 57

Chapter Review 60

Extension: Inverse Variation 67

CHAPTER

1Solving Equations and Inequalities

CONTENTSChapter Opener:

How Can We Care for Our Forests? 56

Lesson 3.1ACTIVITY: Tree Growth 57

Lesson 3.2 INVESTIGATION: From Expression to Equation 61

Lesson 3.3 Using Arrow Diagrams and Tables to Solve Equations 67

Lesson 3.4 R.A.P. 73

Lesson 3.5 INVESTIGATION:Solving Multi-Step Equations 75

Lesson 3.6 INVESTIGATION: Solving Inequalities 82

Lesson 3.7 Compound Inequalities 87

Modeling Project: Algebra’s Next Top Model 91

Chapter Review 92

CHAPTER

3

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C h a p t e r 3 S O LV I N G E Q UAT I O N S A N D I N E Q UA L I T I E S56

One-third of the United States is covered with trees. In some

states, the majority of land consists of forests. For example, in

Virginia, 62% or 16 million acres is forested. Altogether, the

United States contains about 230 billion trees or about 1,000 trees

per person.

We often hear about problems related to the loss of forests, such

as rain forests, throughout the world. In the United States, efforts

are made to maintain a balance between the removal of trees for

commercial and other purposes, and the natural or controlled

regrowth of trees. One company that supplies wood for paper

products, J. D. Irving Limited, has planted over 800 million

trees over the past fi fty years. A nonprofi t organization

called Sustainable Forestry Initiative, Inc., certifi es forest

lands in the United States and Canada to help ensure that

responsible forest management is practiced.

How tall will a tree be after ten years of growth? How

long will it take a tree to reach 50 feet in height?

Mathematical equations can be used to answer

questions like these.

How Can We Care for Our Forests?


56

Lesson Objective • recognize that equations can be

used to help maintain and protect

our country’s forests.

Vocabularynone

DescriptionThis reading helps students see that sustainable forest management involves planting large numbers of trees, and that mathematics can be used to predict the rate at which they will grow.

After students have read the Chapter Opener, lead a class discussion on the importance of forests to their stateand/or community. Ask students toname products that they use that come from trees. Examples could include furniture, paper, and books.

TEACHING TIP

CHAPTER 3 OPENER

5e Engage

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57T R E E G R O W T H Le s s o n 3 . 1

ACTIVITY: Tree GrowthLesson 3.1

Algebraic expressions are some of the basic building blocks of mathematics. They allow us to describe real-world situations in such a way that the power of mathematics can be applied to them. In this Activity, you will write expressions to describe the growth of trees.

New trees arise naturally in a forest, from seeds that come from existing

trees. In the United States, over 2 billion trees are planted each year by

the forest products industry, tree farmers, and government agencies.

Many of these plantings consist of small seedlings or saplings that are

already more than a foot tall.

The fi gure below shows several stages in the growth of a tree. The fi rst

drawing on the left represents a newly planted sapling. This sequence of

drawings shows successive years in the tree’s growth.

Year 0 Year 1 Year 2 Year 3 Year 4 Year 5

1. The scale of the drawings is 1 __ 4 inch � 1 foot. Measure the height of

the drawing of the newly planted sapling on the left to the nearest

eighth of an inch. Then use a proportion to fi nd the actual height

of the tree.

2. Complete the table to the left by measuring each of the drawings

and using a proportion to fi nd the actual height of each tree.

3. Describe any pattern you see in the third column of your table.

4. If the pattern in the table continues, what do you expect the height

of the tree to be after 6 years? After 10 years?

5. Let the variable n represent any number of years of tree growth.

Assuming the pattern continues, what do you expect the height of

the tree to be after n years?

Year Height on Drawing

Actual Height

0

1

2

3

4

5


3. Sample answer: The heights increase by a constant amount of

1 1 __ 2 feet each year.

4. 11 feet; 17 feet5. Sample answer: 2 � 1.5n feet

Lesson 3.1 Activity Answers1. 1 __

2 inch; 2 feet

2. Sample answer:

YearHeight on Drawing

Actual Height

0 1 _ 2 inch 2 feet

1 7 _ 8 inch 3 1 _

2 feet

2 1 1 _ 4 inches 5 feet

3 1 5 _ 8 inches 6 1 _

2 feet

4 2 inches 8 feet

5 2 3 _ 8 inches 9 1 _

2 feet

Lesson Objectives• use arrow diagrams to represent

expressions.

• evaluate expressions.

• write expressions to model real-

world situations.

Vocabulary• algebraic expression

• coeffi cient

• evaluate an expression

• order of operations

Materials List• rulers

DescriptionPreparation:

Have students work in pairs. Provide each pair with a ruler.

During the Activity:

Students should measure the heights of the trees in the drawing to the nearest eighth of an inch and then

use the 1 __ 4 inch � 1 foot scale to

compute the corresponding tree heights.

Closing the Activity:

Make sure students see that the construction of an arrow diagram is based on the Order of Operations Guidelines. Write an expression, such

as x � 4 _____ 7 , on the board. Ask students

to represent the expression with an arrow diagram. Then have them evaluate the expression for different values of x.

57

LESSON 3.15e Engage

Ask students how they converted measurements from the drawing to actual tree heights.

TEACHING TIP



Your answer to Question 5 is an example of an algebraic expression.

An algebraic expression contains one or more variables. It can also

include numbers, grouping symbols, and operation symbols.

Other examples of algebraic expressions include the following:

25w 4t � 3 5(2x � 7y)

6. A number that is multiplied by a variable in an expression is called

the coeffi cient of that variable. Look at the expression you wrote

for Question 5.

a. What is the coeffi cient of n? What does it represent?

b. What is the constant term? What does it represent?

7. An arrow diagram is one way to represent an expression. In an

arrow diagram, arrows are used to show the order in which

operations are performed. For example, this arrow diagram

represents the expression 5x � 3.

Multiply by 5 Subtract 3

5xx 5x – 3

Draw an arrow diagram to represent the expression in your answer

to Question 5.

8. To evaluate, or fi nd the value of, an algebraic expression,

substitute a numerical value for the variable. Then simplify the

resulting expression. Predict the height of the tree after 15 years by

evaluating your expression for n � 15. Use your arrow diagram as

a guide.

ORDER OF OPERATIONS

When you evaluate expressions that contain many mathematical

operations, guidelines must be followed. These guidelines tell you the

order in which the operations must be performed.

Order of Operations Guidelines

To evaluate an expression that contains more than one operation,

the following order is used:

1. Perform the operations within any grouping symbols, which

include parentheses, brackets, and fraction bars.

2. Evaluate powers.

3. Perform multiplications and divisions in order from left to

right.

4. Perform additions and subtractions in order from left to right.

Comap2e_Modeling_Ch03.indd 58 03/02/12 11:55 AM

6a. 1.5; It is the yearly rate of tree growth.

6b. 2; It represents the height of the tree when planted, 2 feet.

7.

Multiply by 1.5 Add 2

1.5n 1.5n + 2n

8. 2 � 1.5(15) � 24.5 feet

58

LESSON 3.1

Point out to students that the Order of Operations Guidelines for algebraic expressions are the same as those for arithmetic.

TEACHING TIP

Remind students that the fraction bar is a grouping symbol, as are parentheses, brackets, and braces.

TEACHING TIP


59T R E E G R O W T H Le s s o n 3 . 1

Evaluate the expression (a � 7) � 4(3 � a) � a3 for a � 2.

Solution:Substitute 2 for a. (a � 7) � 4(3 � a) � a3 � (2 � 7) � 4(3 � 2) � 23

Perform operations within parentheses. � 9 � 4(1) � 23

Evaluate powers. � 9 � 4(1) � 8

Multiply and divide from left to right. � 9 � 4 � 8

Add and subtract from left to right. � 5

Practice for Lesson 3.1

For Exercises 1–2, draw an arrow diagram to represent the

expression. Then evaluate the expression for m � 21.

1. m __ 3

� 12 2. m � 12 _______

3

For Exercises 3–5, evaluate the expression.

3. 4(b � 2) � 3b � b2 for b � 5

4. 3k � k 4 � 5(k � 1) � k for k � –1

5. 5 � 2y � 1

______ 3

� y3 � 10y for y � 4

For Exercises 6–8, use the following information.

A landscaper plants a tree that is 5 feet tall. The tree grows 2 feet taller

each year.

6. Complete the table to show the height of the tree during the fi rst

6 years of growth.

Years of Growth Height of Tree (ft)

0

1

2

3

4

5

6


ADDITIONAL EXAMPLE Evaluate the expression

4t � 8 � 2t 2 � t � 1 _____ 3 for t � 5. �20

Practice for Lesson 3.1 Answers1. 21 _

3 � 12 � 19

Divide by 3 Add 12

m m3

+ 12m3

2. 21 � 12 _______ 3 � 11

Add 12 Divide by 3

m m + 123

m + 12

3. 24. �35. 26

59

LESSON 3.1

For some students, it may be helpful to review some of the properties of real numbers such as the Commutative, Associative, and Distributive Properties. (See Appendix J.)

TEACHING TIP

Exercises 1–2 ask students to draw arrow diagrams and use them to evaluate simple expressions. This is an important preparation for using arrow diagrams to solve two-step equations in Lesson 3.3. While arrow diagrams are of limited utility in representing more involved expressions and equations, they can be very helpful to students who have diffi culty with order of operations and the fundamentals of solving equations.

TEACHING TIP

6. Height of Tree (ft)

5

7

9

11

13

15

17



7. Write an expression for the height of the tree after n years.

8. Use your expression to fi nd the height of the tree after 8 years.


An ocean beach extends out into the ocean a length of 50 feet, but

erosion causes it to lose 2 feet of length a year.

