looking ahead to chapter 3 - somersetcentral.enschool.org

Looking Ahead to Chapter 3Focus In Chapter 3, you will learn how to solve one-and two-step equations and use them

in real-life situations. You will also work with integers and graph equations in the

coordinate plane using the Cartesian coordinate system.

Chapter Warm-upAnswer these questions to help you review skills that you will need in Chapter 3.

Perform the indicated operation.

1. 2. 3.

4. 5. 6.

Use mental math to find the value of x.

7. 8. 9.

Read the problem scenario below.

You start a new job and make $44 each week. Your goal is to save enough money to buy a

new television set that costs $352.

10. Can you buy the television after 6 weeks of working?

11. How many weeks do you need to work so that you have enough money to buy the

television?

12. Suppose that you have saved 75% of the money needed to buy the television.

How much money have you saved?

x100

� 0.2330x � 120x6

� 8

324 � 45(8)12 � 10

78 � 3989

�34

48 � 18

92 Chapter 3 ■ Solving Linear Equations

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solve an equation ■ p. 96

equivalent equations ■ p. 97

one-step equation ■ p. 98

inverse operations ■ p. 98

algebraic check of a

solution ■ p. 99

graphical check of a

solution ■ p. 100

two-step equation ■ p. 102

commission ■ p. 107

commission rate ■ p. 107

percent equation ■ p. 108

integer ■ p. 131

negative integer ■ p. 131

positive integer ■ p. 131

number line ■ p. 131

quotient ■ p. 136

Cartesian coordinate

system ■ p. 140

x-axis ■ p. 140

y-axis ■ p. 140

origin ■ p. 140

coordinate plane ■ p. 140

ordered pair ■ p. 141

x-coordinate ■ p. 141

y-coordinate ■ p. 141

estimate ■ p. 147

Key Terms

Chapter 3 ■ Solving Linear Equations 93

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Solving Linear Equations

3.1 Collecting Road Tolls

Solving One-Step Equations ■ p. 95

3.2 Decorating the Math Lab

Solving Two-Step Equations ■ p. 101

3.3 Earning Sales Commissions

Using the Percent Equation ■ p. 107

3.4 Rent a Car from Go-Go Car

Rentals, Wreckem Rentals, and

Good Rents Rentals

Using Two-Step Equations,

Part 1 ■ p. 113

3.5 Plastic Containers

Using Two-Step Equations,

Part 2 ■ p. 125

3.6 Brrr! It’s Cold Out There!

Integers and Integer Operations ■ p. 131

3.7 Shipwreck at the Bottom of the Sea

The Coordinate Plane ■ p. 139

3.8 Engineering a Highway

Using a Graph of an Equation ■ p. 143

C H A P T E R

3

Most plastic containers are made using polyethylene, which was discovered in 1933.

Polyethylene is also used to make soda bottles and milk jugs. In Lesson 3.5, you will

design shipping boxes for plastic containers.


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Lesson 3.1 ■ Solving One-Step Equations 95

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Collecting Road TollsSolving One-Step Equations

ObjectivesIn this lesson,

you will:

■ Write one-step

equations.

■ Solve one-step

equations.

■ Check solutions

algebraically.

■ Check solutions

graphically.

Key Terms■ solve an equation

■ equivalent equations

■ one-step equation

■ inverse operations

■ algebraic check of a

solution

■ graphical check of a

solution

3.1

SCENARIO When you travel on certain bridges or roadways in

the United States, you are charged a toll or a fee to use the bridge or

roadway. The tolls are used to pay for road maintenance and safety.

Problem 1 Paying a Toll

In Virginia, travelers who cross the George P. Coleman Bridge going

northbound on Route 17 pay a toll. A two-axle vehicle pays $2 to

cross the bridge.

A. How much money is collected in tolls if 40 two-axle vehicles

cross the bridge? Use a complete sentence in your answer.

How much money is collected in tolls if 55 two-axle vehicles






Use complete sentences to explain how you found your answers

in part (A).

B. Let v represent the number of two-axle vehicles that cross the

bridge. Write an algebraic expression that represents the amount

of money collected in tolls.

C. How many two-axle vehicles crossed the bridge if $20 in tolls

were collected from two-axle vehicles? Use a complete

sentence in your answer.

Investigate Problem 11. Just the Math: Solutions of Equations In part (C),

you were actually writing and solving equations. To solve an

equation means to find the value or values of the variable that

make the equation true. Suppose that $800 were collected in

tolls from two-axle vehicles and you want to know how many

two-axle vehicles crossed the bridge. From part (B), you know

that an expression for the amount collected in tolls from two-axle

vehicles is given by 2v.

To determine v, the number of two-axle vehicles that crossed the

bridge when $800 were collected, you can write and solve the

equation .

What is the value of v that makes the equation true?

Use a complete sentence in your answer.

800 � 2v

800 � 2v


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Problem 1 Paying a Toll

How many two-axle vehicles crossed the bridge if $50 in tolls










in part (C).


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Investigate Problem 12. Just the Math: Solving One-Step Equations You may

have used mental math to solve the equation in Question 1, but

in some cases you will want to solve an equation by writing an

equivalent equation that has the variable by itself on one side of

the equation. Two equations are equivalent if they have the

same solution or solutions. There are many ways to write

equivalent equations.

