Expressions, Equations, and Inequalities - Macmillan/McGraw-Hill

CHAPTE R

9

Expressions, Equations, and Inequalities

connectED.mcgraw-hill.com

The

BIG Idea

Investigate

How can I use addition, subtraction, multiplication, and division to solve equations and inequalities?

Animations Vocabulary Math Songs Multilingual eGlossary

Learn Personal Tutor Virtual Manipulatives

Make this Foldable to help you organize information about expressions.

of Order ns o Operati

Ordered Pairs

Equations

Inequalit ies

Audio Foldables

Practice Self-Check Practice eGames Worksheets Assessment

Review Vocabulary bination of numbers, expression expresión A com operation. variables, and at least one

x

2

Key Vocabulary English variable equation coordinate plane ordered pair

414

+

Español variable ecuación plano de coordenadas par ordenado

When Will I Use This?

Your Turn! You will solve thhiis teerrr. problem in the chap

Expressions, Equations, and Inequalities 415

Are You Ready

You have two options for checking Prerequisite Skills for this chapter.

for the Chapter?

Text Option

Take the Quick Check below.

Find the missing number in each fact family. 1. 5 + = 7

2. + 6 = 9

3. + 3 = 15

4. 9 + = 15

5. 6 + = 14

6. + 5 = 7

7. 4 × = 28

8. × 9 = 81

9. × 3 = 30

10. 7 × = 56

11. 8 × = 48

12. × 5 = 30

13. Anderson added the shells shown below to his collection. Now he has 16 seashells. How many seashells did he have at first?

Write an expression for each situation. 14. 7 plus d

15. 5 less than t

16. the sum of 14 and s

17. the product of y and 7

18. 6 less than x

19. f increased by 2

20. the product of 8 and n

21. the sum of 3 and z

22. Hugo has $4 less than Eloy. If m stands for the amount of money Eloy has, write an expression to show how much money Hugo has. If m is $16, how much money does Hugo have?

Online Option 416

Take the Online Readiness Quiz.


Multi-Part Lesson

1

PART

Order of Operations A

Main Idea I will use order of operations to evaluate expressions.

B

C

D

E

Order of Operations The order of operations is a set of rules to follow when more than one operation is used in an expression.

Vocabulary V order of operations

Get ConnectED GLE 0506.3.1 Understand and use order of operations. SPI 0506.3.2 Evaluate multi-step numerical expressions involving fractions using order of operations.

Order of Operations

1. Perform operations in parentheses. 2. Find the value of exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right.

The table shows the number of Calories burned in one minute for two different activities. Nathan swims for 4 minutes and then runs for 8 minutes. How many Calories has Nathan burned in all?

Swimming

12

Running

10

Find the value of the expression 12 × 4 + 10 × 8. 12 × 4 + 10 × 8

Write the expression.

48 + 10 × 8

Multiply 12 and 4.

48 +

Multiply 10 and 8.

128

80

Add 48 and 80.

So, Nathan has burned 128 Calories.

Lesson 1A Order of Operations 417

Parentheses MEASUREMENT The table shows the time Santos spends doing different activities in one day. Find the value of 3 × (60 + 45) to find the number of minutes he spends doing homework and practicing drums in 3 days. Santos’ Activities Activity Homework Practice Drums

Time (minutes) es) 60 45

3 × (60 + 45) Write the expression. 3 × 105 315

Add 60 and 45. Multiply.

In 3 days, Santos spends 315 minutes doing his activities.

Exponents Find the value of each expression. 15 - 32 + 4 15 - 3 2 + 4


15 - 9 + 4

Find 3 2.

6+4

Subtract 9 from 15.

10 32 is 3 squared and 32 = 3 × 3. 23 is 2 cubed and 23 = 2 × 2 × 2.

Add 6 and 4. 2

So, 15 - 3 + 4 = 10. 24 ÷ 6 × (2 3 - 7) 24 ÷ 6 × (2 3 - 7) Write the expression. 24 ÷ 6 × (8 - 7)

Find 2 3.