Years of Erosion Length of Beach (ft)

0

1

2

3

4

5

9. Complete the table to show the length of the

beach during the fi rst 5 years of erosion.

10. Explain the process you used to fi nd the length of the beach after

3 years of erosion.

11. Write an equation that models the length L of the beach after t years.

12. Find the length of the beach after 15 years if erosion continues at

the same rate.


The expression 12,500 � 700n describes the population of a town

n years after the year 2000.

13. What was the town’s population in 2010?

14. What does 12,500 represent in the expression?

15. What does 700 represent in the expression?


A wrestler is trying to lose a little weight in preparation for wrestling

season. The expression 152 � 3t gives his weight (in pounds) during

October, where t represents time in weeks.




7. 5 � 2n or 2n � 58. 5 � 2(8) � 21 feet9. Length of Beach (ft)

50

48

46

44

42

40

10. Sample answer: Start with a length of 50 feet, then subtract2 feet three times.

11. L � 50 � 2t12. 50 � 2(15) � 20 ft13. 19,50014. It represents the population of

the town in 2000.15. It represents the yearly growth

rate of the town.16. It represents his weight at the

beginning of October.17. It represents the number of

pounds he loses each week.

60

LESSON 3.1


61F R O M E X P R E S S I O N TO E Q UAT I O N Le s s o n 3 . 2

INVESTIGATION: From Expression to Equation

Lesson 3.2

Expressions can be used to form equations, which are useful mathematical representations. They allow people to predict what will happen in a certain situation or how to achieve a particular objective. In this lesson, you will use expressions as building blocks for writing equations.

As you saw in Lesson 3.1, the numerical value of an expression depends

on the value or values given to the variables in the expression. If two

expressions are known to have the same value, they can be written as

part of an equation.

An equation is a statement that two expressions are equal.

You found in Lesson 3.1 that the expression 2 � 1.5n represents the

height of the tree after n years. The equation

height � 2 � 1.5n

models the relationship between age and height for all possible ages

and heights of the tree. You could also write this as h � 2 � 1.5n.

1. Coast redwood trees include the tallest individual

organisms that ever existed on Earth. For the fi rst

20 years of its life, a coast redwood might grow

3 feet each year. Write an equation that models

the height h of a coast redwood at age n years

after being grown from a seed.

2. If grown from a sapling that is planted when it is

2 feet tall, what would be an equation for a coast

redwood’s height at any age?

After 20 years, a redwood’s growth rate typically

slows. Suppose that a particular tree begins to grow at

only 1.4 feet a year after year 20. Finding an equation

that models the tree’s height for ages beyond 20 years

requires careful thought.

3. You can often begin modeling a situation like this by

using words to express the total amount described in

the problem. The total height of the coast redwood

includes the growth during both parts of the

growing cycle. Begin an equation with “h �” and use words to describe the two portions of the tree’s growth for any age

greater than 20 years.

h � �


Lesson 3.2 Investigation Answers1. h � 3n2. h � 3n � 23. Sample answer: h � (growth

during fi rst 20 years) � (growth after 20 years)

Lesson Objective• write an equation that models a

real-world situation.

Vocabulary• equation

Materials Listnone

DescriptionThis lesson is designed as a small group investigation (2– 4 students). In Questions 1–6 students build on their work with simple expressions from Lesson 3.1 and construct equations that model a second stage of tree growth.

Question 7 asks students to use their equation to predict tree growth.

Wrapping Up the Investigation:

Ask students to share their models with the class to see if there are any differences. If any groups have written their equation in the simplifi ed form h � 34 � 1.4n, point out that this form would be diffi cult to fi nd directly. This should help convince students of the value of the incremental model-building process in the Investigation.

61

LESSON 3.2

The growth pattern discussed in Question 3 is a simplifi cation. Actual tree growth rates would not change so abruptly.

CONNECTION

Beginning with Question 3, caution students to resist the urge to jump immediately to a symbolic equation. Point out that the process of building a model is the objective here.

TEACHING TIP

5e Explore



4. Your answer to Question 2 about the growth of a tree from a

sapling provides a way to calculate the tree’s height after 20 years.

Find that height and use it to simplify part of your word equation

from Question 3.

5. Another way to describe the growth after 20 years might be “After

20 years, the tree grows 1.4 feet per year.” Revise your equation

to take this fact into account. Include the number 1.4, but use

words to express the number of years after 20 that your expression

will represent.

6. Express the number of years greater than 20 in terms of the total

age n of the tree. Then write a completely symbolic equation that

models the height of the tree for any age n greater than 20 years.

7. Use your equation to fi nd the height of the tree after 45 years.

STRATEGIES FOR CONSTRUCTING MATHEMATICAL MODELS

Some situations are simple enough that they can be easily modeled by

comparing them to known models. Many real-world applications involve

an initial value of some quantity that then grows at a constant rate, just

like the growth of the redwood during its fi rst 20 years. For example,

a baseball weighs about one-third of a pound. So, a 2-pound box

containing n baseballs would have a total weight of w � 2 � 1 __ 3 n pounds.

But other problems may be unique, or at least unlike situations more

familiar to a modeler. The growth of the redwood tree after 20 years

is an example. In such cases some of the following strategies may

be useful.

• Write a simple “word equation.”

Make sure your words state the big idea of the problem. Then gradually

add smaller pieces of the problem until you have a symbolic equation.

• Make a table.

Your table should contain specifi c values for the quantities that you

are investigating. Once your table is complete, look for a pattern.

• Draw a sketch or diagram.

This strategy is often used in situations that include aspects of

geometry.

• Check your model.

When you have found a model you think is a good one, substitute

values for any variables in the model to see if they predict

reasonable results.


4. 3(20) � 2 � 62; h � 62 � (growth after 20 years)

5. Sample answer: h � 62 � 1.4 (number of years greater than 20)

6. h � 62 � 1.4(n � 20)7. 97 feet

62

LESSON 3.2

In Question 6, if students have trouble writing a symbolic equation, suggest that they build a table to look for a pattern in heights for individual years beyond 20.

TEACHING TIP

Discuss the list of strategies for model construction with students. Point out that Questions 3–6 of the Investigation provide an example of the fi rst strategy listed. Remind students that the second strategy listed was used in the Activity in Lesson 3.1.

TEACHING TIP



A girl is 36 inches tall at age 3. She grows 2 1 __ 2 inches each year up to

age 8. Find an equation that models her height h at any age A from

3 to 8 years.

Solution:

Make a table to show the girl’s height

from ages 3 to 8 years.

The pattern in the table shows that

height increases by 2 1 __ 2 inches for

each year beyond 3 years of age.

Write a word equation: “Her height at

any age is 36 inches plus 2 1 __ 2 inches

for each year of age beyond 3.”

The word is can be replaced by an

equal sign. The number of years

beyond 3 can be written as the expression (A � 3). So, the relationship

between age and height can be modeled by the equation

h � 36 � 2 1

__ 2 (A � 3)

Check the model: At age 7, the model predicts a height

h � 36 � 2 1 __ 2 (7 � 3) or 46 inches.

This checks with the corresponding value in the table.

Age (years) Height (inches)

3 36

4 38 1 __ 2

5 41

6 43 1 __ 2

7 46

8 48 1 __ 2


Time (years) Height (feet)

0

1

2

3

4

5


A tree is planted as a 3-foot sapling and grows 1 3 __ 4 feet per year.

1. Complete the table to show the tree’s growth for the fi rst 5 years.

2. Identify the independent and dependent variables. Explain

your choices.

3. Make a scatter plot of the data in your table. Does it make

sense to draw a line through the plot? Explain. If it does, fi nd

the slope of the line.

4. Is this a direct variation function? Explain.


ADDITIONAL EXAMPLEA giant sequoia tree has an initial diameter of 4.320 meters and it grows 0.003 meters in diameter each year thereafter.

a. Find an equation that models the diameter d of the tree after t years.

d � 4.320 � 0.003t

b. Find the diameter of the tree after 100 years.

4.620 m

Practice for Lesson 3.2 Answers1. Height (feet)

3

4 3 _ 4

6 1 _ 2

8 1 _ 4

10

11 3 _ 4

2. Sample answer: Time (years) is the independent variable and height (feet) is the dependent variable, because the height of the tree depends on how much time it has been growing.

3. Sample answer: It makes sense to draw a line, because the rate of change of height should remain constant at all times. The slope is 1 3 __

4 feet/year.

Hei

gh

t (fe

et)

Time (years)

6

8

10

12

14

4

2

0 1 2 3 4 5

4. No, the ratios of time to height are not constant and the graph does not pass through the origin.

63

LESSON 3.2



5. Complete the following sentence to describe the way the

tree’s height changes: “Its height at any time is 3 feet plus

_____________________.”

6. Write an equation that models the height h of the tree after t years.

7. Check your model against the values in your table.

8. Find the height of the tree after 10 years if the same rate of growth

continues.


The table shows the amount of snow that accumulated on the ground

during a snowstorm.

Time(hours)

Snow Height(inches)

0 0

1 2.3

2 4.6

3 6.9

4 9.2

9. Identify the independent and dependent variables. Explain your

choices.

10. Make a scatter plot of the data. Does it make sense to draw a line

through the plot? Explain. If it does, fi nd the slope of the line.

11. Is this a direct variation function? Explain.

12. Write an equation that models the height h of the snow after

t hours.

13. Find the height of the snow after 8 hours if the same rate of

snowfall continues.

14. DVDs can be bought from an Internet retailer for $6.99 each. The

shipping cost for the entire order is $12, no matter how many

DVDs are ordered.

a. Write an equation that models the total cost C of an order

of n DVDs.

b. Find the total cost of an order of 8 DVDs.

