untitled

Name ________________________________________ Date __________________ Class __________________

209

Writing Equations to Represent Situations

Practice and Problem Solving: A/B

Determine whether the given value is a solution of the equation. Write yes or no.

1. x + 11 = 15; x = 4

_________________ 2. 36 − w = 10; w = 20 _________________

3. 0.2v = 1.2; v = 10

_________________ 4. 15 = 6 + d; d = 8 _________________

5. 28 − w = 25; w = 3

_________________ 6. 4t = 32; t = 8 _________________

7.

12

s

= 4; s = 3 _________________ 8.

33

p

= 3; p = 11 _________________

Circle the letter of the equation that each given solution makes true.

9. m = 19 10. a = 16

A 10

+ m = 20 C 7m = 26 A 2a = 18 C 24 − a = 6

B m

− 4 = 15 D

18

m

= 2 B a + 12 = 24 D

4

a

= 4

Write an equation to represent each situation.

11. Seventy-two people signed up for the 12. Mary covered her kitchen floor with

soccer league. After the players were 10 tiles. The floor measures 6 feet long

evenly divided into teams, there were by 5 feet wide. The tiles are each 3 feet

6 teams in the league and x people on long and w feet wide.

each team.

_____________________________________ _____________________________________

Solve.

13. The low temperature was 35°F. This was 13°F lower than the daytime

high temperature. Write an equation to determine whether the high

temperature was 48

°F or 42°F.

_________________________________________________________________________________________

14. Kayla bought 16 bagels. She paid a total of $20. Write an equation to

determine whether each bagel cost $1.50 or $1.25.

_________________________________________________________________________________________

15. Write a real-world situation that could be modeled by the equation

24

y

= 3. Then solve the problem.

_________________________________________________________________________________________

LESSON

11-1

Name ________________________________________ Date __________________ Class __________________

210

Writing Equations to Represent Situations

Practice and Problem Solving: C

Circle the letter of the value that makes each equation true.

1.

18

m

= 15 − 12 2. 6d = 8(12 − 6)

A m = 6 C m = 9 A d = 18 C d = 8

B m = 3 D m = 2 B d = 48 D d = 4

3. x =

14 6

2

−

4.

4

a

= 3(10 ÷ 2)

A x = 6 C x = 16 A a = 15 C a = 40

B x = 8 D x = 4 B a = 60 D a = 20

For Exercises 5–7, use the table at the right that shows

how many minutes certain mammals can stay under water.

5. A sperm whale can stay under water 7 times as long as

x minutes more than a platypus can. Write an equation

that states the relationship of the minutes these two

mammals can stay under water.

____________________________________________

6. A sea cow can stay under water y minutes. This is 11 minutes

longer than one-third the time a hippopotamus can. Write an

equation that states the relationship of the minutes these two

mammals can stay under water. Complete the table with 16 or 56.

____________________________________________

7. Write an equation that includes division that relates the number

of minutes a seal can stay under water to the number of minutes

a sperm whale can stay under water.

____________________________________________

Solve.

8. Mr. Sosha teaches 4 math classes, with the same number of students

in each class. Of those students, 80 are sixth graders and 40 are fifth

graders. Write an equation to determine whether there are 22, 25, or

30 students in each class. How many are in each class?

_________________________________________________________________________________________

9. Write an equation that involves multiplication, addition, contains a

variable, and has a solution of 8.

_________________________________________________________________________________________

LESSON

11-1

Animal Min

Hippopotamus 15

Platypus 10

Sea Cow

Seal 22

Sperm Whale 112

Name ________________________________________ Date __________________ Class __________________

211

4 + 1 = 5

Writing Equations to Represent Situations

Practice and Problem Solving: D

Is the given value of the variable a solution of the equation?

Write yes or no. The first one is done for you.

1. x + 1 = 5; x = 4

_________________ 2. 13 − w = 10; w = 2 _________________

3. 2v = 12; v = 10

_________________ 4.

14

2;

p

=

p = 7 _________________

5. 8 + w = 11; w = 3

_________________ 6. 4t = 20; t = 5 _________________

Circle the letter of the equation that each given solution makes true.

The first one is done for you.

7. x = 5 8. g = 7

A 2 + x = 7 A 9g = 16

B 9 − x = 3 B 8 − g = 1

C 3x = 18 C 11 + g = 17

9. y = 2 10. m = 9

A 7 − y = 1 A m − 4 = 13

B 3y = 6 B 7m = 36

C

10

20

y

=

C

18

2

m

=

11. z = 4 12. a = 8

A 5z = 20 A 2a = 10

B

12

4

z

=

B a + 12 = 20

C z − 3 = 7 C

4

a

=

13. Rhonda has $13. She has one $5 bill, three $1 bills, and one

other bill. Is the other bill a $1 bill or a $5 bill? Explain.

_________________________________________________________________________________________

LESSON

11-1

2 + 5 = 7

yes

________

+

(1)$5

(3)

________

+

=

Other bill

$

________

Name ________________________________________ Date __________________ Class __________________

212

Writing Equations to Represent Situations

Reteach

An equation is a mathematical sentence that says that two

quantities are equal.

Some equations contain variables. A solution for an equation is a

value for a variable that makes the statement true.

You can write related facts using addition and subtraction.

7 + 6 = 13 13 − 6 = 7

You can write related facts using multiplication and division.

3 • 4 = 12

12

3

4

=

You can use related facts to find solutions for equations. If the related

fact matches the value for the variable, then that value is a solution.

A. x + 5 = 9; x = 3 B. x − 7 = 5; x = 12

Think: 9 − 5 = x Think: 5 + 7 = x

4 = x 12 = x

4 ≠ 3 12 = 12

3 is

not a solution of x + 5 = 9. 12 is a solution of x − 7 = 5.

C. 2x = 14; x = 9 D.

3

5

x

=

; x = 15

Think: 14 ÷ 2 = x Think: 3 • 5 = x

7 = x 15 = x

7 ≠ 9 15 = 15

9 is not a solution of 2x = 14. 15 is a solution of x ÷ 5 = 3.

Use related facts to determine whether the given value is a

solution for each equation.

1. x + 6 = 14; x = 8 2. 5;

4

s

=

s = 24 3. g − 3 = 7; g = 11

_________________ _________________ _________________

4. 3a = 18; a = 6 5. 26 = y − 9; y = 35 6. b • 5 = 20; b = 3

_________________ _________________ _________________

7. 15 ;

3

v

= v = 45 8. 11 = p + 6; p = 5 9. 6k = 78; k = 12

_________________ _________________ _________________

LESSON

11-1

Name ________________________________________ Date __________________ Class __________________

213

Writing Equations to Represent Situations

Reading Strategies: Build Vocabulary

You can see part of the word equal in equation. In math, an equation

indicates that two expressions have the same value, or are equal.

The = sign in an equation separates one expression from the other.

The value on each side of the = sign is the same.

Look at the equations below. Notice how the value on each side of

the = sign is the same for each equation:

5 + 7 = 8 + 4 19 − 7 = 12 42 = 3 • 14

If an equation contains a variable, and the variable is replaced by a value

that keeps the equation equal, that value is called a solution of the equation.

Determine whether 80 or 60 is a solution to

15

4

y

=

?

15

4

80

15

4

20 15

y

=

?

15

4

60

15

4

15 15

y

=

“20 is

not equal to 15.” “15 is equal to 15.”

Which are equations? Write yes or no.

1. 7 + 23

?

= 9 + 21 _________________

2. 35 + 15

?

= 45 _________________

3. 28 − 7

?

= 15 + 6 _________________

Replace the given value for the variable. Is it a solution? Write yes or no.

4. d + 28 = 45; d = 17 5.

84

28;

s

= s = 3

________________________________________ ________________________________________

6. 17 = 56 − t; t = 40 7. 86 = 4w; w = 24

________________________________________ ________________________________________

Solve.

8. Use the numbers 2, 11, 13, and 15 to write an equation.

_________________________________________________________________________________________

9. Replace one of the numbers in your equation in Exercise 8 with the

variable y. Determine whether 2, 11, 13, or 15 is a solution of your

equation.

_________________________________________________________________________________________

LESSON

11-1

Name ________________________________________ Date __________________ Class __________________

214

Writing Equations to Represent Situations

Success for English Learners

Problem

Determine whether 61 or 59 is a solution of the equation a + 23 = 82.

1. How do you know that 61 is not a solution of a + 23 = 82?

_________________________________________________________________________________________

2. How can you find out whether 65 is a solution of a + 23 = 82?

_________________________________________________________________________________________

3. Write a real-world situation that could be modeled by a + 23 = 82.

_________________________________________________________________________________________

LESSON

11-1

Name ________________________________________ Date __________________ Class __________________

215

Addition and Subtraction Equations

Practice and Problem Solving: A/B

Solve each equation. Graph the solution on the number line.

1. 6 = r + 2 r =

____

2. 26 = w − 12 w =

____

3.