Solve by writing an equivalent equation. You can do

this by adding 5 to each side of the equation.

Original equation

Add 5 to each side.

Equivalent equation

The solution of is

Solve by writing an equivalent equation. You can do

this by subtracting 7 from each side of the equation.

Original equation

Subtract 7 from each side.

Equivalent equation

The solution of is

Solve by writing an equivalent equation. You can do this

by multiplying each side of the equation by 4.

Original equation

Multiply each side by 4.

Equivalent equation

The solution of is

Solve by writing an equivalent equation. You can do this

by dividing each side of the equation by 3.

Original equation

Divide each side by 3.

Equivalent equation

The solution of is x � _____.3x � 24

� � �

3x

��

24

�

3x � 24

3x � 24

x � _____.x4

� 6

� � �

x4

� � � 6 � �

x4

� 6

x4

� 6

x � _____.x � 7 � 10

� � �

x � 7 � � � 10 � �

x � 7 � 10

x � 7 � 10

x � _____.x � 5 � 8

� � �

x � 5 � � � 8 � �

x � 5 � 8

x � 5 � 8


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Investigate Problem 13. The equations in Question 2 are one-step equations. Write a

complete sentence to explain why you think this is so.

4. In Question 2, you solved the equation by using the

operation of addition, which undoes the operation of subtraction.

Operations that undo each other are inverse operations. For

each equation below, use a complete sentence to identify the

operation you would use to get the variable by itself on one side

of the equation. Then solve the equation.

5. Complete the table of values that shows the relationship between

the amount collected in tolls and the number of two-axle vehicles

that cross the bridge.

Quantity Name

Unit

Expression

2x � 450x � 15 � 45

x � 9 � 14x6

� 5.5

x � 5 � 8

Number of two-axle vehicles Tolls collected

vehicles dollars

40

200

350

480

600


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Investigate Problem 16. Use the grid below to create a graph of the data from the table in

Question 4. First, choose your bounds and intervals. Be sure to

label your graph clearly.

7. Just the Math: Algebraic Check of a SolutionA toll taker collected $250 in tolls. The toll taker thinks he

counted 125 two-axle vehicles crossing the bridge. Did he

count correctly? The answer can be determined by checking

the solution of an equation. To algebraically check whether 125

is a solution of the equation , substitute the value for

the variable in the equation.

Substitute 125 for v:

Because both sides of the equations are equal, 125 is a solution.

Algebraically determine whether 169 is a solution of the equation

. Then, algebraically determine whether 179 is a solu-

tion of the equation . Show all your work.2v � 3582v � 358

250 � 250 2(125) � 250

2v � 250

Variable quantity Lower bound Upper bound Interval

(label) (units)

(lab

el)

(units)


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Investigate Problem 18. Just the Math: Graphical Check of a Solution

To graphically check whether 125 is a solution of the equation

, find the point on your line that corresponds to

125 vehicles. This point also corresponds to 250 dollars in tolls.

So, 125 is a solution of the equation because the tolls

collected for 125 two-axle vehicles is $250.

Graphically determine whether 250 is a solution of the equation

. Use a complete sentence to explain your reasoning.


. Use a complete sentence to explain.

9. Use complete sentences to explain which method of checking

the solution of an equation is better, algebraic or graphical.

2v � 800

2v � 500

2v � 250

2v � 250

Take NoteDraw straight lines to help

you determine whether a

value is a solution of an

equation. Draw a vertical

line up from the value of the

number of two-axle vehicles

until it intersects the line.

Then draw a horizontal line

over from the amount of tolls

collected until it intersects

the line. If you intersect the

line at the same place, then

the value is a solution of

the equation.

t

v50 100 150 200 2500 300 350 400 450 500 550 750600 650 700

100

200

300

400

0

500

600

700

800

900

1000

1100

1200

1300

1400

1500Collecting Road Tolls

Two-axle vehicles (vehicles)

Tolls

co

llecte

d (d

olla

rs) t = 2v

125

250

Take NoteWhenever you see the

share with the class icon,

your group should prepare

a short presentation to share

with the class that describes

how you solved the problem.

Be prepared to ask questions

during other groups’

presentations and to

answer questions during

your presentation.

Lesson 3.2 ■ Solving Two-Step Equations 101

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Decorating the Math LabSolving Two-Step Equations


you will:

■ Solve two-step

equations.

■ Check solutions

algebraically.

■ Check solutions

graphically.

Key Terms■ two-step equation

■ inverse operations

■ algebraic check of a

solution

■ graphical check of a

solution

SCENARIO Your class will decorate the walls of your math

computer lab with posters of great mathematicians. Your teacher

finds a catalog that sells posters for $2 each. There is an additional

charge of $4 to ship an order.

Problem 1 Finding the Total Cost

A. What will the cost be if your class orders 3 posters? Use a

complete sentence in your answer.

What will the cost be if your class orders 5 posters? Use a







in part (A).

B. Let p represent the number of posters ordered. Write an

algebraic expression that represents the total cost of the order.

3.2

Investigate Problem 11. Just the Math: Solving Two-Step Equations When

you were finding the number of posters ordered in part (C), you

were solving a two-step equation. Suppose that your class paid

$54 for posters and you want to know how many posters were

ordered. From part (B), you know that an expression for the total

cost of an order is .