15. A salesperson makes a salary of $800 a week plus a commission of

6 percent on the value of all goods sold.

a. Write an equation that models the salesperson’s total income I for a week in which her sales amount is s (in dollars).

b. Find the salesperson’s total income if she sells $2,000 worth of

goods.


5. Sample answer: 1 3 __ 4 inches for

every year of growth6. h � 3 � 1.75t or h � 1.75t � 37. Sample answer: After 5 years, the

model gives h � 3 � 1.75(5) �11.75, which agrees with the table.

8. h � 3 � 1.75(10) � 20.5 ft9. Sample answer: Time (hours)

is the independent variable and snow height (inches) is the dependent variable, because the height of the snow depends on how much time it has been snowing.

10. Sample answer: Yes, if snow continues to fall, it would gradually build up. The slope is 2.3 inches per hour.

Sno

w H

eig

ht (

inch

es)

Time (hours)

3

4

5

6

7

8

9

10

2

1

0 1 2 3 4 5

11. Sample answer: Yes, the graph is a straight line that passes through the origin. The ratios of height to time are constant.

12. h � 2.3t13. h � 2.3(8) � 18.4 in.14a. C � 6.99n � 1214b. $67.9215a. I � 800 � 0.06s15b. $920

64

LESSON 3.2



16. In one airline’s frequent-fl yer program, a passenger receives two

credits for each round-trip fl ight. Also, 0.5 credits are awarded for

each approved car rental.

a. Write an equation that models the total number of credits C

received if the number of round-trip fl ights is f and the number

of car rentals is r.

b. Find the number of credits received if a person made 5 round-

trip fl ights and rented a car twice.

17. The assembly shop of a custom furniture manufacturer has just

produced a set of 8 chairs. Each chair must be sanded 3 times.

Then 2 coats of fi nish must be applied.

a. Write an equation that models the total time t needed to

complete a set of 8 chairs if it takes s minutes to sand a chair

one time and f minutes to apply a coat of fi nish.

b. Find the time it takes to complete a set of chairs if it takes

35 minutes to sand a single chair one time and 25 minutes to

apply one coat of the fi nish.


In one state, for speeding on a 65-mile-per-hour highway a person may

be fi ned $50 as well as an additional $10 for each mile per hour over the

speed limit.

18. Complete the table to show the total fi ne F for various speeds s.

Speed (mph) Fine ($)

65 0

66

67

68

69

70

19. Identify the independent and dependent variables. Explain your

choices.

20. Complete the following sentence to describe the way the fi ne depends

on speed: “The total fi ne is $50 plus _____________________.”

21. Write an equation that models the total fi ne F (in dollars) for any

speed s that is at least 65 mph.

22. Find the total fi ne if a person is caught going 78 miles per hour.


16a. C � 2f � 0.5r16b. 11 credits17a. t � 8(3s � 2f )17b. 1,240 minutes18.

Fine ($)

0

60

70

80

90

100

19. Sample answer: Speed (mph) is the independent variable and the fi ne ($) is the dependent variable, because the amount of the fi ne depends on how fast a person is traveling.

20. Sample answer: $10 for every mile per hour of speed above65 mph

21. F � 50 � 10(s � 65)22. F � 50 � 10(78 � 65) � $180

65

LESSON 3.2

In Exercise 21, students may be tempted to write “10(miles per hour over 65)” and then have trouble seeing where to go from there. Suggest they pick a speed that is over 65 mph and compute the diff erence to see that (s � 65) is the appropriate expression.

COMMON ERROR



23. Suppose that your scores on the fi rst four tests in a course are 84, 72,

65, and 90. There is one test remaining, and all tests have equal weight.

a. Write an equation that relates your score T on the fi nal test to

your fi nal average A for all 5 tests.

b. Calculate your average if the score on your fi nal test is 94.

24. One cell-phone plan costs $39.99 a month for 450 minutes

of talking and unlimited texting. Each additional minute over

450 costs 45 cents.

a. Write an equation that models the relationship between

the total cost C of the plan and the number of minutes

m talked during a month in which more than 450 minutes

are used.

b. Another plan has the same features and costs. But it also includes

a $9.99 charge for up to 25 megabytes of data transfer. Each

additional megabyte of data costs 20 cents. Revise your answer

to part (a) to refl ect the inclusion of the data feature, using d to

represent the number of megabytes of data transferred. Assume

that at least 25 megabytes of data are transfered.

25. The fi gure below shows a window and its frame.

w

h

a. Write an equation that relates the height h and width w of

the inside dimensions of the window to the total length L of

framing material needed for 20 identical windows in a house.

(Assume that the frame is 1 1 __ 2 inches wide.)

b. How much framing material is needed if each window is 2 feet

wide and 3 feet high?


23a. A � 84 � 72 � 65 � 90 � T

__ 5

23b. 8124a. C � 39.99 � 0.45(m � 450)24b. C � 39.99 � 0.45(m � 450) �

9.99 � 0.20(d � 25)25a. L � 20[2h � 2(w � 3)] or

L � 20[2(h � 3) � 2w]25b. 2,520 in. or 210 ft

66

LESSON 3.2


67U S I N G A R R O W D I AG R A M S A N D TA B L E S TO S O LV E E Q UAT I O N S Le s s o n 3 . 3

Using Arrow Diagrams and Tables to Solve Equations

Lesson 3.3

Once a situation has been modeled by an equation, the power of mathematics can be used to analyze the problem. In this lesson, you will examine some basic techniques for solving equations.

INVERSE OPERATIONS AND ARROW DIAGRAMS

In the last lesson, you found that the equation

h � 3n could be used to model the growth in height

of a coast redwood grown from a seed. If you want

to fi nd how many years it will take for a tree grown

from seed to reach a height of 40 feet, you can use the

equation 3n � 40.

This simple equation can be modeled with an arrow diagram.

Multiply by 3

40n

You can add a reverse arrow to this diagram to show how to solve the

equation for n.

Divide by 3

Multiply by 3

40n

The reversed lower arrow indicates that division by 3 will undo the

multiplication represented by the upper arrow.

Then, to solve the equation 3n � 40 for n, use division (the inverse of

multiplication) to undo the operation on the variable.

3n ___ 3

� 40

___ 3

n � 40

___ 3 or 13

1 __

3

So, it will take 13 1 __ 3 years for the tree to grow to 40 feet.

RecallInverse operations are operations that undo each other. Multiplication and division are inverse operations. Addition and subtraction are also inverse operations.


Lesson Objectives• use arrow diagrams to solve two-

step equations.

• use tables to solve equations.

Vocabulary• equivalent equations

• inverse operations

• Properties of Equality

DescriptionThis lesson introduces the use of arrow diagrams and calculator tables as aids to solving two-step equations. Arrow diagrams are especially helpful for students who have diffi culty deciding which operations to use at any stage of solving an equation.

67

LESSON 3.3

If students need to review solving simple one- and two-step equations, take time to review Appendix L.

TEACHING TIP

5e Explain



TWOSTEP EQUATIONS

The equation h � 2 � 1.5n gives the height of a particular tree grown

from a 2-foot sapling. An arrow diagram can be drawn to represent

this equation.


1.5nn h

To fi nd how many years it will take for a 2-foot sapling to grow to a

height of 35 feet, rewrite the equation as 1.5n � 2 � 35. Then replace h

with 35 in the arrow diagram.


1.5nn 35

Reverse each arrow by using the corresponding inverse operation.

Multiply by 1.5

Divide by 1.5

Add 2

Subtract 2

1.5nn 35

The completed diagram shows that the equation can be solved for n by

fi rst subtracting 2 from 35 and then dividing the result by 1.5.

Arrow diagrams are helpful in solving equations because they tell you

which operations to perform, and in what order.

Original equation 1.5n � 2 � 35

Subtract 2. 1.5n � 2 � 2 � 35 � 2

Simplify. 1.5n � 33

Divide by 1.5. 1.5n ____ 1.5

� 33

___ 1.5

Simplify. n � 22

Check: 1.5(22) � 2 � 35

33 � 2 � 35

35 � 35

?

?


68

LESSON 3.3



a. Draw an arrow diagram that models the solution of the

equation t __ 5

� 1.2 � 3.

b. Solve the equation for t.

Solution:a. Divide by 5

Multiply by 5

Subtract 1.2

Add 1.2

t 3t5

b.

Original equation t __ 5 � 1.2 � 3

Add 1.2. t __ 5 � 1.2 � 1.2 � 3 � 1.2

Simplify. t __ 5 � 4.2

Multiply by 5. 5 � t __ 5

� � 5(4.2)

Simplify. t � 21

Check: 21

___ 5 � 1.2 � 3

4.2 � 1.2 � 3

3 � 3

?

?

USING A TABLE TO SOLVE AN EQUATION

Equations can often be solved by examining a table of values. Some

calculators have a TABLE feature that simplifi es this approach.


ADDITIONAL EXAMPLE a. Draw an arrow diagram that

models the solution of the

equation a � 2.6 ______ 3 � 1.8.

Subtract 2.6

Add 2.6

Divide by 3

Multiply by 3

a – 2.6a 1.8

b. Solve the equation for a. a � 8

69

LESSON 3.3



Use a calculator table to solve the equation t __ 5

� 1.2 � 3 for t.

Solution:Enter the expression

t __ 5 � 1.2 on the function screen Y� of a calculator.

On the TABLE SETUP screen [TBLSET], choose a starting value

(TblStart) and the increment (ΔTbl) for the variable.

Examine the [TABLE] screen to see if the expression equals 3 for

any of the input values that fi rst appear.