1

2

= m −

1

8

m = ____

4.

t + 1 = −3 t = ____

Use the drawing at the right for Exercises 5–6.

5. Write an equation to represent the measures of the angles.

_____________________________________

6. Solve the equation to find the measure of the unknown angle.

_____________________________________

Use the drawing at the right for Exercises 7–8.

7. Write an equation to represent the measures of the angles.

_____________________________________

8. Solve the equation to find the measure of the unknown angle.

_____________________________________

Write a problem for the equation 3 + x = 8. Then solve the

equation and write the answer to your problem.

9. _______________________________________________________________________________________

_________________________________________________________________________________________

LESSON

11-2

Name ________________________________________ Date __________________ Class __________________

216

Addition and Subtraction Equations

Practice and Problem Solving: C

Solve each equation.

1. b + 2.3 = 5.7 b =

____

2. s −

1

3

=

4

9

s = ____

3. 6

1

2

+ n = 12 n = ____

4. 15.35

= z − 1.84 z = ____

5.

d + (−3) = −7 d = ____ 6. 12 = g + 52 g = ____

Use the drawing at the right for Exercises 7–8.

7. Write an equation to represent the measures of the angles.

_____________________________________

8. Solve the equation to find the measure of the unknown angle.

_____________________________________

Write and solve an equation to answer each question.

9. Kayla is 13 years old. Her uncle says that his age minus 22 is equal to

Kayla’s age. How old is Kayla’s uncle?

_________________________________________________________________________________________

10. Gavin wants to buy a jacket that sells for $38.95. An advertisement

says that next week that jacket will be on sale for $22.50. How much

will Gavin save if he waits until next week to buy the jacket?

_________________________________________________________________________________________

11. Sierra sawed 10

1

2

inches off the end of a board. The remaining board

was 37

1

2

inches long. How long was the board that Sierra started with?

_________________________________________________________________________________________

Write a problem for the equation 4.65 = x − 2.35. Then solve the

equation and write the answer to your problem.

12. _______________________________________________________________________________________

_________________________________________________________________________________________

LESSON

11-2

Name ________________________________________ Date __________________ Class __________________

217

Addition and Subtraction Equations

Practice and Problem Solving: D

Solve each equation. Graph the solution on the number line.

The first one is done for you.

1. 5 = r − 1 r = ____

51

11

6

r

=−

++

=

2. 2 = w + 3 w = ____

3. 5

= m + 2 m = ____

4. t

− 5 = 0 t = ____

Use the drawings at the right for Exercises 5–6. The first one has

been done for you.

5. Write an equation to represent the measures of the angles.

_____________________________________

6. Solve the equation to find the measure of the unknown angle.

_____________________________________

7. Mayumi has the boxes shown at the right. The total

number of objects in two of the boxes is the same

as the number of objects in the third box. Write an

equation to show the relationship of the number of

objects in the boxes.

_____________________________________

8. How many objects are in the box marked n?

____ objects

Write a problem for the equation x − 5 = 2. Then solve the equation

and write the answer to your problem.

9. _______________________________________________________________________________________

_________________________________________________________________________________________

LESSON

11-2

6

x + 100 = 180

Name ________________________________________ Date __________________ Class __________________

218

Addition and Subtraction Equations

Reteach

To solve an equation, you need to get the variable alone on one side of

the equal sign.

You can use tiles to help you solve

subtraction equations.

Addition undoes

subtraction, so you

can use addition to

solve subtraction

equations.

One positive tile and

one negative tile

make a zero pair.

Zero pair: +1 + (−1) = 0

Variable add 1 subtract 1

add 1

make

zero

subtract 1

To solve x − 4 = 2, first use

tiles to model the equation.

X − 4 = 2

To get the variable alone, you have to add positive tiles. Remember to

add the same number of positive tiles to each side of the equation.

x − 4 + 4 = 2 + 4

Then remove the greatest possible number of zero pairs from each side of the equal sign.

x = 6

Use tiles to solve each equation.

1. x − 5 = 3 2. x − 2 = 7 3. x − 1 = 4

x =

____ x = ____ x = ____

4. x − 8 = 1 5. x − 3 = 3 6. x − 6 = 2

x = ____ x = ____ x = ____

LESSON

11-2

The remaining

tiles represent

the solution.

x = 6

−1

+1

−1

Name ________________________________________ Date __________________ Class __________________

219

Addition and Subtraction Equations

Reading Strategies: Use a Visual Clue

You can picture balanced scales to solve subtraction equations.

Picture balanced scales for this equation.

Step 1: To find the value of b, get b by itself on the left

side of the equation. So, add 17 to the left side

of the equation.

Step 2: To keep the equation balanced, add 17 to the

right side of the equation as well.

Step 3: Check to verify that b = 82 is the solution.

b − 17 = 65

82 − 17

?

=

65

?

= 659

To get the variable by itself in a subtraction equation,

add the same value to both sides of the equation.

Use n

− 21 = 32 to answer Exercises 1–4.

1. On which side of the equation is the variable?

___________________________

2. What will you do to get the variable by itself?

___________________________

3. What must you do the other side of the

equation to keep it balanced?

___________________________

4. What is the value of n?

___________________________

Use 12

= p − 25 to answer Exercises 5–8.

5. On which side of the equation is the variable?

___________________________

6. What will you do to get the variable by itself?

___________________________

7. What must you do the other side of the

equation to keep it balanced?

___________________________

8. What is the value of p?

___________________________

LESSON

11-2

Name ________________________________________ Date __________________ Class __________________

220

Addition and Subtraction Equations

Success for English Learners

Problem 1

The surfboard is 14 inches

taller than the person.

How tall is the person?

14 82

14 14

68

h

+=

−−

=

Subtract 14 from both sides.

68

+ 14 = 82 9 Check your answer.

Problem 2

21 36

21 21

57

x

−

=

++

=

Add 21 to both sides.

1. Why do you use an addition equation to find the surfer’s height?

_________________________________________________________________________________________

2. How can you check the answer to Problem 2?

_________________________________________________________________________________________

3. Write an addition or a subtraction equation. Explain how to solve your

equation. Give the solution to your equation.

_________________________________________________________________________________________

Surfboard height

Height of person

The person is

68 inches tall.

Think: This is a

subtraction equation.

I add to undo the

subtraction.

Think: This is

an addition

equation.

I subtract to

undo the

addition.

LESSON

11-2

Name ________________________________________ Date __________________ Class __________________

221

Multiplication and Division Equations

Practice and Problem Solving: A/B

Solve each equation. Graph the solution on the number line.

Check your work.

1.

2

e

= 3 e = ____

_________________________________________________________________________________________

2. 20 = 2w w = ____

_________________________________________________________________________________________

3.

1

2

= 2m m = ____

_________________________________________________________________________________________

4.

5

k

= 2 k = ____

_________________________________________________________________________________________

Use the drawing at the right for Exercises 5–6.

5. Write an equation you can use to find the length of the rectangle.

_____________________________________

6. Solve the equation. Give the length of the rectangle.

_____________________________________

Solve.

7. Alise separated her pictures into 3 piles. Each pile contained

9 pictures. How many pictures did she have in all? Write and solve an

equation to represent the problem. State the answer to the problem.

_________________________________________________________________________________________

LESSON

11-3

Name ________________________________________ Date __________________ Class __________________

222

Multiplication and Division Equations

Practice and Problem Solving: C

Solve each equation.

1. 8b = 5.6 b =

____

2. 9 =

3

s

s = ____

3. 2

1

2

= 5n n = ____

4. 15

= 0.2z z = ____

5. 3.5d = 70 d = ____

6.

4

39

t

=

t = ____

7. The perimeter of the square at the right is 48 inches. What is the area

of the square at the right? Explain how you found your answer.

_________________________________________________________________

Write and solve an equation to answer each question.

8. Jose is making model SUVs. Each SUV takes 5 tires. He used 85 tires

for the models. How many model SUVs did Jose make?

_________________________________________________________________________________________

9. Renee talked for 6 minutes on the phone. Nathan talked for n minutes.

Nathan talked three times as long as Renee. How long did Nathan talk?

_________________________________________________________________________________________

10. Sylvia rented a boat for $16.50 per hour. Her total rental fee was

$49.50. For how many hours did Sylvia rent the boat?

_________________________________________________________________________________________

Write a problem for the equation 0.5n = 12.5. Then solve the equation

and write the answer to your problem.

11. _________________________________________________________________________

_________________________________________________________________________________________

LESSON

11-3

Name ________________________________________ Date __________________ Class __________________

223

Multiplication and Division Equations

Practice and Problem Solving: D

Solve each equation. Graph the solution on the number line.

Check your work. The first is done for you.

1. 8 = 2m m =

____

82

22

m

=

4 = m 8 = 2 • 4

9

2.

4

a

= 2 a =

____

_________________________________________________________________________________________

3. 12 = 3s s = ____

_________________________________________________________________________________________

4.