To determine p, the number of posters ordered when the total

cost was $54, you can write and solve the two-step equation

.

This equation is a two-step equation because it requires two

steps in order to solve it. To solve a two-step equation, you need

to perform operations together that “undo” each other. What

operations do you need to undo in the equation to

get the variable by itself on the right side? What are the inverse

operations of these operations?

2p � 4 � 54

2p � 4 � 54

2p � 4


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Problem 1 Finding the Total Cost

C. How many posters were ordered if the cost of the order was $8?


How many posters were ordered if the cost of the order was

$16? Use a complete sentence in your answer.






in part (C).


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Investigate Problem 1The first step is to undo the addition. Write the equivalent

equation that is created when you undo the addition with the

inverse operation of subtraction. Then simplify each side of

the equation.

The second step is to undo the multiplication. Write the

equivalent equation created when you undo the multiplication

with the inverse operation of division. Then simplify each side of

the equation.

So, _____ posters were ordered.

2. For each equation below, use a complete sentence to identify the

operations that you would use to get the variable by itself on one

side of the equation and the order in which you would perform

these operations. Then solve the equation.

x3

� 5 � 1x3

� 2 � 4

1 � 2x � 75x � 3 � 2

Take NoteWhen you undo operations

to solve an equation, you

must follow the order of

operations in reverse order.

This is why you first undo the

addition and then you undo

the multiplication to solve

the equation .2p � 4 � 54


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Investigate Problem 13. Complete the table of values that shows the relationship between

the number of posters ordered and the total cost of the order.

Quantity Name

Unit

Expression

4. Use the grid below to create a graph of the data from the table

above. First, choose your bounds and intervals. Be sure to label

your graph clearly.

Number of posters ordered Total cost

posters dollars

14

18

22

30

40


(label) (units)

(lab

el)

(units)


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Investigate Problem 15. Your teacher tells you that it cost $70 for the posters. One of

your fellow classmates claims that 32 posters were ordered.

Another classmate claims that 35 posters were ordered.

You think that 33 posters were ordered.

Algebraically determine whether 32 and 33 are solutions of the

equation . Use complete sentences to explain.


. Use complete sentences to explain.


Use complete sentences to explain.

6. Determine whether 11 is a solution of the equation

graphically or algebraically. Use complete sentences to explain

the reasons for the choice of your method.

2p � 4 � 26

2p � 4 � 50.

2p � 4 � 70

2p � 4 � 70


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Lesson 3.3 ■ Using the Percent Equation 107

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Earning Sales CommissionsUsing the Percent Equation


you will:

■ Find commissions.

■ Solve percent equations.

■ Solve two-step

equations.

Key Terms■ commission

■ commission rate

■ percent equation

SCENARIO Often, part of a salesperson’s pay is made up of a

commission. A commission is an amount based on a percent of the

salesperson’s total sales. The percent is called the commission rate.

You have started working at a car dealership as a salesperson.

You earn a commission that is 2% of your total sales.

Problem 1 What Is My Commission?

A. During your first week, your total sales are $10,000.

Use a proportion to find your commission for your first week.

Show your work and use a complete sentence in your answer.

B. During your second week, your total sales are $15,000.

Use a proportion to find your commission for your second week.


C. During your third week, your total sales are $25,000.

Use a proportion to find your commission for your third week.


D. During your fourth week, your total sales are $32,000.

Use a proportion to find your commission for your fourth week.


3.3

Investigate Problem 11. Write a proportion that relates the commission, the total sales,

and the commission rate. Use the variable t to represent your

total sales and c to represent the amount of money that you

earn in commission.

Now, use means and extremes to solve the proportion.

Then, write the equation that you can use to find the commission

earned c on the total sales for any value of t. Show all

your work.

2. Just the Math: The Percent Equation The equation you

wrote in Question 1 is a percent equation. Recall that a percent

is a ratio that compares two numbers. A percent equation is an

equation of the form where p is the percent and the

numbers being compared are a and b. Use complete sentences

to explain how the percent equation is equivalent to the

proportion .

3. In each statement below, identify a, b, and p. Then write a

percent equation that represents the situation.

18 is 40% of 45.

Sixty percent of 35 is 21.

ab

�p

100

a �p

100b


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Investigate Problem 14. Write and solve a percent equation to answer each question.


Fifty is what percent of 200? What is 20% of 160?

Eight is 5% of what number? What is 40% of 85?

Problem 2 How Much Money Did I Earn?

You are working at a car dealership as a salesperson. In addition

to your 2% commission, you also earn a base salary of $215

each week.

A. How much money will you earn if your total sales for a week are

$10,000? Show all your work and use a complete sentence in

your answer.

B. How much money will you earn if your total sales for a week are


your answer.

C. How much money will you earn if your total sales for a week are


your answer.

D. How much money will you earn if your total sales for a week are


your answer.

E. Let t represent your total sales for a week and let E represent

your earnings. Write an equation for your earnings in terms of

your total sales.

Take NoteIt is common to write the

expression as a decimal

when using the percent

equation. For instance,

when the percent is 35%,

the percent equation can

be written as a � 0.35b.

p100


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Investigate Problem 21. Use your equation to find your total sales for a week if your

earnings are $300. Show all your work and use a complete


What are your total sales for a week if your earnings are $215?