If not, scroll up or down until the value in the Y1 column equals 3.

The solution is t � 21.

NoteSometimes you may not be able to fi nd the exact number you are looking for in the Y1 column. In such cases, you may have to refi ne your table. (See Exercise 4, page 71.)


ADDITIONAL EXAMPLE Use a calculator table to solve

a � 2.6 ______ 3 � 1.8.

4b. Sample answer: The value of h is 60 for some value of n between 22 and 23.

70

LESSON 3.3

On many calculators, the only variables allowed are X and Y. Remind students that they should interpret calculator output in the context of the given situation.

TEACHING TIP

4c. The value of h is 60 when n � 22.8 years.



PROPERTIES OF EQUALITY

The methods used in this lesson illustrate important algebraic

properties that are used to solve many types of equations. All of the

steps in solving an equation involve changing it to an equivalent

equation. Equations are equivalent if they have the same solution

(or solutions).

The Properties of Equality state that an equation can be changed

into an equivalent equation by

• exchanging the expressions on the two sides of the equation;

• adding the same number to each side of the equation;

• subtracting the same number from each side of the equation;

• multiplying each side of the equation by the same non-zero

number;

• dividing each side of the equation by the same non-zero number.


For Exercises 1–3, complete Parts (a), (b), and (c).

a. Draw an arrow diagram that models the solution of the

equation.

b. Use a calculator table to solve the equation for the given

variable.

c. Use the Properties of Equality to solve the equation

algebraically.

1. Solve 6b � 5 � 19 for b.

2. Solve 3(x � 10) � 6 for x.

3. Solve d � 10

______ 5 � 7 for d.

4. The equation h � 3 � 2.5n models the height h (in feet) of a tree

after n years. Suppose that you want to use a table to determine

how many years it will take the tree to grow to a height of 60 feet.

a. Enter the expression 3 � 2.5n on the function screen of a

calculator.

b. Create a table and scroll to fi nd where the value of h is

approximately 60. Describe what you fi nd.

c. To search for a more accurate answer, go to the TABLE SETUP

screen. Enter a starting value of 22 and an increment of 0.1.

Then return to the table and search for an h-value of 60.

Describe what you fi nd.


3c. Original equation d � 10 ______ 5 � 7

Multiply both sides by 5. d � 10 � 35 Add 10 to both sides. d � 45

4a.

4b.–4c. Answers can be found on page 70.

3a.

Subtract 10

Add 10

Divide by 5

Multiply by 5

d – 10d 7

3b.

Practice for Lesson 3.3 Answers1a.

Multiply by 6

Divide by 6

Subtract 5

Add 5

6bb 19

1b.

1c. Original equation 6b � 5 � 19 Add 5 to both sides. 6b � 24 Divide both sides by 6. b � 4

2a. Add 10

Subtract 10

Multiply by 3

Divide by 3

x + 10x 6

2b.

2c. Original equation 3(x � 10) � 6 Divide both sides by 3. x � 10 � 2 Subtract 10 from both sides. x � �8

71

LESSON 3.3

Exercise 2 If students have trouble fi nding negative solutions, point out that they may sometimes have to scroll upward on their table screens if they start with an x-value of 0.

TEACHING TIP

While an exact solution for Exercise 4 can be found using a calculator table, you may wish to remind students that sometimes only an approximate solution can be found. Such a situation is addressed in Exercise 27 of Lesson 3.5.

TEACHING TIP



For Exercises 5–11, solve the given equation for the variable. Draw

an arrow diagram or use a table, if needed, to help you.

5. The equation L � 50 � 2t models the erosion over time of a beach

that extends 50 feet into the ocean and erodes at a rate of 2 feet per

year. After how many years will the beach extend only 37 feet?

6. The diameter d (in meters) of a giant

sequoia tree after t years is modeled by the

equation d � 4.320 � 0.003t. After how

many years will the tree’s diameter be

4.590 meters?

7. The total cost, including shipping, of buying

n DVDs from a certain Internet retailer is

modeled by the equation C � 6.99n � 12.

How many DVDs can be bought for $80?

8. A salesperson earns a weekly income of

I � 800 � 0.06s dollars on sales with a

value of s dollars. How much must she sell

to earn a total of $1,500 in a week?

9. A boy borrowed $2,200 from his uncle to

buy a billiard table. He plans to pay his

uncle back $150 a week from his summer

job earnings.

a. Write an equation that models the

amount A that the boy will owe his uncle

after w weekly payments.

b. How much will the boy still owe after

8 weeks?

c. How many weeks will it take to pay off

the loan?

10. When real estate is purchased, the value of buildings is determined

by subtracting the value of the land from the purchase price. A

three-unit apartment building is purchased for $800,000. If each

apartment unit is valued at $225,000, then the dollar value of the

land L is given by the equation

800,000 � L

___________ 3 � 225,000

Find the value of the land for this building.

11. The velocity v (in meters per second) of an object that is thrown

straight upward with an initial velocity v0 is given by v � v0 � 9.8t, where t represents time in seconds. If a ball is thrown upward at

30 meters per second, when will its velocity drop to 5 meters per

second? Round to the nearest tenth of a second.


5. 6 1 _ 2 years

6. 90 years7. n � 9.728, so 9 DVDs can be

bought8. $11,666.679a. A � 2,200 � 150w9b. $1,0009c. 14.7 weeks or 15 weeks to pay off

the loan10. $125,00011. after 2.6 seconds

72

LESSON 3.3

In Exercise 7, only whole number values for n make sense. Students may routinely round off 9.728 . . . to 10, not realizing that the cost for 10 DVDs would exceed $80.

COMMON ERROR

In Exercise 9c, point out to students that the loan will be paid off when A � 0.

TEACHING TIP


73R . A . P. Le s s o n 3 . 4

R.A.P.Lesson 3.4

Fill in the blank.

1. The word __________ means parts per hundred.

2. A statement that two expressions are equal is called a(n)

__________.

3. Subtraction and __________ are called inverse operations.

Choose the correct answer.

4. Which percent is equivalent to 3 __ 4 ?

A. 34% B. 133%

C. 0.75% D. 75%

5. Which percent is equivalent to 0.025?

A. 25% B. 2.5%

C. 0.025% D. 0.0025%

Multiply or divide. Write your answer in simplest form.

6. 7 __ 9

� 3

__ 4

� 8

__ 9

7. 3 3 __

5 � 4

2 __

3

8. 9 ___ 10

� 3

__ 5

9. 8 � 1 1 __

3

Add or subtract.

10. 4.5 � 12.14 � 0.78 11. 125.98 � 42.176

12. 0.4 � 0.04 � 0.004 13. 14 � 1.76

Write the phrase as a variable expression.

14. the sum of 6 and a number

15. a number multiplied by 8

16. 8 subtracted from a number

17. twice a number divided by 3

For Exercises 18–21, evaluate the expression.

18. 4d � 3(d � 1) � d2 for d � 3

19. 6x � x2 � y � 8 for x � 3 and y � 4

20. a2 � (2a � 6) � 1 for a � 5

21. 5 � m2 _______

m3 � 5 for m � 2


Lesson Objective• solve problems that require

previously learned concepts and

skills.

Exercise ReferenceExercise 1: Appendix E

Exercise 2: Lesson 3.2


Exercises 4–5: Appendix E

Exercises 6–9: Appendix A

Exercises 10–13: Appendix B

Exercises 14–21: Lesson 3.1


Exercises 23–28: Appendix K

Exercises 29–33: Appendix N

Lesson 3.4 R.A.P. Answers1. percent2. equation3. addition4. D5. B

6. 14 _ 27

7. 16 4 _ 5

8. 3 _

2 or 1 1 _

2

9. 6

10. �6.8611. 83.80412. 0.44413. 12.2414. 6 � n15. 8n16. n � 8

17. 2n _ 3

18. 919. 520. 1021. 3

73

LESSON 3.45e Evaluate



22. Find the slope of a line that passes through the points (4, 8)

and (–3, –6).

Write the number in scientifi c notation.

23. 512,000,000 24. 628 25. 1,802.5

Write the number in standard form.

26. 5.2 � 105 27. 7 � 103 28. 5.227 � 108

For Exercises 29–33, use the line plot.

3433Daily Low Temperature (°F)

3231302928 35 36 37 38 39

29. How many low temperatures are recorded in the line plot?

30. Find the range of the data.

31. What is the median of the data?

32. What is the mode of the data?

33. What is the mean of the data? (Round to the nearest tenth.)


22. 223. 5.12 � 108

24. 6.28 � 102

25. 1.8025 � 103

26. 520,00027. 7,00028. 522,700,00029. 14 low temperatures30. 9 degrees31. 34°F32. 33°F and 37°F33. about 34.1°F

74

LESSON 3.4


75S O LV I N G M U LT I S T E P E Q UAT I O N S Le s s o n 3 . 5

INVESTIGATION: Solving Multi-Step Equations

Lesson 3.5

In Lesson 3.3, you used the Properties of Equality to solve two-step equations. In this lesson, you will apply what you know about solving equations to solving equations with many steps.

MULTISTEP EQUATIONS

Suppose that your school’s environmental

club wants to travel to the state capital

to participate in an Earth Day festival.

Rental of a bus for the day will cost

$895. Lunch and festival activities will

cost $25 per student. Your club advisor

thinks that the fee for participating

should not be more than $60. How

many students must be recruited for

the trip?

1. To solve this problem, notice that the total fees collected must equal

the total cost for the trip. If each of n students pays $60 to attend,

write an expression for the total number of dollars collected.

2. The total cost includes the fi xed cost of $895 for the bus plus the

cost for all the students. Write an expression for the total cost.