2

u

= 5 u =

____

_________________________________________________________________________________________

Use the situation below to complete Exercises 5–8.

The first one is done for you.

Jim knows the length of his garden is 12 feet. He knows the area of the

garden is 60 ft

2

. What is the width of Jim’s garden?

5. Fill in the known values in the picture at the right.

6. Write an equation you can use to solve the problem.

_____________________________________

7. Solve the equation. w = ____

8. Write the solution to the problem.

_____________________________________

LESSON

11-3

4

Name ________________________________________ Date __________________ Class __________________

224

Multiplication and Division Equations

Reteach

Number lines can be used to solve multiplication and division equations.

Solve: 3n = 15

How many moves of 3 does it take to get to 15?

n = 5 Check: 3 • 5 = 159

Solve:

3

n

= 4

If you make 3 moves of 4, where are you on the number line?

n = 12 Check: 12 ÷ 3 = 49

Show the moves you can use to solve each equation. Then give the

solution to the equation and check your work.

1. 3n = 9 Solution: n = ____

Show your check:

2.

2

n

= 4 Solution: n = ____

Show your check:

LESSON

11-3

Name ________________________________________ Date __________________ Class __________________

225

Multiplication and Division Equations

Reading Strategies: Use a Flowchart

A flowchart gives you a plan. You can use a flowchart to solve equations.

Solve:

6

x

= 4

Plan

Solve: 4n = 12

Think: Multiplying by 6

undoes dividing by 6.

1

Decide on what operation to use.

Think: Dividing by 4

undoes multiplying by 4.

6

x

• 6 = 4 • 6

2

Do the same on both sides.

4

n

=

12

4

x = 24

3

Solve the equation.

n = 3

24

6

= 49

4

Check the solution.

4

• 3 = 129

Use the flowchart to solve each equation.

Plan

Solve: 3r = 24

1

Decide on what operation to use.

2

Do the same on both sides.

3

Solve the equation.

4

Check the solution.

Plan

Solve:

8

b

= 16

1

Decide on what operation to use.

2

Do the same on both sides.

3

Solve the equation.

4

Check the solution.

LESSON

11-3

1

Decide how

to get the

variable b

y

itself.

3

Solve the

equation.

2

Do the same

on both sides

of the e

q

uation.

4

Check the

solution.

1.

2.

Name ________________________________________ Date __________________ Class __________________

226

Multiplication and Division Equations

Success for English Learners

Problem 1

Some armadillo mothers had Each mother had 4 babies.

these babies. How many mothers were there?

4m = 32

432

44

m

=

m = 8

There are 8 mothers.

Problem 2

6

x

= 12

6

x

•

= 12 • 6 Multiply each side by 6.

x = 72

72

÷ 6 = 12 9 The answer checks.

1. Explain how to check the solution to Problem 1.

_________________________________________________________________________________________

2. Solve

3

n

= 2. Show your work.

Check your work.

3. Solve 5t

= 20. Show your work.

Check your work.

LESSON

11-3

4 babies for

each mother

32 babies

To undo

multiplication,

use division.

4m = 32

Think: This is a division

equation. I multiply to undo

the multiplication.

Name ________________________________________ Date __________________ Class __________________

227

Writing Inequalities

Practice and Problem Solving: A/B

Complete the graph for each inequality.

1. a > 3 2. r ≤ –2

Graph the solutions of each inequality. Check the solutions.

3. w ≥ 0

Check:

___________________________

4. b ≤ −4

Check:

___________________________

5. a < 1.5

Check:

___________________________

Write an inequality that represents each phrase. Draw a graph

to represent the inequality.

6. The sum of 1 and x is less than 5. 7. 3 is less than y minus 2.

_____________________________________ _____________________________________

Write and graph an inequality to represent each situation.

8. The temperature today will be at least 10°F.

_________________________

9. Ben wants to spend no more than $3.

_________________________

Write an inequality that matches the number line model.

10.

_________________________

11.

_________________________

LESSON

11-4

Name ________________________________________ Date __________________ Class __________________

228

Writing Inequalities

Practice and Problem Solving: C

Circle the values that are solutions for each inequality.

1. a > −2 2. r ≤ 2

−3.5 −1 0 4

1

4

−3.5 −1 0 4

1

4

Graph the solutions of each inequality. Check the solutions.

3. 4 ≥ y

Check: ___________________________

4. b ≤ 0.5

Check: ___________________________

5. a < 1 − 3

Check: ___________________________

Write and graph an inequality to represent each situation. Then

determine if 36 is a possible solution. Write yes or no.

6. The temperature today will be at least 35°F.

________________________

7. Monica wants to spend no more than $35.

_________________________

Write an inequality that matches the number line model. Then write

a situation that the inequality could represent.

8.

_________________________

_________________________________________________________________________________________

9.

_________________________

_________________________________________________________________________________________

LESSON

11-4

Name ________________________________________ Date __________________ Class __________________

229

Writing Inequalities

Practice and Problem Solving: D

Complete the graph for each inequality. The first one is done for you.

1. a > 2 2. r ≤ −1

Graph the solutions of each inequality. Check the solutions. The first

one is done for you.

3. m ≥ −2

Check:

___________________________

4. d ≤ 3

Check:

___________________________

5. s < −3

Check:

___________________________

Write an inequality that represents each phrase. Draw a graph to

represent the inequality. The first one is done for you.

6. x is less than 4 7. −1 is greater than y

_____________________________________ _____________________________________

Write and graph an inequality to represent each situation. The first

one is done for you.

8. Today’s temperature is greater than 0°F.

_________________________

9. Lyle paid more than $2 for lunch.

_________________________

LESSON

11-4

0 ≥ −2; this is true

x < 4

t > 0

Name ________________________________________ Date __________________ Class __________________

230

Writing Inequalities

Reteach

An equation is a statement that says two quantities are equal. An

inequality is a statement that says two quantities are not equal.

A solution of an inequality that contains a variable is any value or

values of the variable that makes the inequality true. All values that

make the inequality true can be shown on a graph.

Inequality Meaning Solution of Inequality

x > 3

All numbers greater

than 3

The open circle at 3 shows that the value 3 is

not included in the solution.

x ≥ 3

All numbers greater

than or equal to 3

The closed circle at 3 shows that the value 3

is included in the solution.

x < 3

All numbers less than

3

x ≤ 3

All numbers less than

or equal to 3

Graph the solutions of each inequality.

1. x > −4

• Draw an open circle at −4.

• Read x > −4 as “x is greater than −4.”

• Draw an arrow to the right of −4.

3. a > −1

2. x ≤ 1

• Draw a closed circle at 1.

• Read x ≤ 1 as “x is less than or equal to 1.”

• Draw an arrow to the left of 1.

4. y ≤ 3

Write an inequality that represents each phrase.

5. the sum of 2 and 3 is less than y

_____________________________________

6. the sum of y and 2 is greater than or equal to 6

_____________________________________

LESSON

11-4

Name ________________________________________ Date __________________ Class __________________

231

Writing Inequalities

Reading Strategies: Understand Symbols

An inequality is a comparison of two unequal values. This chart will help you

understand both words and symbols for inequalities.

The team has scored fewer than 5 runs in

each game.

“Fewer than 5” means “less than 5.”

Symbol for “less than 5”: < 5

No more than 8 people can ride in the

elevator.

“No more than 8” Means “8 or less than 8.”

Symbol for “less than or equal to 8”: ≤ 8

More than 25 students try out for the team

each year.

“More than 25” means “a number greater

than 25.”

Symbol for “greater than 25”: > 25

There are at least 75 fans at each home

game.

“At least 75” means “75 or more” or “a

number greater than or equal to 75.”

Symbol for “greater than or equal to 75”: ≥ 75

Use the chart to answer each question.

1. What is an inequality?

_________________________________________________________________________________________

2. Explain the difference between the symbols < and ≤.

_________________________________________________________________________________________

3. Explain the difference between the symbols > and ≥.

_________________________________________________________________________________________

4. Write an inequality to describe the number of students in each

homeroom: There is a limit of 30 students for each homeroom.

_________________________________________________________________________________________

5. Is 28 a solution to the inequality you wrote in Exercise 4? How do

you know?

_________________________________________________________________________________________

LESSON

11-4

Inequalities

Name ________________________________________ Date __________________ Class __________________

232

Writing Inequalities

Success for English Learners

Problem 1

w ≤ 4

Word Phrase Symbol

is less than or equal to

≤

is greater than or equal to

≥

is greater than

>

is less than

<

Problem 2

w > −2

1. In Problem 1, is 4 part of the solution set? How do you know?

_________________________________________________________________________________________

2. In Problem 2, is −2 part of the solution set? How do you know?

_________________________________________________________________________________________

3. When graphing an inequality with a ≥ sign, should you use an empty

or a solid circle? Why?