Show all your work and use a complete sentence in your answer.








in Question 1.

2. What is the independent variable quantity in your equation?

What is the dependent variable quantity in your equation?


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the earnings and the total sales for a week.

Quantity Name

Unit

Expression



your graph clearly.


Total sales Earnings

0

235

5000

12,000

635

(label) (units)

(lab

el)

(units)


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Investigate Problem 25. You have heard that a different sales job offers a base salary that

is lower than your base salary, but the commission rate is higher.

Yet another sales job offers a base salary that is higher than your

base salary, but the commission rate is lower. A third sales job

offers only a straight salary. Under which conditions will each

amount of pay be the best? Would you stay at the dealership

that you are at, or try to get a job at a different dealership?

Write a paragraph to answer the questions.

Lesson 3.4 ■ Using Two-Step Equations, Part 1 113

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Rent a Car from Go-Go Car Rentals,Wreckem Rentals, and Good Rents RentalsUsing Two-Step Equations, Part 1


you will:

■ Write and solve two-step

equations.

■ Determine the better deal.

■ Compare three problem

situations.

Key Term■ two-step equation

3.4

Investigate Problem 11. Use the equation that you wrote in part (F) to find the number of

miles driven in one day if the total rental cost is $30. Show all

your work and use a complete sentence in your answer.

SCENARIO You are taking a road trip and you need to rent

a car. You look on the Internet to compare car rental prices. The first

company you find, Go-Go Car Rentals, charges $28 each day along

with a charge of $.25 per mile.

Problem 1 Renting from Go-Go Car Rentals

A. What will the cost be for one day if you drive the car 10 miles?


B. What will the cost be for one day if you drive the car 50 miles?


C. What will the cost be for one day if you drive the car 536 miles?


D. What will the cost be for one day if you drive the car 750 miles?


E. Use complete sentences to explain how you determined the cost

of the car rental in parts (A) through (D).

F. Let t represent the total cost of the car rental in dollars and let mrepresent the number of miles driven. Write an equation that

gives the total cost in terms of the number of miles driven.

Investigate Problem 1Find the number of miles driven in one day if the total rental

cost is $45. Show your work and use a complete sentence

in your answer.

Find the number of miles driven in one day if the total rental

cost is $67.50. Show your work and use a complete sentence

in your answer.

Find the number of miles driven in one day if the total rental cost is

$121.25. Show your work and use a complete sentence in your answer.


in Question 1.


the total cost and the number of miles driven.

Quantity Name

Unit

Expression



your graph clearly.


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Take NoteWhen you create your table,

be sure to only consider

distances that are reasonably

traveled in one day.

Miles driven Total cost

miles dollars



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Investigate Problem 1

4. Go-Go Car Rentals offers you a special deal. The cost for each

day is $35, but you get 500 miles for free. For any number of

miles driven above 500, you pay $.25 per mile. If you plan to

drive 20 miles in one day, is this deal better? Show all your work

and use complete sentences to explain your reasoning.

If you plan to drive 400 miles in one day, is the special deal

better? Show all your work and use complete sentences

to explain.


better? Show your work and use complete sentences

to explain.

(label) (units)

(lab

el)

(un

its)


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Problem 2 Renting from Wreckem Rentals

Another rental company, Wreckem Rentals, charges $29.95 each day

along with a charge of $.23 per mile.














miles you drove in one day if the total cost of the rental is $30.64.




in your answer.


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Investigate Problem 2Find the number of miles driven in one day if the total rental


in your answer.



in your answer.


in Question 1.



Quantity Name

Unit

Expression

3. Use the grid on the next page to create a graph of the data

from the table above. First, choose your bounds and intervals.

Be sure to label your graph clearly.

Take NoteRemember that when you

create your table, you should

only consider distances that

are reasonably traveled in

one day.



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4. Wreckem Rentals offers you a special deal. The cost for each

day is $32, but you get 250 miles for free. For any number of

miles driven above 250, you pay $.23 per mile. If you plan to

drive 20 miles in one day, is this deal better? Show all your work

and use complete sentences to explain your reasoning.


better? Show your work and use complete sentences to explain.


better? Show your work and use complete sentences to explain.

(label) (units)

(lab

el)

(un

its)


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Problem 3 Renting from Good Rents Car Rentals

The last rental company you consider, Good Rents Car Rentals,

charges $21.65 each day along with a charge of $.27 per mile.














miles you drove in one day if the total cost of the rental is $29.75.


Find the number of miles driven in one day if the total rental cost

is $33.26. Show all your work and use a complete sentence in

your answer.


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Investigate Problem 3Find the number of miles driven in one day if the total rental cost


your answer.

Find the number of miles driven in one day if the total rental cost


your answer.


in Question 1.



Quantity Name

Unit

Expression

3. Use the grid on the next page to create a graph of the data

from the table above. First, choose your bounds and intervals.

Be sure to label your graph clearly.

Take NoteRemember that when you

create your table, you should

only consider distances that

are reasonably traveled in

one day.



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4. Good Rents Car Rentals offers you a special deal. The cost for

each day is $28, but you get 150 miles for free. For any number

of miles driven above 150, you pay $.27 per mile. If you plan to

drive 20 miles in one day, is this deal better? Show all your

work and use complete sentences to explain your reasoning.



to explain.



to explain.