24 ÷ 6 × 1

Subtract 7 from 8.

4

× 1 4

Divide 24 by 6. Multiply.

So, 24 ÷ 6 × (2 3 - 7) = 4.

418


Find the value of each expression. expression See Examples 1–4 1 4 1. 12 - 2 × 5

2. 4 × (15 - 3)

3. 6 2 + 30 ÷ 2

4. 5 × (92 - 18)

5. 12 + 4 2 - 11

6. 15 ÷ 3 × (5 2 - 5)

7. Giselle bought three DVDs that each cost $12. She also had a coupon for $10 off her total purchase. Find the value of 3 × 12 - 10 to find her final cost. 8. The table shows the number of minutes Curtis read in five days. Find the value of 25 × 3 + 20 × 2 to find how many minutes he read. 9. Hector buys 5 books on the Internet. The cost of shipping the books is $3 plus $1 for each book purchased. Find the value of 3 + 1 × 5 to find the cost of shipping 5 books. 10.

E

Curtis’s Reading Time Time Day (minutes) Monday Tuesday Wednesday Thursday Friday

25 20 25 25 20

TALK MATH Explain why it is important to follow

the order of operations when simplifying 15 + 3 × 4.

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Evaluate E l t each h expression. i See Examples 1–4 11. (15 - 5) × (3 3 + 3)

12. 58 - 6 × 7

13. 32 + 4 × 8

14. 4 × 2 4 + 12

15. 150 × (13 - 11)

16. 63 ÷ 9 - (2 2 + 3)

17. 30 - (5 3 - 100)

18. 7 × 10 + 3 × 30

4 19. _ × (6 2 - 7) 2

20. Measurement The total distance around the garden shown is 2 times the length plus 2 times the width. What is the total distance around the garden?

6 ft

21. Vikram counted the number of cans turned in on Monday morning. His results are shown. Each

8 ft

represents 5 cans.

How many cans did Vikram count? Write an expression. Then evaluate the expression.

Lesson 1A Order of Operations 419

22. Three students are on the same team for a relay race. They finish the race in 54.3 seconds. The runners’ times are shown on the table. Find the value of 54.3 - (18.8 + 17.7) to find the time of the third runner.

Relay Times Runner

Time (seconds)

1

18.8

2

17.7

3

?

23. Ryan and Maggie are splitting the cost of a $12 pizza. They also have a $2-off coupon. Find (12 - 2 ) ÷ 2 to find the cost each person will pay.

Algebra Temperature can be measured in degrees Fahrenheit (°F) or in degrees Celsius (°C). When you know a temperature in degrees Fahrenheit, you can find the temperature in degrees Celsius by using the expression 5 × (F - 32) ÷ 9. Find each temperature in degrees Celsius. 24. 41°F 25. 68°F 26. 95°F 27. If the temperature of a cup of hot chocolate is 104°F, what is the temperature of the cup of hot chocolate in degrees Celsius? 28. Guess, check, and revise to find the temperature in degrees Fahrenheit that would equal 0°C.

29. OPEN ENDED Write an expression using only multiplication and subtraction so that its value is 25. 30. CHALLENGE Use each of the numbers 2, 3, 4, and 5 exactly once to write an expression that equals 5. 31.

E

WRITE MATH Should you ever add or subtract before you

multiply in an expression? Explain your reasoning.

420


Test Practice 32. Each of Alisha’s 4 photo albums holds 24 vertical photos and 24 horizontal photos. Which shows one way to find the total number of photos Alisha’s photo albums can hold?

34. A seating area has 8 rows, with 15 chairs in each row. If 47 seats are occupied, which of the following shows how to find the number of empty chairs? A. Add 47 to the product of 15 and 8.