_________________________________________________________________________________________

4. Graph the solutions of x < −2.

LESSON

11-4

Solid Circle Arrow to the left

Empty Circle Arrow to the right

Name ________________________________________ Date __________________ Class __________________

233

Equations and Relationships

Challenge

Write and solve an equation to find the unknown

measurement. Then use your answer to find the

perimeter of each field or court.

1. Equation to find area: _________________

Unknown measurement:

________________

Equation to find perimeter:

P =

___________________________

Perimeter of court:

_________________

2. Equation to find area:

_________________

Unknown measurement:

________________

Equation to find perimeter:

P =

___________________________

Perimeter of field:

_________________

3. Equation to find area: _________________

Unknown measurement: ________________

P = ___________________________

Perimeter of rink:

_________________

4. Equation to find area: _________________

Unknown measurement: ________________

P = ___________________________

Perimeter of diamond:

_________________

MODULE

11

Remember

Area = length • width or A = l • w

Perimeter is the distance around

or P = 2l + 2w

Name ________________________________________ Date __________________ Class __________________

234

Graphing on the Coordinate Plane

Practice and Problem Solving: A/B

Give the coordinates of the points on the coordinate plane.

1. A (

____ , ____)

2. B (

____ , ____)

3. C (

____ , ____)

4. D (

____ , ____)

5. E (

____ , ____)

6. F (

____ , ____)

Plot the points on the coordinate plane.

7. G (2, 4)

8. H (−6, 8)

9. J (10, −12)

10. K (−14, −16)

11. M (0, 18)

12. P (−20, 0)

Describe how to go from one store to the next on the map. Use words

like left, right, up, down, north, south, east, and west. Each square on

the coordinate plane is a city block.

13. The computer store, A, to the food store, B.

_____________________________________

14. The computer store, A, to the hardware

store, C.

_____________________________________

15. The hardware store, C, to the food store, B.

_____________________________________

LESSON

12-1

Name ________________________________________ Date __________________ Class __________________

235

Graphing on the Coordinate Plane

Practice and Problem Solving: C

Label the axes to locate the points on the coordinate planes.

1. A(−6, 15), B(3, −9), C(−9, −9) 2. D(0, 6), E(−12, 6), F(18, 0)

Start with the given point. Give the quadrant in which you end up after

following the directions. Then, give the coordinates of the point where

you end up.

3. X(5, −8) Go down 5, left 7, and down 6 more.

Quadrant:

________ ; Point: X(________, ________)

4. Y(−2, 6) Go up 3, right 5, and up 4 more.

Quadrant:

________; Point: Y(________, ________)

5. Z(0, −5) Go left 5, up 4, right 7, and down 3.

Quadrant:

________; Point: Z(________, ________)

Give the coordinates of a point that would form a right triangle with

the points given. Use the grids for reference. Tell what you know

about one of the coordinates of your new point.

6. P(2, 4), Q(2, 8), R(

________, ________) 7. S(−3, −5), T(4, −5), U(________, ________)

_________________________________________________________________________________________

LESSON

12-1

Name ________________________________________ Date __________________ Class __________________

236

Graphing on the Coordinate Plane

Practice and Problem Solving: D

Use the coordinate plane for Exercises 1–3. Give the letter of the

correct answer. The first one is done for you.

1. Which point is located in Quadrant I?

A point Q

B point P

C point X

____

2. Which point is located in Quadrant IV?

A point X

B point Y

C point P

____

3. Which point is located in Quadrant II?

A point Q

B point Y

C point X

____

Use the coordinate plane for Exercises 4–7. The first one is done for you.

4. What are the coordinates of point A?

________________________________________

5. What are the coordinates of point B?

B (

________, ________)

6. What are the coordinates of point C?

C (

________, ________)

7. What are the coordinates of point D?

D (

________, ________)

LESSON

12-1

C

Go over 3 to the right and down 1,

so the x-coordinate is 3 and the

y-coordinate is −1, or A(3, −1).

Name ________________________________________ Date __________________ Class __________________

237

Graphing on the Coordinate Plane

Reteach

Each quadrant of the coordinate plane has a unique combination of

positive and negative signs for the x-coordinates and y-coordinates as

shown here.

Quadrant x-coordinate y-coordinate

I + +

II − +

III − −

IV + −

Use these rules when naming points on the coordinate plane.

Example 1

Draw the point A(1, −3) on the coordinate grid.

Solution

According to the table, this point will be in

Quadrant IV.

So, go to the right (+) one unit, and go down

(−) three units.

Example 2

What are the coordinates of point B?

Solution

According to the table, this point will have

a negative x-coordinate and a positive

y-coordinate.

Point B is 3 three units to the left (−) and

four units up (+). So the coordinates of point

B are (−3, 4).

Add the correct sign for each point’s coordinates.

1. (

____ 3, ____ 4) in 2. (____ 2, ____ 5) in 3. (____ 9, ____ 1) in

Quadrant II Quadrant IV Quadrant I

4. In which quadrant is the point (0, 7) located? Explain your answer.

_________________________________________________________________________________________

LESSON

12-1

Name ________________________________________ Date __________________ Class __________________

238

Graphing on the Coordinate Plane

Reading Strategies: Build Vocabulary

This lesson introduces words used to graph numbers. Mathematics uses

these words to build new concepts. It is important to remember and to use

them. Look at this example. Read each definition, and find it on the picture.

A. The coordinate plane includes all of the

parts marked on the picture.

B. The axes are the darker number lines.

C. The x-axis goes left to right, whereas the

y-axis goes up and down.

D. The axes intersect at the origin, which is

marked with an “O”.

E. The scale on the number line is always

important in using a coordinate plane.

Here, every square on the grid is 2 units.

F. The axes divide the coordinate plane into

four quadrants. Quadrant I is upper right,

Quadrant II is upper left, Quadrant III is

lower left, and Quadrant IV, which is read

“quadrant four,” is lower right.

G. Pairs of numbers, called ordered pairs,

are represented on the coordinate plane

as points and in the format P(a, b), where

P is the point’s label, a is a value on the

x-axis, and b is a value on the y-axis.

H. The numbers a and b in the format (a, b)

are called coordinates. The a is called

the x-coordinate and the b is called the

y-coordinate.

Write a letter that indicates each of the following in the diagram above.

1. point on x-axis 2. x-coordinate of Q 3. y-coordinate of Q 4. point on y-axis

________________ ________________ _______________ ________________

5. point in Quadrant I 6. ordered pair for Q 7. point in Quadrant III 8. origin

________________ ________________ _______________ ________________

LESSON

12-1

Name ________________________________________ Date __________________ Class __________________

239

Graphing on the Coordinate Plane

Success for English Learners

Problem

1. If an ordered pair has an x

-value of 0, which direction do you move

from the origin?

_________________________________________________________________________________________

2. A negative y-coordinate means that a point may lie in which two

quadrants?

_________________________________________________________________________________________

3. Does it matter which number comes first in an ordered pair? Explain.

_________________________________________________________________________________________

LESSON

12-1

Name ________________________________________ Date __________________ Class __________________

240

Independent and Dependent Variables in Tables and Graphs

Practice and Problem Solving: A/B

Name the dependent variable and the independent variable in each

problem.

1. A food service worker earns $12 per hour. How much money, m, does

the worker earn on a shift of h hours?

Dependent variable:

_________________; independent variable: _________________

2. A large 2-topping pizza, L, costs $2 more than a medium 3-topping

pizza, M.

Dependent variable:

_________________; independent variable: _________________

The table shows the electric current produced by a solar cell in

different amounts of sunlight (light intensity). Answer the questions

using the data.

Light

intensity

150 300 450 600 750 900

Current

10 30 45 60 75 90

3. What is the dependent variable? 4. What is the independent variable?

________________________________________ ________________________________________

5. What do you predict the current will be in the absence of sunlight?

Explain.

_________________________________________________________________________________________

6. What do you predict the current will be if the light intensity is 1,000?

Explain.

_________________________________________________________________________________________

A race car driver’s time in seconds to

complete 12 laps is plotted on the graph.

7. Which axis shows the dependent variable?

________________________________________

8. Why does the graph begin at x = 1?

________________________________________

LESSON

12-2

Name ________________________________________ Date __________________ Class __________________

241

Independent and Dependent Variables in Tables and Graphs

Practice and Problem Solving: C

Use the situation below to complete Exercises 1–4.

The commuter bus system collected the data in the table below. All of the

data were collected under the same conditions: dry roads, no accidents or

traffic jams, same distance each trip, and no mechanical problems with the

bus on each trip.

Number of passengers per trip, n

30 35 40 45 50

Average speed, km per hour, s

60 58 55 55 52

Liters of biodiesel fuel used, f

45 48 50 52 54

1. Assume that more passengers cause the bus to travel slower. Of these

two factors, which would be the dependent and independent variables?

Dependent variable:

_________________; independent variable: _________________

2. Assume that an average slower speed causes the bus to consume

more fuel. Describe the relationship between bus speed and fuel

consumption.

_________________________________________________________________________________________

3. What can you say about the relationship between the number of

passengers and the fuel consumption?