(label) (units)

(lab

el)

(un

its)


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Investigate Problem 35. Summarize the work that you have done so far in the

table below.

6. Use the bounds and intervals shown below to create graphs of

all three situations on the grid.

Go-Go Rentals Wreckem RentalsGood Rents

Rentals

Quantity Name

Unit

Expression


Miles driven 0 300 20

Total cost 0 150 20

(label) (units)

(lab

el)

(units)


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Investigate Problem 37. Compare the three companies’ prices from your Investigate

Problem 3 Question 6 graph. Prepare a written report that

indicates which company will cost the least and for which

numbers of miles. Can you tell whether the cost will ever be

the same for all three companies? If so, for which number

of miles? Use complete sentences to explain.


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Plastic ContainersUsing Two-Step Equations, Part 2


you will:

■ Write and use two-step

equations.

■ Compare three problem

situations.

Key Term■ two-step equation

SCENARIO Your job at the Nikkerware Company is to create

new packaging and shipping boxes for its plastic containers. The

company makes all different shapes and sizes of plastic containers.

To ship the containers, the lids are removed to allow for stacking.

The company wants to design shipping boxes that will hold two

dozen stacks of the plastic containers without the lids, regardless of

the size or shape of the container.

Problem 1 Different Shapes and Sizes

The table below shows the data gathered from measuring the heights

of different-sized stacks of the various plastic containers.

If you have some stackable plastic containers in your classroom,

you should add the data about them to this table.

A. What is the difference in heights between a stack of two round

containers and a stack of one round container? Use a complete


B. What is the difference in heights between a stack of three

round containers and a stack of two round containers?


C. What is the difference in heights between a stack of four

round containers and a stack of three round containers?


3.5

Number of

containersStack height (centimeters)

Round Square Rectangular

1 5.4 4.5 14.6

2 5.9 6.2 14.9

3 6.4 7.9 15.2

4 6.9 9.6 15.5

5 7.4 11.3 15.8

Investigate Problem 11. What is the height of 10 round containers? Show all your work

and use a complete sentence in your answer.

What is the height of 15 round containers? Show all your work


What is the height of 20 round containers? Show all your work



in Question 1.


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Number of

containers

Stack height

(cm)

1 5.4 4.514.6

2 5.9 6.214.9

3 6.4 7.915.2

4 6.9 9.615.5

5 7.4 11.315.8

Problem 1 Different Shapes and Sizes

D. What is the difference in heights between a stack of five

round containers and a stack of four round containers?


E. How does the stack height change when one round container is

added to a stack of round containers? Use a complete sentence

in your answer.

F. How does the stack height change when one square container is

added to a stack of square containers? Use complete sentences

to explain how you found your answer.

G. How does the stack height change when one rectangularcontainer is added to a stack of rectangular containers?

Use complete sentences to explain how you found your answer.

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Investigate Problem 12. If you use c to represent the number of containers in a stack of

round containers, what does the expression represent?

Use a complete sentence to explain.

3. Let h represent the height of a stack. Write an equation that

gives the height of a stack of round containers in terms of the

number of containers c in the stack.

4. Use your equation to find the height of a stack of two dozen

round containers. Show your work and use a complete sentence

in your answer.

5. Let c represent the number of containers in a stack of squarecontainers and let h represent the height of the stack. Write an

equation that gives the height of the stack in terms of the number

of containers in the stack.

6. What is the height of a stack of two dozen square containers?

Show all your work and use a complete sentence to explain

your reasoning.

7. Let c represent the number of containers in a stack of rectangularcontainers and let h represent the height of the stack. Write an

equation that gives the height of the stack in terms of the number

of containers in the stack.

8. What is the height of a stack of two dozen rectangular containers?

Show all your work and use a complete sentence to explain

your reasoning.

c � 1


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Investigate Problem 19. How are the three equations you wrote similar? Why are these

equations similar? Use complete sentences to explain.

10. How are the three equations you wrote different? Why are these

equations different? Use complete sentences to explain.

11. In all equations, what are the variable quantities? Use a


What is the relationship between these quantities? In other

words, which variable quantity depends on the other?

12. Does it make sense for the company to have a stack of zero

containers? Why or why not? Use complete sentences to

explain your reasoning.

Use your answer to explain why each of these graphs on the

next page starts at 1.

13. Use the grid on the next page to create graphs for each

container shape’s stack height in terms of the number of

containers used. Use the bounds and intervals indicated in

the table. Be sure to label your graph clearly.


Containers 0 75 5

Stack height 0 75 5

Number of

containers

Stack height

(cm)

1 5.4 4.514.6

2 5.9 6.214.9

3 6.4 7.915.2

4 6.9 9.615.5

5 7.4 11.315.8

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14. The square container is the shortest container. Find the number(s)

of containers for which the height of a stack of square containers

is the shortest. Use complete sentences to explain why the

square container stack height is not always the shortest.

15. Find the number(s) of containers for which a stack of round

containers is the shortest. Use complete sentences to explain

why a stack of any number of round containers does not remain

the shortest.

(label) (units)

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Investigate Problem 116. Find the number(s) of containers for which a stack of rectangular

containers is the shortest. Use complete sentences to explain

why a stack of any number of rectangular containers does not

remain the shortest.