3. Form an equation from your expressions in Questions 1 and 2.

The solution to this equation represents the number of students n that

must be recruited in order to pay for the trip. Notice that this equation is

different from the kind you solved in Lesson 3.3. The variable n appears

on both sides of the equation.

4. You can explore a solution to the equation using a calculator. Enter

your expressions from Questions 1 and 2 separately as Y1 and Y2

on the function screen of a calculator.

5. Create a table on your calculator using an increment of 1 for n.

Search for a value of n that makes the two expressions

approximately equal. Describe what you see.

6. Interpret your answer to Question 5 in the context of this situation.

7. There are two terms in the equation from Question 3 that contain n.

One is on the left side of the equation and one is on the right side.

To solve the equation algebraically for n, you must fi nd an equivalent

equation in which n appears on only one side of the equation.

Perform an operation that will do this while leaving the constant

term (895) by itself.

RecallA term in an expression or equation is a variable, a number, or a product or quotient of variables or numbers.

Comap2e_Modeling_Ch03.indd 75 03/02/12 7:32 PM6. Sample answer: For 25 students, the total cost of the trip is $1,520, but the total payments are only $1,500. For 26 students, the total cost is $1,545 and the total payments are $1,560. So, when the fee to participate is $60, 26 students are needed to pay for the trip. There will be $15 left over.

7. Subtract 25n; 60n � 25n � 895

4.

5. Sample answer: The values of the expressions are closest for n � 26.

Lesson Objectives• combine like terms in an

expression.

• solve equations.

• use equations to solve real-world

problems.

Vocabulary• like terms

• term

Materials Listnone

DescriptionThis lesson is designed as a small group investigation (2–4 students). Students should work through Questions 1–9 to explore solving equations that involve more than two steps. In this case they will be combining like terms.

Because of the importance of these equation-solving skills and the large number of exercises, you may wish to devote two days to this lesson.


Make sure students understand that terms are like terms only if their variable parts are exactly the same.

Lesson 3.5 Investigation Answers1. 60n2. 895 � 25n3. 60n � 895 � 25n

75

LESSON 3.5

Notice that an arrow diagram is not useful for the equation in Question 3. These diagrams are rarely helpful for solving equations that require more than two steps.

TEACHING TIP

For Question 5, students need not try for greater accuracy than the nearest whole number, since fractions make no sense in this context.

TEACHING TIP

5e Explore



8. Each term containing n can be thought of as a count of n’s.

Simplify the equation by combining the two terms into one term

giving the total count of n’s.

9. Solve your equation for n. Write a sentence explaining the

meaning of the solution.

It is not unusual for an equation to contain several terms that may have to be

combined. The Properties of Equality can then be used to solve the equation.

COMBINING LIKE TERMS

If the variable parts of two terms are exactly the same, the terms are

called like terms. For example, in the expression 5x � 3x � 9y, the

5x and the 3x are like terms. In the expression 3t2 � 4t � 10, there are

no like terms: t2 and t are not the same.

The expression 5a � 2a contains two like terms. You can use the

Distributive Property to combine like terms.

5a � 2a � (5 � 2)a � 7a

Combine like terms in each expression.

a. 8p � 6pb. 3b � 4c � 7bc. 10t � 3w � t � 6w � 2t

Solution:a. The variable parts are the same for both terms, so subtract

the coeffi cients.

8p � 6p � (8 � 6)p � 2p

b. First, rewrite the expression so that like terms are together.

3b � 4c � 7b � 4c � 3b � 7b

� 4c � (3 � 7)b � 4c � 4b

c. Again, rewrite the expression so that like terms are together.

10t � 3w � t � 6w � 2t � 10t � t � 2t � 3w � 6w

� (10 � 1 � 2)t � (�3 � 6)w � 9t � 3w


8. 35n � 895

9. n � 895 ___

35 or about 25.57; Sample

explanation: Since the number of students must be a whole number, at least 26 students must make the trip when the fee to participate is set at $60.

ADDITIONAL EXAMPLE Combine like terms in each expression.

a. 12d � 7d 5d

b. 5m � 2n � 4m m � 2n

c. 14x � 2y � 3z � x � 6y 15x � 4y � 3z

76

LESSON 3.5


77

Solve 3p � 9 � 2p � 21 for p.

Solution:First, combine like terms. Reverse the usual order of operations.

Undo the subtraction by adding 9 to both sides of the equation.

Then divide both sides of the equation by 5.

Original equation 3p � 9 � 2p � 21

Combine like terms. 5p � 9 � 21

Add 9 to both sides. 5p � 9 � 9 � 21 � 9

Simplify. 5p � 30

Divide both sides by 5. 5p

___ 5 �

30 ___

5

Simplify. p � 6

Check:

3(6) � 9 � 2(6) � 21

18 � 9 � 12 � 21

9 � 12 � 21

21 � 21

?

?

?

HINTS FOR SOLVING EQUATIONS

The operations of addition, subtraction, multiplication, and division can

be performed on both sides of an equation to produce an equivalent

equation with the same solution.

• Simplify the expression on each side of the equation.

• Use addition and/or subtraction to isolate the variable in a single

term on one side of the equation.

• Use multiplication and division to solve the equation.

• If an equation contains fractions, it is often helpful to eliminate the

fractions by multiplying both sides of the equation by a common

denominator.

S O LV I N G M U LT I S T E P E Q UAT I O N S Le s s o n 3 . 5


ADDITIONAL EXAMPLE

Solve 7b � 12 � 3b � 4 for b. � 4 _ 5

77

LESSON 3.5

Discuss the Hints for Solving Equations with students and point out specifi c examples of each. Also point out that not all the strategies are needed for every equation.

TEACHING TIP



Solve 5x � 3(7 � 2x) � 23 for x.

Solution:Original equation 5x � 3(7 � 2x) � 23

Distributive Property. 5x � 3(7) � 3(�2x) � 23

Simplify. 5x � 21 � 6x � 23

Combine like terms. 11x � 21 � 23

Add 21 to both sides. 11x � 23 � 21

Simplify. 11x � 44

Divide both sides by 11. 11x ____ 11

� 44

___ 11

Simplify. x � 4

Check:

5(4) � 3[7 � 2(4)] � 23

20 � 3(�1) � 23

20 � 3 � 23

23 � 23

?

?

?

Solve 5B � 3 � 2B � 11 for B.

Solution:Begin by isolating the variable on one side of the equation.

Original equation 5B � 3 � 2B � 11

Subtract 2B from both sides. 5B � 3 � 2B � 2B � 11 � 2B

Combine like terms. 3B � 3 � 11

Subtract 3 from both sides. 3B � 3 � 3 � 11 � 3

Simplify. 3B � 8

Divide both sides by 3. 3B ___ 3 �

8 __

3

Simplify. B � 8 __

3

Check:

5 � 8 __ 3 � � 3 � 2 � 8 __

3 � � 11

40

___ 3 � 3 �

16 ___

3 � 11

49

___ 3

� 49

___ 3

Notice that if you use 2.67 to approximate 8 __ 3 , the left and right sides of the

equation are not exactly equal. Only 8 __ 3 is an exact solution to the equation.

?

?


ADDITIONAL EXAMPLE Solve 12t � 2(3t � 1) � 44 for t. 7

ADDITIONAL EXAMPLE Solve 8q � 10 � q � 4 for q. 2

78

LESSON 3.5


79



In one state, the fi ne F for speeding on a 65-mile-per-hour highway is

modeled by the equation F � 50 � 10(s � 65), where s is the speed.

1. Write an equation that you could solve to fi nd the speed that

would result in a fi ne of $270.

2. To use a table to solve for s, what would you use for a starting

value for s? Explain.

3. Use a table to fi nd the speed for a fi ne of $270.

4. Use the Properties of Equality to solve algebraically for s.


One cell-phone plan costs $39.99 a month for 450 minutes of talking

and unlimited texting. Each additional minute over 450 costs 45 cents.

The total monthly cost for any number of minutes m that is greater than

450 can be modeled by C � 39.99 � 0.45(m � 450).

5. Write an equation that you could solve to fi nd how many

talk minutes were used in a month for which the total bill is

$136.74.

6. To use a table to solve for m, what would you use for a starting

value for m? Explain.

7. Use a table to fi nd the number of talk minutes for a bill of $136.74.

8. Use the Properties of Equality to solve algebraically for m.

For Exercises 9–12, combine like terms.

9. 8s � 12 � 3s � 20

10. p � 3q � 2p2 � 8p � q

11. M2 � 2M � 3M2 � M

12. 6x2 � x2y � xy2 � 4x2

For Exercises 13–19, solve for the variable, then check.

13. 5y � 9 � 3y

14. 4R � 8 � 6R � 4

15. 18 � 10(w � 1) � 6w

16. 7z � 15 � 6(z � 4)

17. 4(y � 5) � 2y � 28

18. 3 � 5(t � 6) � 8t � (4 � 2t)

19. 8.6 � 2.1m � 3.7m � 4.2



9. 5s � 3210. 2p2 � 7p � 2q11. 4M2 � M12. 2x2 � x2y � xy2

13. y � 9 _

2 or 4 1 _

2

14. R � 615. w � �216. z � 317. y � 4

18. t � � 23 _ 15

or t � �1.53

19. m � 8

7. 665

8. Original equation

39.99 � 0.45(m � 450) � 136.74 Subtract 39.99 from both sides.

0.45(m � 450) � 96.75 Divide both sides by 0.45.

m � 450 � 215 Add 450 to both sides. m � 665

Practice for Lesson 3.5 Answers1. 50 � 10(s � 65) � 2702. Sample answer: You could start with

66, since there will be no fi ne unless the speed is more than 65 mph.