A. 24 + (24 + 4)

B. Add 15 to the product of 47 and 8.

B. (24 × 24) + 4

C. Subtract 47 from the product of 15 and 8.

C. 24 × (24 × 4)

D. Subtract 15 from the product of 47 and 8.

D. (24 + 24) × 4 33. What is the value of the expression? 3 × (2 3 - 1) + 8 F. 22

H. 29

G. 25

I. 30

35. Katie uses the order of operations to evaluate the following expression. 15 + (20 - 4 × 4) What should be the last step Katie performs? F. 20 - 4

H. 15 + 4

G. 4 × 4

I. 15 + 64

Order of Operations An expression may include a fraction bar as a grouping symbol. Evaluate the numerator first and then the denominator. Finally, divide.

Evaluate

3+6×2 _ . 10 ÷ 2

3 +6×2 3 + 12 _ =_ 5 10 ÷ 2 15 _ = 5 =3

Evaluate each expression. (16 + 4) - 11 36. __ 32

Multiply in the numerator. Divide in the denominator. Add in the numerator. Divide.

37.

(13 - 9) × (2 + 3) __ 2 2 +1

To assess mastery of SPI 0506.3.2, see your Tennessee Assessment Book.

421

Multi-Part Lesson

1

Order of Operations

PART

A

Main Idea I will evaluate expressions with a variable using the order of operations.

B

C

D

E

Evaluate Expressions A variable , like x, is a letter or symbol used to represent an unknown amount that can vary. An expression , like x + 2, is a combination of variables, numbers, and at least one operation.

Vocabulary V

x+2

variable

vvariable

operation

expression coefficient

When you replace a variable with a number, you can find the value of the expression. This is called evaluating the expression.

Get ConnectED GLE 0506.3.2 Develop and apply the concept of variable. GLE 0506.3.3 Understand and apply the substitution property. SPI 0506.3.1 Evaluate algebraic expressions involving decimals and fractions using order of operations.

SPORTS Alex scored 4 goals. Theresa scored g more goals than Alex. Write an expression using the variable g. Evaluate the expression if g = 7 to find the number of goals Theresa scored. 4+g


4+7

Replace g with 7.

11

Add 4 and 7.

So, Theresa scored 11 goals.

Evaluate Expressions Evaluate the expression 15 - (x + 5) if x = 8. 15 - (x + 5)


15 - (8 + 5)

Replace x with 8.

15 - 13 2

Add 8 and 5. Subtract 13 from 15.

If x = 8, 15 - (x + 5) = 2.

422


Algebraic expressions do not usually contain a multiplication sign ×. Here are some ways to show multiplication. 2n means 2 times n.

4(x + y) means 4 times (x + y).

In the expressions 2n and 4(x + y), the numbers 2 and 4 are called coefficients. A coefficient is the numerical factor of a term containing a variable.

Write and Evaluate Expressions FOOD Terrell made x sandwiches. He used 2 slices of bread for each sandwich he made. If x = 6, how many slices of bread did Terrell use? 2x


2×6

Replace x with 6.

12

Multiply 2 and 6.

So, Terrell used 12 slices of bread. Evaluate 4(x 2 + y) if x = 3 and y = 1. 4(x 2 + y) Write the expression. You can also use parentheses to show multiplication. 4(10) = 4 × 10

4(3 2 + 1) Replace x with 3 and y with 1. 4(9 + 1) Evaluate the power first: 32 = 3 × 3 or 9. 4(10) 40

Add 9 and 1. Multiply.

So, if x = 3 and y = 1, 4(x 2 + y) = 40.

9. See Examples 11–44 Evaluate each expression if a = 4 and b = 9 1. 4 + a

2. b + 30 ÷ 6

3. (b - a) + 6

4. a 2 - 5

5. 3a - 2

6. 6b + 3 3

7. MONEY Arturo buys x containers of the ice cream shown to take to his friend’s party. How much money did Arturo spend if he bought 4 containers of ice cream and a cake that costs $6? 8.

E

TALK MATH Write a real-world problem that contains the

variable y. Then evaluate your expression if y = 6.