17. Write a memo to your supervisor in which you recommend

the best shipping carton sizes for each type of container.

Use complete sentences to explain your reasoning.

Lesson 3.6 ■ Integers and Integer Operations 131

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Brrr! It’s Cold Out There!Integers and Integer Operations


you will:

■ Graph integers on a

number line.

■ Compare integers.

■ Add, subtract, multiply,

and divide integers.

■ Write fractions as

decimals.

■ Write decimals as

fractions.

Key Terms■ integer

■ negative integer

■ positive integer

■ number line

■ quotient

SCENARIO You are working on a report about winter weather

and you are studying the wind-chill temperature. The wind-chill

temperature is a measure of how wind speed affects low tempera-

tures and how the combination of wind and low temperature feels to

a human being.

You find a table of some wind chill temperatures when the wind

speed is 5 miles per hour.

Problem 1 Wind Chill Temperatures

The numbers in the table are integers. The numbers …, –4, –3, –2,

–1, 0, 1, 2, 3, 4, … are integers. Integers include zero, negative

integers (integers less than zero), and positive integers (integers

greater than zero).

A. Use a complete sentence to describe the effect that the wind has

on actual temperatures.

B. One way to show the relationship between integers is to graph

them on a number line. A number line is a line that extends in

both directions forever. One point is assigned a value of zero

and a given length is assigned a length of one unit. Positive

integers are assigned to the units on the right of zero and

negative integers are assigned to the units on the left of zero.

Use the number line below to graph each of the wind chill

temperatures in the table. The first wind chill temperature is

done for you.

C. On the number line, the values of integers increase as you move

from left to right. Write the wind chill temperatures in order from

least to greatest.

1211109876543210−1−2−3−5−6−7−8−9−10 −4

3.6

Actual temperature

(degrees Fahrenheit)15 10 5 0 –5

Wind chill temperature

(degrees Fahrenheit)12 7 0 –5 –10

Take NoteThe symbol … is an ellipsis.

It implies the continuation

of a pattern.

For instance,

1, 2, 3, 4, …

implies that you should

continue counting by 1s.

Take NoteTwo integers are opposites if

they are on opposite sides of

0, but are the same distance

from 0. For example, 4 and

–4 are opposites. You will

learn more about opposites

in Chapter 4.

Investigate Problem 11. When the actual temperature is 10°F and the wind speed is

15 miles per hour, the wind chill temperature is –18°F. When the

wind speed is 30 miles per hour at the same actual temperature,

the wind chill temperature is –33°F. Which wind chill temperature

is higher? Use a complete sentence to explain.

When the actual temperature is 5°F and the wind speed is


wind speed is 35 miles per hour at the same actual temperature,

the wind chill temperature is –21°F. Which wind chill temperature

is lower? Use a complete sentence to explain.

2. When the actual temperature is 25°F and the wind speed is

15 miles per hour, the wind chill temperature is 2°F. When

the wind speed decreases to 10 miles per hour, the wind chill

temperature increases by 8 degrees. Use a number line to find

the wind chill temperature when the wind speed is 10 miles per

hour. Use a complete sentence to explain.

When the actual temperature is 10°F and the wind speed is


wind speed decreases to 5 miles per hour, the wind chill

temperature increases by 5 degrees. Use a number line to find

the wind chill temperature when the wind speed is 5 miles per

hour. Use a complete sentence to explain.

−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8

−1 0 1 2 3−2 4 5 6 7 8 9 1310 11 12−4 −3

−23 −22 −21 −20 −19−24 −18 −17 −16 −15 −14 −13 −9−12 −11 −10−26 −25

−31 −30 −29 −28 −27−32 −26 −25 −24 −23 −22 −21 −17−20 −19 −18−34 −33


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Investigate Problem 13. Just the Math: Adding Negative Integers

In Question 2, you were adding a positive integer to another

integer. Complete the following instructions to add a negative

integer to an integer.

To add 4 and –7, start at 4. Then move 7 units to the left.

Represent this on the number line by graphing 4 on the number

line and then by drawing an arrow that starts at 4 and ends

7 units to the left of 4. Where are you on the number line?

Write an integer addition equation that represents this situation.

To add –2 and –5, start at –2. Then move 5 units to the left.

Represent this on the number line by graphing –2 on the number

line and then by drawing an arrow that starts at –2 and ends

5 units to the left of –2. Where are you on the number line?

Write an integer addition equation that represents this situation.

Complete the rule below for using a number line to add two

integers.

On a number line, move to the ________ when you add a

positive integer, and move to the ________ when you add a

negative integer.

4. When the actual temperature is 15°F and the wind speed is

10 miles per hour, the wind chill temperature is 3°F. When the

wind speed increases to 20 miles per hour, the wind chill

temperature decreases by 5 degrees. Use a number line to

find the wind chill temperature when the wind speed is 20 miles

per hour. Use complete sentences to explain how you found

your answer.

−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8

−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8

−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8


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Investigate Problem 1When the actual temperature is 0°F and the wind speed is


wind speed increases to 10 miles per hour, the wind chill tempera-

ture decreases by 5 degrees. Use a number line to find the

wind chill temperature when the wind speed is 10 miles per hour.