_________________________________________________________________________________________

4. What effect does the number of passengers have on bus speed and

fuel consumption?

_________________________________________________________________________________________

In the graph, the independent variable is the

x-axis and the dependent variable is the y-axis.

Use the graph to answer Exercises 5–6.

5. Describe and compare how the dependent

variables shown by lines A and B change as

the independent variables change.

________________________________________

6. Describe and compare how the dependent

variables shown by lines B and C change as

the independent variables change.

________________________________________

LESSON

12-2

Name ________________________________________ Date __________________ Class __________________

242

Independent and Dependent Variables in Tables and Graphs

Practice and Problem Solving: D

Answer the questions for each real-world situation. The first one is

done for you.

1. The table gives the amount of water in a water tank as it is being filled.

Gallons

50 100 150 200 250

Time (min)

10 20 30 40 50

a. Why is gallons the dependent variable?

_____________________________________________________________________________________

b. Divide gallons by time in each pair of cells. What do you get?

_____________________________________________________________________________________

c. If the time is 60 minutes, how would you get the gallons? What

would you get?

_____________________________________________________________________________________

2. The table shows how to change miles to

kilometers. Divide kilometers by miles for

each of the four mileage numbers. How many

kilometers per mile do you get?

________________________________________

(km)

3.22 4.83 6.44 8.05

(mi)

2 3 4 5

Answer each question using the graph.

The first one is done for you.

3. How many sandwiches are available at the

start of the business day?

________________________________________

4. Which axis shows the dependent variable,

sandwiches?

________________________________________

5. How many sandwiches are left after

20 minutes?

________________________________________

LESSON

12-2

300

It depends on how long the water has been filling the tank.

50 ÷ 10 = 100

÷

20 = 150 ÷ 30

=

200

÷

40

=

250

÷

50

=

5

;

5

Multiply 60 times 5, which gives 300 gallons.

Name ________________________________________ Date __________________ Class __________________

243

Independent and Dependent Variables in Tables and Graphs

Reteach

In a table, the independent variable is often represented by x. The

dependent variable is often represented by y. Look at this example.

x

0 1 2 3 4 5 6 7

y

4 5 6 7 8 9 10 ?

What y value goes for the question mark?

Step 1 Notice that 4 is added to each value of x to give the y value.

Step 2 So, add 4 to 7. What does this give? 4 + 7 = 11

On a chart or graph,

• the x-axis is usually used for the

independent variable, and

• the y-axis is usually used for the

dependent variable.

Look at the example.

How does y depend on x?

Step 1 Each value of y is the opposite of

the value of x.

Step 2 What equation shows this fact?

y = −x

Give the relationship between x and y.

1.

_________________________________________________________________________________________

2. a. What is y when x = 2?

___________________________

b. What value of x gives y = −2?

___________________________

c. Write the equation for the graph.

___________________________

LESSON

12-2

x

1 2 3 4 5

y

3 4 5 6 7

Name ________________________________________ Date __________________ Class __________________

244

Independent and Dependent Variables in Tables and Graphs

Reading Strategies: Cause and Effect

It can sometimes be useful to think of the independent variable as the

cause of an event. This cause has an effect on the dependent variable.

This type of thinking can be helpful in doing some real-world problems.

Example 1

A middle-school science student did an experiment in which different

amounts of water were added on a one-time basis to a solution to see what

effect it would have on the solution’s concentration. Here are the results.

Water (milliliters)

5 10 15 20

Change in dilution

2 5 10 15

As more water is added (the “cause”), the concentration dilutes.

The amount of water is the independent variable. The amount of dilution

is the dependent variable.

Example 2

The chart shows how the yield of a crop per

acre changes as the number of insect pests

counted per acre increases.

• If the vertical axis (left) is the crop yield,

what is happening as the number of insects

(horizontal axis) increases?

• The crop yield continues to increase but not

as fast as at the beginning.

• The number of insects is the independent

variable (the cause), and the crop yield is

the dependent variable (the effect).

Identify the cause and the effect in each problem.

1. After a storm, the number of bottles of drinking water available per

family decreases as the number of families requesting assistance

increases.

_________________________________________________________________________________________

2. The number of hours it takes to vote increases as the number of voters

per hour increases.

_________________________________________________________________________________________

3. The car’s mileage, or miles per gallon, increases as its speed goes

from 20 miles per hour to 40 miles per hour.

_________________________________________________________________________________________

LESSON

12-2

Name ________________________________________ Date __________________ Class __________________

245

Independent and Dependent Variables in Tables and Graphs

Success for English Learners

Problem 1

Let

ℓ = length.

Let w = width.

Problem 2

In Problem 1, the length, l, “depends” on the width, w.

Independent variable

width, w

Dependent variable

length, l

The variables are related by the formula, l = 5 w.

1. Suppose the width of the rectangle in Problem 1 is 10 inches. What is

the length of the rectangle?

_________________________________________________________________________________________

2. The table shows the money the school band members collected for

washing cars on four different days.

Cars washed

12 15 20 30

Money collected

$120 $150 $200 ?

What is the dependent variable? ___________________________

What is the independent variable?

___________________________

How much money is collected for washing 30 cars?

_________________________________________________________________________________________

LESSON

12-2

Name ________________________________________ Date __________________ Class __________________

246

Writing Equations from Tables

Practice and Problem Solving: A/B

Write an equation to express y in terms of x. Use your equation to

complete the table.

1.

x

1 2 3 4 5

___________________________

y

7 14 21 28

2.

x

2 3 4 5 6

___________________________

y

−3 −2 −1 0

3.

x

20 16 12 8 4

___________________________

y

10 8 6 4

4.

x

7 8 9 10 11

___________________________

y

11 12 13 14

Solve.

5. Henry records how many days he rides his bike and how far he rides

each week. He rides the same distance each time. He rode 18 miles in

3 days, 24 miles in 4 days, and 42 miles in 7 days. Write and solve an

equation to find how far he rides his bike in 10 days.

Number of days, d

3 4 7 10

Number of miles, m

18

Equation relating d and m is ________________________________________.

The number of miles Henry rides his bike in 10 days is

_______________.

6. When Cabrini is 6, Nikos is 2. When Cabrini is 10, Nikos will be 6.

When Cabrini is 16, Nikos will be 12. When Cabrini is 21, Nikos will be

17. Write and solve an equation to find Nikos’ age when Cabrini is 40.

Cabrini’s age, x

6 10 16 21 40

Nikos’ age, y

2

Equation relating x and y is _________________________________________.

When Cabrini is 40 years old, Nikos will be

__________________________.

LESSON

12-3

Name ________________________________________ Date __________________ Class __________________

247

Writing Equations from Tables

Practice and Problem Solving: C

Write an equation to express y in terms of x. Use your equation to

complete the table.

1.

x

1 2 3 4 5

___________________________

y

1 4 9 16

2.

x

32 28 24 16

___________________________

y

−8 −6 −5 −4

3.

x

8 6 4 2

___________________________

y

4 3.2 1.6 0.8

4.

x

1 3 4 5

___________________________

y

7 12 17 22

Solve.

5.

9

32

5

FC=+

is an equation that models the relationship in the table.

Equivalent Temperatures

Celsius, (°C) −15 −10 −5 0 5

Fahrenheit, (°F) 5 14 23 32 41

What does each variable represent?___________________________________________________.

What is the temperature in °F when it is 20°C?

___________________________________________.

Is the ordered pair (30, 86) a solution for the equation? Justify your answer.

_________________________________________________________________________________________

6. Use the table of values and the equation in Exercise 5 to write an equation

for which F is the independent variable and C is the dependent variable.

An equation relating F and C is

______________________________________________________.

What is the temperature in °C when it is 59°F? Justify your answer.

_________________________________________________________________________________________

LESSON

12-3

Name ________________________________________ Date __________________ Class __________________

248

Writing Equations from Tables

Practice and Problem Solving: D

Write an equation to express y in terms of x. The first one is done

for you.

1.

x

0 1 2 3

___________________________

y

2 3 4 5

2.

x

5 10 15 20

___________________________

y

1 2 3 4

3.

x

3 4 5 6

___________________________

y

9 12 15 18

4.

x

7 8 9 10

___________________________

y

5 6 7 8

Solve. The first one is done for you.

5. When George works 8 hours he earns $80. When George works

10 hours he earns $100. When George works 12 hours he earns $120.

Complete the table. Circle the letter of the equation that relates the

dollars George earns, y, to the number of hours he works, x.

Number of hours, x

8 10 12

Dollars earned, y

80

100 120

A y = x ÷ 10 C y = 10x

B y = x + 72

6. When Javier is 2, Arianna is 5. When Javier is 3, Arianna is 6.

When Javier is 8, Arianna will be 11. When Javier is 20, Arianna is 23.

Complete the table. Circle the letter of the equation that relates the age

of Arianna, y, to the age of Javier, x.