Lesson 1B Order of Operations 423

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Evaluate l each h expression i if y = 3 and d z = 8. See Examples 1–4 9. y + 14 12. 29 - z

10. (z + 10)

11. 15 - y

13. z 2 ÷ 4

14. 18 ÷ y

Evaluate each expression if f = 3 and g = 8. 15. g × 5

16. 4 + (f × g)

17. f + g

18. f + 2 4

19. 4( g - 5)

20. f 2 - 1

21. Deirdre has $256 in her savings account. She adds x dollars to her account on Friday. If x = $50.25, write an expression to find the amount of money in Deirdre’s account. Then evaluate. 22. In h hours, a car travels 220 miles. If h = 4, write an expression to find the distance the car travels in 1 hour. Then evaluate. 23. Measurement To find the area of a square, you can use the formula s 2. What is the area of the square shown at the right? 24. Randall had 127 songs on his MP3 player. He deleted x songs. If x = 15, write an expression to find how many songs he has left. Then evaluate.

s = 12 cm

25. Measurement To find the perimeter of a rectangle, you can use the expression 2() + 2(w). Find the perimeter if = 10 inches, and w = 8 inches.

w

Evaluate each expression if a = 0.4, b = 6.3, and c = 10.05. 26. a + b

27. c - a

28. b - 6

29. (9 + c) - b

30. (c - b) + 5

31. (a + b + c) - 7

Evaluate each expression if m =

_1 , n = _1 , and p = _3 . 2

4

5

32. m - n

33. n + p

1 34. _ + p

35. m + n + p

36. p - m

37. (m + n) - p

424


5

38. OPEN ENDED Write an algebraic expression that uses the variable b and an exponent with more than one operation. 39. WHICH ONE DOESN’T BELONG? If m = 4, identify the algebraic expression that does not belong.

m2 40.

E

10 + m

20 - m

4×m

WRITE MATH Explain why the expression 3 less than x is

written as x - 3 and not 3 - x.

Test Practice 41. A taco costs $6. Chandler has a coupon for $1 off. The expression 6n - 1 represents the cost of buying any number n of tacos. How much would it cost Chandler to buy 3 tacos? Food Item

$7

Taco

$6

Enchilada

$8

40 ÷ 5 + (15 - 5) What should be the last step Layla performs? A. 15 - 5

Price ce

Burrito

43. Layla uses the order of operations to evaluate the expression below.

B. 8 + 10 C. 40 ÷ 5 D. 8 + 15

A. $15

C. $17

B. $16

D. $18

42. Evaluate the expression a + b if a = 4.5 and b = 7.2. F. 10.5 G. 11.7 H. 12.1 I. 12.2

44. Which of the following could the expression 4x NOT represent? F. the number of total sides on x number of four-sided figures G. the total cost of four baseball bats if they cost x dollars each H. the total number of wheels on x number of cars I. the number of vacation days that Libby has if she has 4 more than Kelsey

Lesson 1B Order of Operations 425

Multi-Part Lesson

1

Order of Operations

PART

A

B

C

D

Problem-Solving Strategy:

Make a Table

Main Idea I will solve problems by making a table. Nestor is saving money to buy a new camping tent. Each week he doubles the amount he saved the previous week. If he saves $1 the first week, how much money will Nestor save in 7 weeks?

Understand

What facts do you know?

• Each week he doubles the amount he saved the previous week. • The first week he saved $1. What do you need to find? • How much money he will have saved in 7 weeks.

Plan

You can make a table to solve the problem.

Solve

Draw a table with two rows as shown. In the first row, list each week. Then complete the table by doubling the amount he saved the previous week. Week

1

2

3

4

5

6

7

Amount Saved

$1

$2

$4

$8

$16

$32

$64

×2 ×2 ×2 ×2 ×2 ×2 Next, add the amount of money he saved each week. $1 + $2 + $4 + $8 + $16 + $32 + $64 = $127 So, Nestor will save $127 in 7 weeks.

Check

Check to see if the amount saved doubled each week. Use estimation to check for reasonableness. Round each two-digit number to the nearest $10. $1 + $2 + $4 + $8 + $20 + $30 + $60 = $125

GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution. Also addresses GLE 0506.3.2, GLE 0506.3.3.