3. The speed is 87 mph.

4. Original equation 50 � 10(s � 65) � 270 Subtract 50 from

both sides. 10(s � 65) � 220 Divide both sides by 10. s � 65 � 22 Add 65 to both sides. s � 875. 39.99 � 0.45(m � 450) � 136.746. Sample answer: You could start

with a number higher than 450, since the bill for 450 minutes would only be $39.99.

79

LESSON 3.5

Exercises 6 and 7 If students fi nd it tedious to scroll all the way down the table, suggest that they start with an increment of 10 and then refi ne their search.

TEACHING TIP

If students struggle with Exercise 14, you may want to have them use algebra tiles to help them visualize the operations involved in solving equations.

TEACHING TIP

In Exercise 19, students may fi nd it helpful to begin by multiplying every term on both sides of the equation by 10 to eliminate decimal numbers.

TEACHING TIP



20. In Lesson 3.2, you found that the equation h � 62 � 1.4(n � 20)

can be used to model the height of a coast redwood for ages

greater than 20 years. How many years would it take a coast

redwood to grow to a height of 125 feet?

21. For people born after 1975, average life expectancy L can be found

using the equation L � 72.6 � 0.17(B � 1975), where B represents

year of birth. For what birth year is the average life expectancy

75 years?

22. Lab technicians often mix two solutions to make one solution with a

required concentration. To fi nd the volume V of a 75% solution that

must be added to 2 liters of a 30% solution in order to make a 50%

solution, solve the equation 0.50(2 � V) � 0.30(2) � 0.75V.

23. A square sandbox is made with lumber that is 1.5 inches thick.

s

a. If the outside dimensions of the sandbox are of length s (in

inches), write an equation that models the total length L of

lumber needed.

b. How long is a side of the largest sandbox that can be made

from a single 16-foot long piece of lumber?

24. Consider the equation y � 1

_____ 2 � 5 �

2 __

3 y. To avoid having to

combine fractions, you can multiply both sides of the equation by a

common denominator.

a. What is a common denominator for the fractions in the

equation?

b. Multiply both sides of the equation by your answer to Part (a)

and simplify the result.

c. Solve your equation from Part (b) for y.

25. Solve 2 � x _____

5 �

4x � 3 ______

2 for x, then check.

26. Solve 2n � 3

______ 4 �

n � 5 _____

3 � 1 for n, then check.


20. 65 years21. 198922. 1.6 L23a. L � 2s � 2(s � 3) or L � 4(s � 1.5)

23b. 49 1 _ 2 inches or 4 feet 1 1 _

2 inches

24a. Sample answer: 624b. 3(y � 1) � 30 � 4y24c. y � �27

25. x � � 1 _ 2

26. n � 17 _ 2

80

LESSON 3.5

If students struggle with Exercise 23b, point out that their equation in Part (a) expresses the total length L in inches and Part (b) gives the length of the piece of lumber in feet. Hence, they must convert 16 feet to 192 inches before using their equation.

COMMON ERROR

Exercise 24 If students get an answer of �2, remind them that each term in the equation must be multiplied by 6.

COMMON ERROR


81

27. The equation 5B � 3 � 2B � 11 was solved algebraically in Example 4

(on p. 78). But if you try using a table to solve the equation, you will

not be able to fi nd the exact solution even if you refi ne the TABLE

SETUP [TBLSET] by using a smaller increment. Explain.


The equation a � 3 contains an absolute value expression. Recall

that a � 3 means a � 3 or �a � 3, depending on whether the value

inside the absolute value symbols is positive or negative. The graph

of a � 3 is

0 2 4 6–6 –4 –2

The graph shows that a � 3 can be interpreted as follows:

“a is a number whose distance from 0 on a number line is 3 units.”

To solve an absolute value equation such as � x � 7 � � 4, you fi rst need

to rewrite the equation as a compound sentence:

x � 7 � 4 or �(x � 7) � 4

Then both of these equations must be solved in order to fi nd the

complete solution to � x � 7 � � 4. For this equation, the solution set is

x � 3 or x � 11.

28. Graph the solution set for � x � 7 � � 4 on a number line.

29. Write a sentence that explains the meaning of � x � 7 � � 4 in terms

of a number line.

For Exercises 30–31, complete Parts (a) and (b).

a. Solve the equation.

b. Graph the solution set on a number line.

30. � x � 7 � � 4

31. � D � 8 � � 36



27. Sample answer: The solution to

the equation is 8 __ 3 . This fraction

cannot be written as an exact decimal. Using a table, you could determine that the solution is between 2.6 and 2.7, but you could never fi nd a decimal value of B that makes both sides of the equation identical. The answer would have to be found to some required accuracy.

28.

50 10

29. Sample answer: x is a number whose distance from 7 on a number line is 4.

30a. x � �3 or �1130b.

–10–15 –5 0

31a. D � �44 or 2831b.

0–50 50

81

LESSON 3.5

Exercises 28–31 involve absolute value. You may wish to review the concept of the absolute value of a number (or refer students to Appendix G) before assigning these exercises.

TEACHING TIP



INVESTIGATION: Solving InequalitiesLesson 3.6

In the previous lessons in this chapter, you explored using expressions and equations to model real-world situations. You also learned to evaluate the expressions and solve the equations. In this lesson, you will apply your knowledge to solving inequalities.

In Lesson 3.5, you used the equation

60n � 895 � 25n to determine the number

of students that must be recruited for the

Earth Day trip. This problem could have been

thought of in a different way. Any number of

students could have gone on the trip as long

as there was at least enough money to pay

for the bus and other expenses.

So instead of using an equation, the problem is

more accurately modeled with the inequality:

60n � 895 � 25n

Methods for solving inequalities are slightly different from methods for

solving equations. To see how they differ, start with a simple example.

1. Pick any two different positive numbers. Then use those numbers

and either a less-than symbol (�) or a greater-than symbol ()

to write a true statement.

2. Now add the same positive number to each side of your inequality.

Does the inequality remain true?

3. Add the same negative number to each side of your original

inequality. Is this new inequality statement true?

4. Subtract the same number from each side of your inequality. Does

the inequality remain true?

5. Multiply each side of your original inequality by the same positive

number. Is this new inequality true?

6. Divide each side of your original inequality by the same positive

number. Does the inequality remain true?

7. Now multiply each side of your original inequality by the same

negative number. Is this new statement true?

8. If you reverse the direction of the inequality symbol in your

statement from Question 7, is your inequality true?

9. What do you think will happen if you divide each side of your

original inequality by a negative number? Check to see if you

are correct.

RecallRecall the defi nitions of the following inequality symbols:

Symbol Meaning

� is less than

is less than or equal to

is greater than

� is greater than or equal to

� is not equal to


Lesson Objectives• solve inequalities.

• graph inequalities.

Vocabulary• Properties of Inequality

Materials Listnone

DescriptionThe main purpose of the Investigation in this lesson is to provide students with a review of the Properties of Inequality. This lesson can be used as a whole-class investigation, or students can work together in small groups as they review.


Discuss Question 11 a–d with the class to make sure everyone has correctly identifi ed the Properties of Inequality.

Lesson 3.6 Investigation Answers1. Sample answer: 15 102. Yes. 3. Yes. 4. Yes. 5. Yes.6. Yes. 7. No.8. Yes.9. The new inequality is false.

82

LESSON 3.65e Explore


83S O LV I N G I N E Q UA L I T I E S Le s s o n 3 . 6

10. Now reverse the direction of the inequality in your statement from

Question 9. Is your inequality true?

11. Summarize your fi ndings from Questions 1–10 by fi lling in the

blanks.

a. If any number is added to each side of a true inequality, then

the resulting inequality is ____________.

b. If any number is subtracted from each side of a true inequality,

then the resulting inequality is ____________.

c. If each side of a true inequality is multiplied or divided by the

same positive number, the resulting inequality is ____________.

d. If each side of a true inequality is multiplied or divided by the

same negative number, the direction of the inequality symbol

must be _______________ so that the resulting inequality is

also true.

The solution set for an inequality is usually a range of numbers rather

than a single number. But the method for solving an inequality, such as

solving 60n � 895 � 25n for n, is very similar to the method for solving

its associated equation.

Original inequality 60n � 895 � 25n

Subtract 25n from both sides. 60n � 25n � 895 � 25n � 25n

Simplify. 35n � 895

Divide both sides by 35. 35n ____ 35

� 895

____ 35

Simplify. n � 895

____ 35

or n � 25.57

Any value for n that is a whole number greater than 25.57 satisfi es

the inequality. So at least 26 students must go on the trip in order

to meet expenses; but 27, 28, or more students will also satisfy the

requirement.

To solve an inequality, use the Properties of Inequality that are

summarized as follows:

Properties of Inequality

• Any number can be added to or subtracted from both sides

of an inequality without changing the direction of the

inequality.

• When both sides of an inequality are multiplied or divided by a

positive number, the direction of the inequality is unchanged.

• When both sides of an inequality are multiplied or divided by a

negative number, the direction of the inequality is reversed.


10. Yes.11a. also true11b. also true11c. also true11d. reversed

83

LESSON 3.6



Solve the inequality 8 � 3x 2 for x.

Solution: Original inequality 8 � 3x 2

Subtract 8 from both sides. 8 � 3x � 8 2 � 8

Simplify. �3x �6

Divide both sides by �3 and reverse

�3x ____

�3

�

�6 ___

�3

the direction of the inequality.

Simplify. x � 2

As with equations, inequalities should be checked. One way to check if

a solution makes sense is to choose any number in the solution interval.

Then substitute the number for the variable in the original inequality

and see if it is satisfi ed. In this example, you could try the number 1

because it is less than 2.