Javier’s age, x

2 3 8 20

Arianna’s age, y

5

A y = x ÷ 2 C y = 2x

B y = x + 3

When Javier is 30 years old, Arianna will be

__________________________.

LESSON

12-3

y = x + 2

Name ________________________________________ Date __________________ Class __________________

249

Writing Equations from Tables

Reteach

The relationship between two variables in which one quantity depends

on the other can be modeled by an equation. The equation expresses

the dependent variable y in terms of the independent variable x.

To write an equation from a table of values, first

x

0 1 2 3 4 5 6 7 compare the x- and y-values to find a pattern.

y

4 5 6 7 8 9 10 ? In each, the y

-value is 4 more than the x-value.

Then use the pattern to write an equation expressing y in terms of x.

y = x + 4

You can use the equation to find the missing value in the table. y = 7 + 4

To find y when x = 7, substitute 7 in for x in the equation. y = 11

So, y is 11 when x is 7.

Write an equation to express y in terms of x. Use your equation to find

the missing value of y.

1.

x

1 2 3 4

5 6

___________________________

y

3 6 9 12 15 ?

2.

x

18 17 16 15 14 13

___________________________

y

15 14 13 ? 11 10

To solve a real-world problem, use a table of values and an equation.

When Todd is 8, Jane is 1. When Todd is 10, Jane will be 3. When

Todd is 16, Jane will be 9. What is Jane’s age when Todd is 45?

Todd, x

8 10 16 45 Jane is 7 years younger than Todd.

Jane, y

1 3 9 ? So y = x − 7. When x = 45, y = 45 − 7. So, y = 38.

Solve.

3. When a rectangle is 3 inches wide its length is 6 inches. When it is

4 inches wide its length will be 8 inches. When it is is 9 inches wide its

length will be 18 inches. Write and solve an equation to complete the table.

Width, x

3 4 9 20

___________________________

Length, y

6

When the rectangle is 20 inches wide, its length is ______________________.

LESSON

12-3

Name ________________________________________ Date __________________ Class __________________

250

Writing Equations from Tables

Reading Strategies: Analyze Information

A table is useful for changing cups to ounces.

Cups Ounces

1 8

2 16

3 24

4 32

5 40

Use the table above to complete Exercises 1–3.

1. How many ounces are in 1 cup?

_________________________________________________________________________________________

2. How many ounces are in 3 cups?

_________________________________________________________________________________________

3. If “6 cups” were added to the table, how many ounces would be listed?

_________________________________________________________________________________________

An equation shows the relationship between cups and ounces.

ounces = 8 • cups

y = 8x

Independent Variable

x

1 2 3 4 5

Dependent Variable

y

8 16 24 32 40

The number of ounces depends on the number of cups. The value of y

depends on the value of x.

Use the table above to complete Exercises 4–6.

4. Which variable stands for ounces? for cups?

_________________________________________________________________________________________

5. What is the value of y when x = 2?

_________________________________________________________________________________________

6. Use the equation to find the number of ounces when the number of cups is 15.

_________________________________________________________________________________________

LESSON

12-3

Name ________________________________________ Date __________________ Class __________________

251

Writing Equations from Tables

Success for English Learners

Problem 1

The number of inches is 12 times the number of feet.

The independent

variable is the x-value of the equation. number of feet x

The dependent

variable is the y-value of the equation. number of inches y

Write an equation

to show the relationship between x and y. y = 12x

Problem 2

Mike has 8 feet of rope. How many inches of rope does he have?

Solve an equation

to find a value.

Substitute

the value for x into the equation, then solve for y.

y = 12x

x = 8 Substitute 8 for x. y = 12(8) = 96

Mike has 96 inches of rope.

1. What does an equation with x and y show?

_________________________________________________________________________________________

2. What does it mean to substitute a value into the equation?

_________________________________________________________________________________________

3. Suppose Mike has 5 feet of rope. How many inches of rope does

he have?

_________________________________________________________________________________________

LESSON

12-3

Independent variable

x-value

The number of inches depends on

how man

y

feet there are.

Name ________________________________________ Date __________________ Class __________________

252

Representing Algebraic Relationships in Tables and Graphs

Practice and Problem Solving: A/B

An antiques dealer has 24 clock radios to sell at a 12-hour-long

antique-radio sale. Use the graph to complete the table.

1. Complete the table with the data from the graph.

Radios remaining

24 ? ? ? ? ? ?

Hours completed

0 2 4 6 8 10 12

2. What are the dependent (y) and independent (x) variables?

dependent:

___________________________; independent: ___________________________

3. Write ordered pairs for the points on the graph and in the table.

_________________________________________________________________________________________

4. How many radios are sold every two hours? _________________

5. What happens to the total number of radios every two hours?

_________________________________________________________________________________________

6. If h is hours and n is the number of radios remaining, complete the equation:

n =

_________________× h + _________________

7. Why is the sign of the number that is multiplied by hours, h, negative?

_________________________________________________________________________________________

LESSON

12-4

Name ________________________________________ Date __________________ Class __________________

253

Representing Algebraic Relationships in Tables and Graphs

Practice and Problem Solving: C

Use the graph to answer the questions.

1. A paleontologist is counting fossilized remains of extinct plants at a

geological site. Complete the table with data from the graph.

Plant fossils counted, f

____ ____ ____ ____ ____ ____

Elapsed days of dig, d

1 2 3 4 5 6

2. There are three rates at which the fossils are being counted: Rate A for

Days 1 and 2, Rate B for Days 2 − 4, and Rate C: for Days 5 − 6. What

is happening to the number of fossils counted as each day passes?

_________________________________________________________________________________________

3. Which rate describes the period of time over which the number of

fossils counted decreases at the greatest rate? Explain your answer.

_________________________________________________________________________________________

4. Give the numerical value of each of the rates, A, B, and C. Your

answer should be negative and expressed in units of “fossils counted

per day” or “fossils/day.”

Rate A: ____________; Rate B: ____________; Rate C: ____________

LESSON

12-4

Name ________________________________________ Date __________________ Class __________________

254

Representing Algebraic Relationships in Tables and Graphs

Practice and Problem Solving: D

Complete the tables. Then, write the ordered pairs. Finally, fill in the

blanks to give the algebraic relationship of x and y. The first problem

has been done for you.

1.

x

0 1 2 3

y

4 7

10 13

_________________________________________________________________________________________

y = 3 x + 4

2.

x

0 1 2 4

y

0 −4

____

−16

_________________________________________________________________________________________

y = __________ x + __________

3.

x

0 2

____

6

y

5 11 17

____

_________________________________________________________________________________________

y = __________ x + __________

Write the ordered pairs of three points on the graph. Then, write

the algebraic relationship of x and y. The first one is done for you.

4.

____________________________

y = 0.5 x + 0

5.

____________________________

y =

__________ x + __________

6.

____________________________

y =

__________ x + __________

LESSON

12-4

(0, 4), (1, 7), (2, 10), (3, 13)

(−8, −4), (4, 2), (8, 4)

Name ________________________________________ Date __________________ Class __________________

255

Representing Algebraic Relationships in Tables and Graphs

Reteach

The x- and y-values in an algebraic relationship should be related in the same way when new

values of x or y are used. This pattern should be seen in a table of values and from a graph

of the x and y values.

Example 1

What is the relationship of the x and y values

in the table?

x

2 4 6 8 10

y

6 12 18 24 30

Solution

First, check to see if there is a simple addition,

multiplication, division, or subtraction

relationship between the x and y values.

Here, the y values are 3 times the x values.

This means that the algebraic relationship is

y = 3x.

Example 2

What is the relationship between x and y

represented by the graph.

Solution

First, notice that the line through the points

crosses the y-axis at y = 2. This means that

part of the relationship between x and y is

given by y =

____ + 2.

Next, notice that the line through the points

goes over to the right by one unit as it

“rises” by 3 units. This means that any

x value is multiplied by 3 over 1 or 3 units

as the line goes from one point to another.

This is written as y = 3x.

Combine these two observations:

y = 3x and y = 2 give y = 3x + 2.

Both parts are needed to completely

describe the relationship shown.

1. Find the relationship of x and y in the table.

x

0 1 3 6 7

y

1.5 2 3 4.5 5

y = ____________ x + ____________

2. Find the relationship of x and y from

a graph of a line that crosses the

y-axis at y = 6 and that goes to the

left 2 units and rises 3 units.

y =

____________ x + ____________

LESSON

12-4

Name ________________________________________ Date __________________ Class __________________

256

Representing Algebraic Relationships in Tables and Graphs

Reading Strategies: Reading a Table

In order to write a rule that gives an algebraic relationship, you sometimes

need to use a table.

Car washers tracked the number of cars they washed and the total amount

of money they earned. They charged the same price for each car they

washed. They earned $60 for 20 cars, $66 for 22 cars, and $81 for 27 cars.

Use the information to make a table and write an equation.

Make a table.

Cars washed (c)

20 22 27

Money earned (m)

60 66 81

The money earned is three times the number of cars washed.