426


Refer to the problem on the previous page. 1. Explain why you multiplied each week’s savings by 2 to solve the problem.

3. Find the amount of money Nestor will save in 9 weeks.

2. Explain why making a table made this problem easier to solve.

4. Suppose Nestor tripled the amount of money he saved each week. How many weeks will it take him to save $120?

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Solve. Use the make a table strategy. 5. Algebra Betsy is saving to buy a new sound system. She saves $1 the first week, $3 the second week, $9 the third week, and so on. How much money will she save in 5 weeks? 6. Kendall is planning to buy the laptop shown below. Each month she doubles the amount she saved the previous month. If she saves $20 the first month, in how many months will Kendall have enough money to buy the laptop?

7. Measurement Liseli is 3 years old. Her mother is 35 years old. How old will Liseli be when her mother is exactly five times as old as she is? 8. Melissa bought packages of pencils for $3 each. Each package contains 12 pencils. If she spent $15 on pencils, how many pencils did she buy?

9. Measurement A recipe for cupcakes calls for 3 cups of flour for every 2 cups of sugar. How many cups of sugar are needed for 18 cups of flour? 10. Mrs. Piant’s yearly salary is $42,000 and increases $2,000 per year. Mr. Piant’s yearly salary is $37,000 and increases $3,000 per year. In how many years will Mr. and Mrs. Piant make the same salary? 11. Geometry Mr. Ortega is making a model of a staircase he is going to build. Use the picture below to find how many blocks Mr. Ortega will need if the staircase has 12 steps.

12.

E

WRITE MATH Write a real-world

problem that you can solve using the make a table strategy. Explain why making a table is the best strategy to use when solving your problem.

Lesson 1C Order of Operations 427

Multi-Part Lesson

1

PART

Order of Operations A

Main Idea I will complete function tables.

Vocabulary V ffunction

B

C

D

E

Function Tables Did you know that a giraffe sleeps an average of 2 hours each day? There is a relationship between the number of days and the number of hours of sleep. We call this relation a function.

input output function rule function table

Get ConnectED GLE 0506.3.2 Develop and apply the concept of variable. GLE 0506.3.3 Understand and apply the substitution property.

A function is a relationship between two variables in which one input quantity is paired with exactly one output quantity. The input is the quantity put into a function. The end amount is the output . The function rule is the operation performed on the input value. You can use a function table to organize input-output values.

ANIMALS How many hours of sleep will a giraffe get in 1, 2, 3, 4, and 5 days? Make a function table. In words, the rule is multiply the number of days by 2. As an expression, the rule is 2d. Days

Input (d)

2d

Output

1

2 ×1

2

2

2 ×2

4

3

2 ×3

6

4

2 ×4

8

5

2 ×5

10

In 5 days, a giraffe will sleep about 10 hours.

428


Hours of sleep

Use a Function Table MONEY The cost of renting a popcorn machine is $30 plus $8 per hour. Find the function rule. Then make a function table to find the cost of renting the popcorn machine for 4, 5, and 6 hours.

The expression 8x means 8 times the value of x.

The function rule is 30 + 8h. First multiply 8 by the input value. Then, add 30. Number of hours

Input ((h) (h)

30 + 8h

Output

4

30 + (8 × 4)

62

5

30 + (8 × 5)

70

6

30 + (8 × 6)

78

Cost

Copy and complete each function table for each real-world real world situation. See Examples 1 and 2 1. Desmond has 9 more model airplanes than his brother. Input (x)

x+9

2. Each comic book at the comic shop costs $4.

Output

Input (x)

6

5

9

6

12

7

3. Kristen is buying magazines that cost $3 each. She has a coupon for $2 off the total purchase. Input (x)

3x - 2

Output

4. Bryan and Jeanie split the cost of a package of photo paper. Each package costs $4.

Output

Input (x)

2

1

3

2

4

3

5. Hiro charges $8 for each dog he washes. Find the function rule. Then make a function table to find how much money he would make if he washes 4, 5, or 6 dogs.