Use complete sentences to explain how you found your answer.

5. Just the Math: Subtracting Negative Integers Recall

that when you add a positive integer on a number line you move

to the right, when you add a negative integer on a number line,

you move to the left. When you subtract a positive integer, which

way do you think that you should move on a number line?

When you subtract a negative integer on a number line, which

way should you move? Use complete sentences to explain

your reasoning.

Complete the following instructions to subtract a negative integer

from an integer.

To subtract –4 from 8, start at 8. Then subtract –4. Where are

you on the number line?

Write an integer subtraction equation that represents this

situation.

To subtract –5 from –3, start at –3. Then subtract –5. Where are

you on the number line?

Write an integer subtraction equation that represents this

situation.

−6 −5 −4 −3 −2−7 −1 0 1 2 3 4 85 6 7−9 −8

3 4 5 6 72 8 9 10 11 12 13 1714 15 160 1

−17 −16 −15 −14 −13−18 −12 −11 −10 −9 −8 −7 −3−6 −5 −4−20 −19


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Investigate Problem 1Complete the rule below for using a number line to subtract two

integers.

On a number line, move to the ________ when you subtract a

positive integer, and move to the ________ when you subtract a

negative integer.

6. Just the Math: Multiplying Integers Recall that you

can think of multiplication as repeated addition. Complete each

statement below.

What is the effect of multiplying a positive integer by –1?

Use complete sentences in your answer.

What do you think will be the effect of multiplying a negative

integer by –1?

Find each product.

What do you think is the product of –1 and –1? Use a complete

sentence to explain your reasoning.

Find each product.

0(�4) � �

�1(�4) � �1(�4) � �

�2(�4) � �2(�4) � �

�3(�4) � �3(�4) � �

(�1) (4) (�1)(�1) (4)

4(�2) � (�2) � (�2) � (�2) � (�2) � �

(�1) (5) � (�1) � (�1) � (�1) � (�1) � (�1) � �

2(�1) � (�1) � (�1) � �

3(�5) � (�5) � (�5) � (�5) � �


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Investigate Problem 1Complete the statements below about the product of

two integers.

The product of two positive integers is a __________ integer.

The product of two negative integers is a __________ integer.

The product of a positive integer and a negative integer is a

__________ integer.

The product of a negative integer and a positive integer is a

__________ integer.

7. Just the Math: Dividing Integers You can find the

quotient of two integers by using what you know about multipli-

cation. For instance, if we divide 35 by 5, the result is 7 because

. Find each quotient by writing a related multiplica-

tion problem.

because

because

because

because

Complete the statements below about the quotient of

two integers.

The quotient of two positive integers is a __________ integer.

The quotient of two negative integers is a __________ integer.

The quotient of a positive integer and a negative integer is a

__________ integer.

The quotient of a negative integer and a positive integer is a

__________ integer.

8. Perform the indicated operations.

14 � (�2)�56 � (�7)�18 � 3

(�1)3(�3)24(�9)

�5 � (�6)�3 � (�4)7 � (�12)

�6 � � � 48.48 � (�6) � �

�6 � � � �48.�48 � (�6) � �

6 � � � �48.�48 � 6 � �

6 � � � 48.48 � 6 � �

7 � 5 � 35

Take NoteRecall that the result of

dividing two numbers is

the quotient.


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Problem 2 Celsius Temperature Scale

As part of your report, you will also write your information about

wind chill temperatures in degrees Celsius. An equation that relates

a temperature C in degrees Celsius to a temperature F in degrees

Fahrenheit is

A. Use the equation to find the temperature in degrees Celsius that

corresponds to a temperature of 23°F. Show all your work and

use a complete sentence in your answer.

B. Use the equation to find the temperature in degrees Fahrenheit

that corresponds to a temperature of –10°C. Show all your work


F �95

C � 32.

Investigate Problem 21. Just the Math: Writing a Fraction as a Decimal

When you use the equation above to write 12°C in degrees

Fahrenheit, you get To write this

temperature properly, we need to write it as a decimal instead of

as a fraction. To write a fraction as a decimal, use long division.

53.6

18

3 0

Use long division to write each fraction as a decimal.

45

72

15

255)268.0

F �95

(12) � 32 �2685

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Investigate Problem 22. Just the Math: Writing a Decimal as a Fraction

To write a decimal as a fraction, first determine the name for

the last decimal place of the number that is to the right of the

decimal point. Then use the name to write the decimal as

the appropriate fraction.

hundredths

Write each decimal as a fraction. If possible, simplify

your answer.

0.9 0.08 0.125

3. Write the temperature equation so that the fraction is a decimal.

Is this equation easier to use? Use complete sentences to


4. Convert each temperature to degrees Celsius. Show all

your work.

5ºF –4ºF

Convert each temperature to degrees Fahrenheit. Show all

your work.

8°C –2°C

0.17 �17100

Take NoteRemember that the first

decimal place to the right

of the decimal point is

tenths, the second decimal

place is hundredths, and

the third decimal place is

thousandths.

Lesson 3.7 ■ The Coordinate Plane 139

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Shipwreck at the Bottom of the SeaThe Coordinate Plane


you will:

■ Draw a coordinate

system.

■ Write ordered pairs.

■ Plot points in a

coordinate plane.