Check: 8 � 3(1) > 2

8 � 3 > 2

5 > 2

GRAPHING INEQUALITIES

The solution set in Example 1 can be graphed on a number line.

But whereas a single number is graphed as a point, the graph of an

inequality is an interval on the line. The graph of x < 2 is shown below.

0 5–5

The open circle at x � 2 indicates that 2 is not included in the solution

interval. The arrow means that the interval extends indefi nitely to the

left without limit.

The graph of x � 1 is shown below.

0–5 5

Notice that the endpoint at x � 1 is treated differently. Because the

interval includes 1, a fi lled circle is placed at x � 1.

?

?


ADDITIONAL EXAMPLE Solve the inequality 5 � 4x³ � 1 for x. x 1

84

LESSON 3.6

Point out to students that any number in the solution interval (except an endpoint of a closed interval) can be used to check whether the direction of the inequality symbol in the solution is correct.

TEACHING TIP


85

a. Solve the inequality 4y � 3(y � 6) 16 � y.


Solution:a.

Original inequality 4y � 3( y � 6) 16 � y

Distributive Property 4y � 3y � 18 16 � y

Simplify. y � 18 16 � y

Add y to both sides. y � 18 � y 16 � y � y

Simplify. 2y � 18 16

Subtract 18 from both sides. 2y � 18 � 18 16 � 18

Simplify. 2y �2

Divide both sides by 2. 2y

___ 2

�2 ___

2

Simplify. y �1

b.

0 5–5


For Exercises 1–6, solve the inequality for the variable. Check your

solution and graph it on a number line.

1. 8 � 5t � 6 2. 4x � 7 � 3 � 2x

3. �5m � 3 18 4. 10p 30p � 1

_______ 2

5. 8h � 3 � 3(h � 4) 6. �8q � 24 � �20

For Exercises 7–9, complete Parts (a) and (b).

a. Write an inequality to model the sentence.

b. Solve the inequality.

7. Eleven is no more than the sum of twice a number n and fi ve.

8. One-half a number n, increased by fi ve, is at least twenty.

9. The sum of six times a number n and twelve is more than the

product of that number and three.

S O LV I N G I N E Q UA L I T I E S Le s s o n 3 . 6


ADDITIONAL EXAMPLE a. Solve the inequality 6z � 1 � 4(3� 2z)

for z. z 11 _ 2


50 10

Practice for Lesson 3.6 Answers

1. t 14 _ 5

0–5 5

2. x � 5 _ 3

0 1 2 3–1

3. m � �3

0–5 5

4. p � 1 _ 10

0–1 1

5. h � �3

–5–10 0

6. q 5.5

50 10

7a. 11 2n � 57b. n � 3

8a. 1 _ 2 n � 5 � 20

8b. n � 309a. 6n � 12 3n9b. n �4

85

LESSON 3.6

In Exercises 3 and 6, students may need to be reminded that the direction of the inequality symbol must be reversed when an inequality is multiplied or divided by a negative number.

COMMON ERROR



10. A gallon of a certain brand of paint will cover a maximum area of

450 square feet. The inequality 450n � 2(860) models the number

of one-gallon cans n of paint needed to give two coats of paint to

860 square feet of wall space. Solve for n.

11. The U.S. Postal Service will accept a package for mailing only if the

sum of its length and girth (distance around) is not more than

108 inches. The girth of a package equals twice its width plus twice

its thickness.

Length

Girth

a. Write an inequality that models the Postal Service requirement.

Use l for length, w for width, and t for thickness.

b. If a package is 50 inches long and 22 inches wide, how thick

can it be?

12. A student wants to buy chips and soda for his party. Each large bag

of chips costs $2.40, and each large bottle of soda costs $2.00. He

wants to spend no more than $48.00.

a. Write an inequality that models the number of bags of chips C

and bottles of soda S that he can buy.

b. If he buys 10 bags of chips, solve for S to fi nd how many bottles

of soda he could buy.

13. Consider the inequality 14 � 3x 2. Explain how a table can be

used to solve the inequality.


10. n � 3.8 gallons, so a minimum of 4 cans of paint are needed.

11a. l � 2w � 2t 10811b. t 7 inches12a. 2.40C � 2.00S 48.0012b. S 12 bottles13. Sample answer: A table can be

made that includes values of the expression 14 � 3x. The value of the expression is 2 when x � 4. But for values of x less than 4, the value of the expression is greater than 2.

86

LESSON 3.6


87CO M P O U N D I N E Q UA L I T I E S Le s s o n 3 . 7

Compound InequalitiesLesson 3.7

You have learned to solve equations and inequalities and use them to model real-world situations. In this lesson, you will use what you know to write, solve, and graph compound inequalities.

WRITING COMPOUND INEQUALITIES

Two inequalities that are joined by the word and or the word or form a compound inequality.

Compound Inequalities

Joined by and

Compound Inequalities

Joined by or

x �1 and x 3

A solution to a compound inequality joined by

the word and is any number that makes both

inequalities true.

Its graph is the intersection of the graphs of

the two inequalities. To fi nd the intersection

of the two graphs, graph each inequality and

fi nd where the two graphs overlap.

The graph of x �1 is

0 42–4 –2

and the graph of x 3 is

0 42–4 –2

So, the graph of x �1 and x 3 is

0 42–4 –2

x 0 or x 2

A solution to a compound inequality joined by

the word or is any number that makes at least one of the inequalities true.

Its graph is the union of the graphs of the

two inequalities. To fi nd the union of the two

graphs, graph each inequality.

The graph of x 0 is

0 42–4 –2

and the graph of x 2 is

0 42–4 –2

So, the graph of x 0 or x 2 is

0 42–4 –2

These inequalities can be combined and

written without using and.

�1 � x 3

These inequalities cannot be combined. They

must be written using the word or.


Lesson Objectives• write compound inequalities.

• graph compound inequalities on a

number line.

• solve compound inequalities.

Vocabulary• compound inequality

DescriptionThis lesson builds on Lesson 3.6 and discusses writing, graphing, and solving compound inequalities. Absolute value inequalities are included in Exercises 11–13.

87

LESSON 3.7

Students may need to be reminded of the meaning of “intersection” and “union.” Explain that every point included in the intersection of two graphs must appear on both of the graphs, but a point in the union of two graphs has to appear on at least one of the graphs.

TEACHING TIP

Inequalities are sometimes stated using the word “but,” as in “x is greater than �1 but less than 3.” In this context, “but” is interpreted as being operationally equivalent to “and.”

TEACHING TIP

5e Explain



Write an inequality for each statement.

a. n is between �2 and 5.

b. m is at least 0 and less than 10.

c. d is less than 3 or greater than 8.

d. h is at most 1 or at least 5.

Solution:

a. �2 � n � 5

b. 0 m � 10

c. d � 3 or d 8

d. h 1 or h � 5

Graph each compound inequality on a number line.

a. �2 b � 3

b. x � �3 or x 0

Solution:

a.

0 42–4 –2

b. 0 42–4 –2

Solve �5 � t � 1 � 11 for t.

Solution:Write the compound inequality as two inequalities joined by the word and.

Original inequalities �5 � t � 1 and t � 1 � 11

Solve each inequality. �5 � 1 � t � 1 � 1 and t � 1 � 1 � 11 � 1

Simplify. �6 � t and t � 10

The solution set is all real numbers greater than �6 and less than 10. This can be written as �6 � t � 10.

SOLVING COMPOUND INEQUALITIES

To solve a compound inequality, solve each inequality separately.


ADDITIONAL EXAMPLE Write an inequality for each statement.

a. t is less than 4 and greater than 0. 0 � t � 4

b. p is greater than 1 and no more than 5. 1 � p 5

c. q is less than �2 or greater than 1. q � �2 or q 1

d. b is between 0 and 6 inclusive. 0 b 6

ADDITIONAL EXAMPLEGraph each compound inequality on a number line.

a. �4 t � 1

0–5 5

b. w 1 or w � 2

0–5 5

ADDITIONAL EXAMPLE Solve �6 2(d � 3) 4 for d.0 d 5

88

LESSON 3.7


89

a. Solve 3x � 4 � �2 or 2x � 8 � 10 for x.

b. Graph the solution.

Solution: a.

Original inequalities 3x � 4 � �2 or 2x � 8 � 10

Subtract. 3x � 4 � 4 � �2 � 4 or 2x � 8 � 8 � 10 � 8

Simplify. 3x � �6 or 2x � 2

Divide. 3x ___ 3 �

�6 ___

3 or

2x ___ 2 �

2 __

2

Simplify. x � �2 or x � 1

The solution set is all real numbers less than –2 or greater than or equal to 1.

b. 0 42–4 –2


For Exercises 1–4, write an inequality for each statement. Then

graph the solution set on a number line.

1. y is greater than or equal to –2 and less than 0.

2. q is either less than 1 or at least 4.

3. a is at least 5 and no more than 8.

4. z is between –3 and 3.

For Exercises 5–7, solve the compound inequality.

5. �2a � 3 �1 or 3a � 1 � 5

6. �2 4(x � 6) 7

7. 3n � 2 � 10 and 2n � 1 7

8. A tree has a medium growth rate if its rate of increase in height r is

more than 12 inches per year and no more than 24 inches per year.

a. Write a compound inequality that describes the medium

growth rates.

b. Graph the inequality.

CO M P O U N D I N E Q UA L I T I E S Le s s o n 3 . 7


ADDITIONAL EXAMPLE a. Solve 3 � 2b � 1 or b � 6 4 for x.

b. Graph the solution.a. b 1 or b 10b.