20 × 3 = 60

22 × 3 = 66

27 × 3 = 81

Write an equation.

m = 3c

1. What is the value of m when there are no cars washed?

________________________________________

2. What is the value of m when 100 cars are washed?

________________________________________

3. Complete the table. Then write an equation to represent the table.

Tickets (t)

8 10 12 14 16

Total cost (c)

40 50 60

_______________________________________

4. Complete the table. Then write an equation to represent the table.

x

4 8 12 16 20 24

y

1 2 3 4

________________________________________

LESSON

12-4

Name ________________________________________ Date __________________ Class __________________

257

Representing Algebraic Relationships in Tables and Graphs

Success for English Learners

Problem 1

Find the algebraic rule from a table.

x

0 1 2 3

y

4 6 8 10

Combine the two steps: y = 2x + 4

Problem 2

The line crosses the y-axis at 2.

The line goes over 4 and

up 3.

y = ______ + 2

y =

3

4

x + 2

Give the algebraic rule.

1. (0, 2), (1, 3), (2, 4) 2. (0, 1), (2, 5), (4, 9)

________________________________________ _____________________________________

If x = 0, that value of y

goes in the rule:

y

= ____ + 4

LESSON

12-4

What happens to x before

it is added to 4 to give 6?

Multiplied by 2: 2 × 1

y = 2x

Where does

the line cross

the y-axis?

How far over

and how far up?

Name ________________________________________ Date __________________ Class __________________

258

Relationships in Two Variables

Challenge

Exploring Temperature Data

This activity illustrates the difference between experimental and theoretical data.

1. Complete the tables. Graph the Table 1 data as individual points.

Show the data in Table 2 as a straight line.

Table 1

data read from

the thermometer

°C °F

10

11

12

13

14

15

Table 2

data computed

from the equation

9

32

5

FC=×+

°C °F

10

11

12

13

14

15

2. Describe the difference between the two data sets and explain why they differ.

_________________________________________________________________________________________

MODULE

12

393

UNIT 5: Equations and Inequalities

MODULE 11 Equations and

Relationships

LESSON 11-1

Practice and Problem Solving: A/B

1. yes

2. no

3. no

4. no

5. yes

6. yes

7. yes

8. yes

9. B

10. D

11. Sample equation: 6x = 72

12. Sample equation: (6)(5) = (10)(3)(w)

13. Sample equation: x − 13°F = 35°F;

x = 48°F

14. Sample equation: 16x = $20; x = $1.25

15. Sample problem: Twenty-four people

were divided evenly into y teams. There

were 3 people on each team. Determine

whether there were 8 teams or 6 teams.

Answer: There were 8 teams.

Practice and Problem Solving: C

1. A

2. C

3. D

4. B

5. Sample equation: 7(10 + x) = 112

6. Sample equation: y = 11 +

1

3

(15); y = 16;

sea cow

= 16

7. Sample equation: 22

=

112

4

− 6

8. Sample equation: 4

x = 80 + 40; x = 30

9. Sample equation: 3

x + 5 = 29

Practice and Problem Solving: D

1. yes

2. no

3. no

4. yes

5. yes

6. yes

7. A

8. B

9. B

10. C

11. A

12. B

13. (1) $5

+ (3)$1 + Other bill = $13;

$5

+ $3 + x = $13; x = $5; The other bill

must be a $5 bill.

Reteach

1. yes

2. no

3. no

4. yes

5. yes

6. no

7. yes

8. yes

9. no

Reading Strategies

1. yes

2. no

3. yes

4. yes

5. yes

6. no

7. no

8. Sample equation: 2

• 13 = 15 + 11

9. Sample answer: 2

•

13 = 15 + y;

y

= 11

394

Success for English Learners

1. Because when the variable in the

equation is replaced with 61, it does not

make a true statement.

2. Substitute 65 for a and check to see if the

equation is true.

3. Sample answer: Andrea is given $82 to

buy fruit for the class picnic. She spends

some of the money on apples and $23 on

bananas. Determine whether she spent

$61 or $59 on apples.

LESSON 11-2

Practice and Problem Solving: A/B

1. r = 4

2. w = 38

3. m =

5

8

4.

t = −4

5. x + 139 = 180

6. x = 41°

7. x + 18 = 90

8. x = 72°

9.

x = 5; Sample answer. John has some

CDs. If he buys 3 more CDs, he will have

8 CDs. How many CDs did he start with?

John started with 5 CDs.

Practice and Problem Solving: C

1. 3.4

2.

7

9

3. 5

1

2

4. 17.19

5. −4

6. −40

7.

x + 22 = 90

8.

x = 68°

9. Sample answer:

u − 22 = 13; u = 35;

Kayla’s uncle is 35 years old.

10. Sample answer: 38.95

− 22.50 = g;

g = 16.45; Gavin will save $16.45.

11. Sample answer:

s − 10

1

2

= 37

1

2

;

s = 48; The board Sierra started with was

48 inches long.

12.

x = 7; Sample answer: Andy ran 4.65

kilometers. Pam said that if she had run

2.35 fewer kilometers, she would have run

as far as Andy. How far did Pam run?

Answer: Pam ran 7 kilometers.

Practice and Problem Solving: D

1. r = 6

2. w = −1

3.

m = 3

4.

t = 5

5.

x + 100 = 180

6.

x = 80°

7. 23

+ n = 40

8. 17

9.

x = 7; Sample answer: Joan has some

pencils. If she gives away 5 pencils, she

will have 2 pencils left. How many pencils

did Joan start with? She started with

7 pencils.

Reteach

1. 8

2. 9

3. 5

4. 9

5. 6

6. 8

Reading Strategies

1. Left

2. Add 21.

395

3. Add 21.

4. 53

5. Right.

6. Add 25.

7. Add 25.

8. 37

Success for English Learners

1. Because the surfer’s height, h, plus

14 inches is equal to the height of the

surfboard.

2. Substitute 57 for x in the original equation

and see if that makes the equation true.

3. Sample answer: x − 12 = 10; Add 12 to

both sides; x = 22.

LESSON 11-3

Practice and Problem Solving: A/B

1. e = 6

2. w = 10

3. m =

1

4

4. k = 10

5. Sample answer: 8x = 72

6. x = 9; 9 m

7.

a

3

= 9; a = 27; 27 pictures

Practice and Problem Solving: C

1. 0.7

2. 27

3.

1

2

4. 75

5. 20

6.

4

3

or 1

1

3

7. A

= 144 in.

2

; P = 4s; 48 = 4s, so s = 12.

A = s

2

, A = 12

2

= 144

8. 17 model SUVs; Sample equation:

5

m = 85, m = 17

9. 18 min; Sample equation:

n

3

= 6, n = 18

10. 3 h; Sample equation: 16.50b

= 49.50,

b

= 3

11. n

= 25; Sample answer: Maria used 12.5

meters of material to make doll clothes for

a charity project. Each piece of clothing

used 0.5 meter of material. How many

pieces of clothing did Maria make? She

made 25 pieces of clothing.

Practice and Problem Solving: D

1. m = 4

2. a

= 8

3. s

= 4

4. u

= 10

5. Area—60 ft

2

; length—12 ft

6. Sample answer: 60 = 12w

7. 5

8. Jim’s garden is 5 feet wide.

Reteach

1. n = 3; 3 • 3 = 99

2. n

= 8; 8 ÷ 2 = 49

Reading Strategies

1. Divide by 3;

3

r

=

24

;

3

r = 8; 3 • 8 = 249

2. Multiply by 8;

•

8

b

= 16 • 8;

b

= 128;

128

8

= 16 9

396

Success for English Learners

1. Substitute 8 for m. Check whether that

equation is true. 4 • 8 = 329

2.

n

3

= 2;

• 3

3

n

= 2 • 3; n = 6; 6 ÷ 3 = 2

9

3. 5t = 20;

t5

5

=

20

5

; t = 4; 5 • 4 = 20

9

LESSON 11-4

Practice and Problem Solving: A/B

1.

2.

3.

Possible check: 1 ≥ 0 is true.

4.

Possible check: −5 ≤ −4 is true.

5.

Possible check: 1 < 1.5 is true.

6. Sample inequality: 1 + x < 5

7. Sample inequality: 3 < y − 2

8. Sample inequality: t ≥ 10

9. Sample inequality: b ≤ 3

10. Sample inequality: x ≥ −2

11. Sample inequality: x < −2

Practice and Problem Solving: C

1. −1, 0, 4

1

4

2. −3.5, −1, 0

3. Sample check: 4 ≥ 1

4. Sample check: −4 ≤ 0.5

5. Sample check: −3 < 1 − 3

6. Sample inequality: 35 ≤ t; yes

7. Sample inequality: m ≤ 35; yes

8. Sample answer: x > 25; there are more

than 25 students in the school band.

9. Sample answer: x ≤ −2.75; the highest

temperature today was –2.75°C.

Practice and Problem Solving: D

1.

2.