Key Terms■ Cartesian coordi-

nate system

■ x-axis

■ y-axis

■ origin

■ coordinate plane

■ ordered pair

■ x-coordinate

■ y-coordinate

3.7

SCENARIO You work for a company that salvages shipwrecks.

Items such as pieces of the ship and cargo from the ship are called

debris. Part of your job is to examine the debris field, or the area in

which items that are related to the shipwreck can be found. Once the

debris field is identified, a grid is placed over the field so that the

relative positions of the items can be recorded.

Problem 1 Searching the Debris Field

The figure below shows the debris field and the accompanying grid at

a wreck that you are exploring.

Use complete sentences to describe the location of the anchor’s

center in relation to the ship’s center.

Use complete sentences to describe the location of the center of the

gold coins in relation to the ship’s center.

Investigate Problem 11. Just the Math: Cartesian Coordinate System In the

previous lesson, you used number lines to graph integers. If we

intersect two number lines at right angles as shown below, we

form the Cartesian coordinate system. The horizontal number

line is the x-axis and the vertical number line is the y-axis.

The point at which the axes cross is the origin. The intersection

of these two number lines forms a coordinate plane. Label the

x-axis, the y-axis, and the origin in the coordinate plane below.

x−6 −5 −4 −3 −2−7 −1 O 1 2 3 4 5 6 7

−3

y

−7

−6

−5

−4

−2

−1

1

2

3

4

5

6

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Problem 1 Searching the Debris Field

Use complete sentences to describe the location of the cannonball’s

center in relation to the ship’s center.

Use complete sentences to describe the location of the flag’s center

in relation to the ship’s center.

Take NoteThe plural of axis is axes.

Lesson 3.7 ■ The Coordinate Plane 141

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Investigate Problem 12. Just the Math: Ordered Pairs Each point on a graph can

be represented by an ordered pair. An ordered pair consists of

two numbers. The first number, the x-coordinate, indicates how

far to the left or right the point is from the (vertical) y-axis. The

second number, the y-coordinate, indicates how far up or down

the point is from the (horizontal) x-axis. An ordered pair is written

as (x-coordinate, y-coordinate), such as (3, –5).

For each item, write the ordered pair that represents the location

of the item’s center.

Draw the x- and y-axes on the grid below so that the origin

is the ship’s center. For each item, plot the ordered pair that

represents the location of the item’s center on the grid.


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Investigate Problem 13. Plot and label each point in the coordinate plane.

A(5, 2)

B(–3, 0)

C(6, –3)

D(1, 6)

E(–4, 4)

F(–2, –6)

x−6 −5 −4 −3 −2−7 −1 O 1 2 3 4 5 6 7

−3

y

−7

−6

−5

−4

−2

−1

1

2

3

4

5

6

7

Lesson 3.8 ■ Using a Graph of an Equation 143

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Engineering a HighwayUsing a Graph of an Equation


you will:

■ Write and use a two-step

equation.

■ Use a graph to estimate

solutions of equations.

Key Term■ estimate

3.8

Investigate Problem 11. Find the number of miles that are completed in 10 days.


Find the number of miles that are completed in 50 days.


Find the number of miles that are completed in 1 year.


Find the number of miles that are completed in 2 years.


SCENARIO You are a civil engineer who is overseeing a large

interstate-highway construction project that will connect the cities of

Pittsburgh, Pennsylvania, and Cincinnati, Ohio. Construction began

several months ago, and approximately 87 miles of highway have

been completed. The total length of the highway will be 267 miles.

You estimate that new construction can proceed at the rate of

one-fifth mile per day.

Problem 1 Construction Work

A. What are the two quantities that are changing?

B. Write an equation to model the situation.

Investigate Problem 1Use a complete sentence to explain how you found your answers

in Question 1.

2. Find the number of days it will take to complete 100 miles of

highway. Show all your work and use a complete sentence in

your answer.

Find the number of days it will take to complete 150 miles of


your answer.

Find the number of days it will take to complete the entire


your answer.


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Investigate Problem 1Use complete sentences to explain how you found your answers

in Question 2.

Write an equation that relates the two quantities that are

changing.

3. Use your equation to find the number of days it will take to

complete 50 miles of highway. Show all your work and use

a complete sentence to explain your reasoning.

Use complete sentences to explain the solution of the equation

in terms of the problem situation.

4. How many days before new construction began was the project

started? Show all your work and use complete sentences to



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the number of days and the number of miles that are completed.

Quantity Name

Unit

Expression



your graph clearly.

Take NoteWhen you are choosing

your bounds, remember to

consider when the highway

was started and how many

days it will take to complete

the highway.


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Investigate Problem 17. Use your graph to estimate when 30 miles of highway were

completed. Use a complete sentence in your answer.

Use your graph to estimate when 60 miles of highway were






Use your graph to estimate the number of miles that

were completed 50 days before new construction started.



were completed 100 days before new construction started.



were completed 200 days after new construction started.



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Investigate Problem 18. Some of the other engineers who are working on this project also

responded to the question and several of them had responses

that were different from yours. Use complete sentences to

explain why this could occur.

9. Prepare an overview of the project that will convince the project

manager that this project is on schedule and that the project is

running smoothly.

looking ahead to chapter 3 - somersetcentral.enschool.org

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