0 5 10–5

Practice for Lesson 3.7 Answers1. �2 y � 0; also y � �2 and y � 0

0–4 –2 2

2. q � 1 or q � 4

0 4–2 2

3. 5 a 8; also a � 5 and a 8

3 6 90

4. �3 � z � 3; also z �3 and z � 3

0–5 5

5. a � �1 or a � 2

6. 11 _ 2 x 31 _

4

7. no solution8a. r 12 and r 24; also 12 � r 248b.

20 250 5 10 15

89

LESSON 3.7

In Exercises 2 and 5, students may be tempted to try to combine the inequalities in a manner similar to that for “and” inequalities. But 4 q � 1 makes no sense, since 4 is greater than 1. Solutions to these exercises must be stated as two separate inequalities.

COMMON ERROR



9. Most states have building codes that provide standards for

construction. For public buildings in Massachusetts, the height of

a step in a staircase must be at least 4 inches but cannot be greater

than 7 inches.

a. Write a compound inequality that describes the height h of a

legal stair step in Massachusetts.

b. Graph the inequality.

10. On many interstate highways, a driver must drive at least 40 mph

and no more than 70 mph. If a car is traveling 32 miles per hour,

what increase in speed would allow the car’s speed to be in the

legal limit?


In order to solve an absolute value inequality, such as � x � 3 � � 6, you

need to rewrite the inequalities as a compound sentence.

• If the absolute value is on the left of the inequality symbol and the

symbol is � or , use and to write the compound sentence.

• If the absolute value is on the left of the inequality symbol and the

symbol is or �, use or to write the compound sentence.

So, � x � 3 � � 6 can be rewritten as a compound sentence with and.

x � 3 � 6 and �(x � 3) � 6

Then both of these inequalities must be solved in order to fi nd the

complete solution to � x � 3 � � 6. When you solve this inequality, you get

x � 3 and x �9. This solution set can also be written as �9 � x � 3.

Solve the inequality and graph the solution set.

11. � n � 5 � 9

12. � 4x � 3 � 2

13. 5 � t � 1 � 10

RecallThe quantity x equals either x or �x, depending on whether x is positive or negative.


9a. h � 4 and h 7; also 4 h 79b.

0 8642

10. 8 mph s 38 mph11. n � �4 or n 14

–5–10 200 5 10 15

12. x � 1 _ 4 and x 5 _

4 ; also 1 __

4 x 5 _

4

0 1 2–1

13. t � �3 or t 1

0–6 –2–4 642

90

LESSON 3.7

Exercises 11–13 build on both the concept of absolute value and the skills involved with solving absolute value equations. If both have not been discussed in class, return to Lesson 3.5 and revisit Exercises 28 and 29 with students.

TEACHING TIP

If students struggle with Exercise 11, help them understand the meaning of an inequality like |x � 3| � 6 by having them graph the solution. Point out that this inequality can be interpreted as all points that are within a distance of 6 units from x � �3.

COMMON ERROR


Algebra’s Next Top ModelIn this chapter, you have written expressions, equations, and

inequalities to model many types of situations that were

presented to you. Now it is time to apply your modeling skills to

situations of your own choosing.

Remember where the parts of expressions come from.

• Constant terms represent quantities that do not change in a

given situation. For example, the initial value of the height

of a tree when planted is a constant value in an expression.

• A variable represents a quantity that can change and take

on different values. For tree growth, n or t may represent

the variable number of years after planting.

• If only the number above a certain value is needed, a constant

may be subtracted from the variable. For example, (s – 65)

counts the number of miles per hour of speed over 65.

• The coeffi cient of a variable often represents a constant rate

of change. If a tree grows 2 feet each year, then the term 2n

represents the total growth over n years. The coeffi cient 2

represents the rate of growth per year.

Also, remember that an equation results from setting two

expressions equal to each other.

Think about situations in your experience that involve

changing quantities. You may want to talk to people you know

and fi nd out the kinds of questions they deal with in their

job. For example, you might ask truck farmers what kinds of

considerations help them determine their profi t when they take

goods to market. Or contact tree nurseries in your state to fi nd

out about their procedures and how they plan for future tree

growth. Be creative and fi nd situations that are important to you

or to people in the area where you live.

Choose three situations that can be modeled by equations. For

each situation,

• write a question someone might ask that has a numerical

answer;

• write an equation that models the situation;

• explain the meaning of any constants, variables, or

coeffi cients in your model; and

• use your model to make predictions for several reasonable

values of your variable or variables.

Modeling Project

CHAPTER

3

91M O D E L I N G P R O J E C T Chapter 3


Materials Listnone

DescriptionIn this project students are asked to think creatively about ways in which real situations can be modeled with mathematics. This project can be completed by individual students, although they may prefer to work in pairs if their project involves research with local companies.

Sample AnswersStudents’ reports may vary, depending on the situations they choose to investigate. If students have diffi culty thinking of situations to model, you may wish to give them suggestions. In addition to the nursery and truck farming suggestions on the student page, other situations might include modeling cell phone plans, gym membership fees, small business costs or profi ts, real estate offi ce expenses and commissions, etc.

Regardless of the situations chosen, students’ models should refl ect an understanding of their work with expressions and equations in this chapter. Not all models need to be linear models, but students should justify the components of their models in any case.

91

MODELINGPROJECT

5e Elaborate



Chapter 3 ReviewYou Should Be Able to:

Lesson 3.1• use arrow diagrams to represent

expressions.

• evaluate expressions.

• write expressions to model real-world

situations.

Lesson 3.2• write an equation that models a real-

world situation.

Lesson 3.3• use arrow diagrams to solve two-step

equations.

• use tables to solve equations.

Lesson 3.4• solve problems that require previously

learned concepts and skills.

Lesson 3.5• combine like terms in an expression.

• solve equations.

• use equations to solve real-world

problems.

Lesson 3.6• solve inequalities.

• graph inequalities.

Lesson 3.7• write compound inequalities.

• graph compound inequalities on a

number line.

• solve compound inequalities.

Key Vocabulary

algebraic expression (p. 58)

coeffi cient (p. 58)

evaluate an expression (p. 58)

order of operations (p. 58)

equation (p. 61)

inverse operations (p. 67)

equivalent equation (p. 71)

Properties of Equality (p. 71)

term (p. 75)

like terms (p. 76)

Properties of Inequality (p. 83)

compound inequality (p. 87)

Chapter 3 Test ReviewFor Exercises 1–2, draw an arrow diagram to represent the expression.

1. 4y � 6

2. 4(y � 6)


Chapter 3 Test Review Answers1.

Multiply by 4 Subtract 6

4y 4y – 6y

2.

Subtract 6 Multiply by 4

y – 6 4(y – 6)y

92

CHAPTER REVIEW

5e Evaluate


93C H A P T E R R E V I E W Chapter 3

3. Evaluate the expression 3t � 4(t � 5) � t __ 2 for t � 6.

4. A student opens a college savings account with a gift from her

grandmother. She then makes regular deposits every month from her

babysitting earnings. The expression 80n � 500 models the total balance

in her account after n months.

a. Interpret the meaning of the number 80 in the expression.

b. Interpret the meaning of the number 500 in the expression.

5. A vendor at the state fair will buy 600 cans of

juice for $140. He will charge $1.50 for each can

he sells. If he has to pay a $25 fee to the fair,

which of these expressions represents the amount

of money he makes, after expenses, for selling n

cans of juice?

A. 600n – 140 – 25

B. 140n – 600 – 25

C. 1.5n – 600 – 25

D. 1.5n – 140 – 25

6. Main fl oor seats for the school musical are sold for $8 and balcony seats

for $5. Write an equation that models total ticket revenue R if m main

fl oor seats and b balcony seats are sold.

7. A furniture company makes a variety of

wooden bookcases. The height of each shelf

space is 13 inches. The top and bottom

pieces are each 2 inches thick, and the pieces

separating the shelf spaces are each 3

__ 4 -inch

thick. Write an equation that models the total

height h of a bookcase that contains n shelf

spaces.

8 a. Draw an arrow diagram that models the

solution of �3y � 8 � 13.

b. Solve the equation for y.

9. Which expressions are not like terms?

A. �5xy and 4xy B. �4a2 and �4a

C. 7xy and 9yx D. �3r and 5r

h


3. 174a. the amount deposited each month4b. the original gift from her

grandmother5. D6. R � 8m � 5b

7. h � 13n � 3 _ 4 (n � 1) � 2(2)

8a.

Multiply by –3

Divide by –3

Subtract 8

Add 8

–3yy 13

8b. y � �79. B

93

CHAPTER REVIEW



For Exercises 10–11, combine like terms.

10. 4x � 2y � x � 6y

11. 3ab � 9a � 5ba � 1

For Exercises 12–15, solve, then check.

12. 5h � 3(h � 2) � 8 13. 6(x � 3) � 4(x � 1) � 2

14. 6n � 3 ______

5 � 1 � 4n 15. 0.12r � 3 � 0.04r � 5

For Exercises 16–17, use the equation 10 � (t � 6) � 2(4t � 1) � 7t.

16. Use a table to solve the equation for t.

17. Solve the equation algebraically.

18. Write an inequality for the following statement. Then graph it.

k is either less than 0 or greater than 2.

For Exercises 19–21, solve for the variable.

19. 2(d � 6) 10d

20. 3r 5(r � 7) � 1

21. �2 � 5m � 8 � 4


10. 3x � 4y11. 8ab � 9a � 112. h � 113. x � �12

14. n � 4 _ 13

15. r � �10016. t � 9

17. t � 918. k � 0 or k 2

0–5 5

19. d � � 3 _ 2

20. r � 1721. �2 � m � � 4 _

5

94

CHAPTER REVIEW


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