3. Sample check: 0 ≥ −2; this is true.

4. Sample check: −2 ≤ 3; this is true.

5. Sample check: −4 < −3; this is true.

6. Sample inequality: x < 4

7. Sample inequality: −1 > y

8. Sample inequality: t > 0

9. Sample inequality: m > 2

397

Reteach

1.

2.

3.

4.

5. Sample inequality: 2 + 3 < y

6. Sample inequality: y + 2 ≥ 6

Reading Strategies

1. An inequality is a comparison of two

unequal values.

2. The symbol < means less than and the

symbol ≤ means less than or equal to.

3. The symbol > means greater than and the

symbol ≥ means greater than or equal to.

4. Sample inequality: x ≤ 30

5. 28 is a solution to the inequality because

28 ≤ 30 is true.

Success for English Learners

1. Yes, 4 ≤ 4 is true.

2. No, −2 > −2 is not true.

3. Solid circle; the inequality ≥ means it can

be equal to or greater than.

4.

MODULE 11 Challenge

1. 4,700 = 94w; w = 50 ft; P = 2 • 94 + 2 • 50;

P = 288 ft

2. 8,250 = 75l; l = 110 m; P = 2 • 75 + 2 • 110;

P = 370 m

3. 1,586 = 26l; l = 61 m; P = 2 • 26 + 2 • 61;

P = 174 m

4. 8,100 = 90w; w = 90 ft; P = 2 • 90 + 2 • 90

(or 4

• 90); P = 360 ft

MODULE 12 Relationships In

Two Variables

LESSON 12-1

Practice and Problem Solving: A/B

1. A(−12, 14)

2. B(8, 8)

3. C(−12, −4)

4. D(0, −14)

5. E(14, 0)

6. F(0, 0)

7–12.

13. Answers will vary. Sample answer: “Go 6

blocks south.”

14. Answers will vary. Sample answer: “Go 5

blocks east and 1 block south.”

15. Answers will vary. Sample answer: “Go 5

blocks south and 5 blocks west.”

Practice and Problem Solving: C

1. Axes labeling may vary. Sample answer:

398

2. Axes labeling may vary. Sample answer:

3. Quadrant III; X(−2, −19)

4. Quadrant I; Y(3, 13)

5. Quadrant IV; Z(2, −4)

6. Answers will vary. Sample answer: One of

the coordinates of the new point must be

4 or 8. P(2, 4), Q(2, 8), R(5, 8).

7. Answers will vary. Sample answer: One of

the coordinates of the new point must be

−3 or 4. S(−3, −5), T(4, −5), U(4, 5).

Practice and Problem Solving: D

1. C

2. C

3. B

4. A(3, −1)

5. B(2, 4)

6. C(–3, 0)

7. D(1, −1)

Reteach

1. (−3, +4)

2. (+2, −5)

3. (+9, +1)

4. The point (0, 7) is not in a quadrant; it is

on the positive y-axis between quadrants I

and II.

Reading Strategies

Some answers will vary. Sample answers

are given.

1. J

2. r

3. s

4. K

5. Sample answer: L

6. (r, s)

7. Sample answer: M

8. O

Success for English Learners

1. Up or down, if the y-value is non-zero.

2. Quadrants III or IV

3. Yes, unless the x- and y-values are

equal.

LESSON 12-2

Practice and Problem Solving: A/B

1. m, money; h, hours worked

2. L, cost of large pizza; M, cost of medium

pizza

3. Current

4. Light intensity

5. Answers will vary. Sample answer: close

to zero.

6. Answers will vary. Sample answer: 100;

c =

10

L

.

7.

y-axis

8. There is no lap time until the driver drives

the first lap, x

= 1.

Practice and Problem Solving: C

1. Speed; number of passengers

2. Answers will vary. Sample answer: the

slower the bus goes, the more fuel it uses.

3. Answers will vary. Sample answer: the

more passengers the bus carries, the

more fuel that is consumed.

4. Answers will vary. Sample answer:

Students should recognize that the fuel

consumption is related to both the number

of passengers and the bus speed. The

three variables interact in a complex way

that is not completely clear from or

explained by this data.

5. For each increase in the independent

variable, the dependent variables changes

more for line A than it does for line B,

except when the value of the independent

variables is zero, in which case the value

399

of line A’s dependent variable and line

B’s dependent variable are the same

(22.5 units).

6. For each change in the independent

variable, the dependent variable increases

by the same amount. However, the value

of line B’s dependent variable will always

be 22.5 units more than the corresponding

value of line C’s dependent variable.

Practice and Problem Solving: D

1. a. It depends on how long the water has

been filling the tank.

b. 50 ÷ 10 = 100 ÷ 20 = 150 ÷ 30 =

200 ÷ 40 = 250 ÷ 50 = 5; 5

c. Multiply 60 times 5, which gives 300 gal.

2. 1.61 km per mi

3. 300 sandwiches

4. vertical axis or y-axis

5. 150

Reteach

1. Add 2 to x to get y or x + 2 = y.

2. a. y = 4

b. x = −1

c. y = 2x

Reading Strategies

1. Cause: increasing number of families

requesting assistance; effect: fewer

bottles of drinking water per family.

2. Cause: increasing number of voters per

hour; effect: the number of hours it takes

to vote increases.

3. Cause: car speed; effect: increasing

mileage or miles per gallon.

Success for English Learners

1. 50 in.

2. Money collected; cars washed; $300

LESSON 12-3

Practice and Problem Solving: A/B

1. y = 7x; 35

2. y = x − 5; 1

3. y = x ÷ 2; 2

4. y = x + 4; 15

5. 24, 42, 60; m = 6d; 60 mi

6. 6, 12, 17, 36; y = x − 4; 36 years old

Practice and Problem Solving: C

1. y = x

2

; 25

2. y = x ÷ −4; 20, −7

3. y = 0.4x; 10, 2.4

4. y = 5x + 2; 2, 27

5. F represents °F, C represents °C; 68°F;

Yes it is a solution because

9

(30) 32 86.

5

F =+=

6.

5

(–32)

9

CF=

;

5

(59 – 32) 15

9

C ==

, so

the temperature is 15°C.

Practice and Problem Solving: D

1. y = x + 2

2. y = x ÷ 5

3. y = 3x

4. y = x − 2

5. 100, 120; C

6. 6, 11, 23; B; 33 years old

Reteach

1. y = 3x; y = 18

2. y = x − 3; y = 12

3. 8, 18, 40; y = 2x; 40 in.

Reading Strategies

1. 8

2. 24

3. 48

4. y stands for ounces; x stands for cups

5. y = 16

6. y = 8(15), So 120 ounces is the same as

15 cups.

Success for English Learners

1. The equation shows the relationship

between x and y.

2. To substitute a value means to replace

the variable in the equation with the value

given for it.

400

3. y = 12(5), y = 60; So Mike has 60 inches

of rope.

LESSON 12-4

Practice and Problem Solving: A/B

1. 22, 20, 18, 16, 14, 12

2. radios; hours

3. (0, 24), (2, 22), (4, 20), (6, 18), (8, 16),

(10, 14), (12, 12)

4. 2

5. The total number decreases.

6. −1; 24

7. Because the total number of radios

decreases by h, the number of hours.

Practice and Problem Solving: C

1. 27, 12, 9, 6, 4, 3

2. The number of plant fossils counted daily

is decreasing.

3. Rate A for Days 1 and 2 is the greatest

decrease in fossils counted.

4. −15 fossils counted per day; −3 fossils

counted per day; −1 fossil counted

per day.

Practice and Problem Solving: D

1. 10, 13; (0, 4), (1, 7), (2, 10), (3, 13);

y = 3x + 4

2. 8; (0, 0), (1, 4), (2, 8), (4, 16);

y = 4x

3. 4, 23; (0, 5), (2, 11), (4, 17), (6, 23);

y = 3x + 5

4. (4, 2), (8, 4); y = 0.5x + 0 or

y = 0.5x

5. (0, 4), (4, 8); y = x + 4

6. (0, 8), (2, 4); y = −2x + 8

Reteach

1. y = 0.5x + 1.5

2. y =

3

2

−

x + 6

Reading Strategies

1. 0

2. 300

Tickets (t)

8 10 12 14 16

Total cost (c)

40 50 60 70 80

t = 5c

x

4 8 12 16 20 24

y

1 2 3 4 5 6

y =

1

4

x

Success for English Learners

1. y = x + 2

2.

y = 2x + 1

MODULE 12 Challenge

1. tables and graph

Table 1

°C °F

10 50

11 52

12 54

13 55

14 57

15 59

Table 2

°C °F

10 50

11 51.8

12 53.6

13 55.4

14 57.2

15 59

3.

4.

401

2. Sample answer: The data for Table 2 lie

along the straight line because they are

computed from the equation. For Table 1,

four of the data points are either above

or below the line, although they are close

to it. The data for Table 1 are

approximations because the thermometer

can only be read to about the nearest half

degree.