4x

6.

E

4x ÷ 2

Output

TALK MATH Explain what the

function rule 9n - 4 means. Then find the output value if n = 12.

Lesson 1D Order of Operations 429

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C Copy and d complete l t each h ffunction ti ttable bl ffor each h real-world l ld situation. See Examples 1 and 2 7. Aidan and his friend Tobias play on the school basketball team. Aidan scored 9 points less than Tobias. Input ( p)

p-9

8. It costs $5 a month to be a member of an internet movie company and $1 for every movie you rent.

Output

Input (x)

19

3

20

5

21

7

5x + 1

Find the function rule. Then make and complete a function table. See Examples 1 and 2 9. Ginny had a coupon for $5 off any item at Music Mania. Find the final cost of items that cost $20, $25, and $30. 10. Measurement A textbook weighs about 6 pounds. Find the total weight of 5, 7, and 9 textbooks. 11. Three friends share the cost of renting video games that cost $6 each. How much would one of the friends pay if they rented 2, 3, and 4 games.

12. OPEN ENDED Write a function rule involving both addition and multiplication. Choose three input values and find the output values. 13. FIND THE ERROR Desiree is writing a function rule for the expression 5 less than y. Help find and correct her mistake.

5-y

14.

E

WRITE MATH Write a real-life problem that can be represented

by a function table.

430


Output

Test Practice 15. Rex is buying stickers. The table shows the price of different numbers of stickers. Price Number of Stickers

$0.50 $1.00 $1.50 $2.00 25

50

75

16. A milkshake costs $3. The function rule 3n represents the cost of buying any number of milkshakes. Which shows 3n in words? F. n more than 3

G. 3 more than n

How many stickers will he get for $2?

H. 3 times n

A. 75

I. 3 less than n

B. 80

17. Find the missing value in the table below.

C. 100 D. 125

Input (x)

4

5

6

Output

32

40

48

7

A. 63

C. 56

B. 58

D. 50

Function Tables When you know the rule and output of a function, you can use the work backward strategy to find the input. Input (x)

Find the input for the function table shown. If the output is found by subtracting 7, then the input is found by adding 7.

x-7

Output

10 11 12

So, the missing inputs are 10 + 7 or 17, 11 + 7 or 18, and 12 + 7 or 19.

Find the input for each function. 18.

Input (x)

x+3

Output

5 6 7

19.

Input (x)


5x

Output

15 20 25

431

Multi-Part Lesson

2

PART

Identify and Plot Ordered Pairs A

Main Idea I will name points on a coordinate plane.

Vocabulary V coordinate plane origin ordered pair x-coordinate y-coordinate

Get ConnectED GLE 0506.4.3 Describe length, distance relationships using the first quadrant of the coordinate system. Also addresses GLE 0506.1.5.

B

C

Ordered Pairs A coordinate plane is formed when two number lines intersect. One number line has numbers along the horizontal x-axis (across) and the other has numbers along the vertical y-axis (up). The point where the two axes intersect is the origin . An ordered pair is a pair of numbers that is used to name a point. The first number is the x-coordinate and corresponds to a number on the x-axis.

(3, 2)

y

7 6 5 4 3 2 1

y-axis

origin

O

1 2 3 4 5 6 7 x

The second number is the y-coordinate and corresponds to a number on the y-axis.

Name the ordered pair for point A. Step 1

x-axis

y

5 4 3 2 1

Start at the origin (0, 0). Move right along the x-axis until you are under point A. The x-coordinate of the ordered pair is 5.

A

O

1 2 3 4 5 x

Step 2 Move up until you reach point A. The y-coordinate is 4. So, point A is named by the ordered pair (5, 4). MAPS Name the location of Amy’s house. Step 1

Start at the origin (0, 0). Move right along the x-axis until you are under Amy’s house. The x-coordinate of the ordered pair is 3.

Step 2 Move up until you reach Amy’s house. The y-coordinate is 5. So, Amy’s house is located at (3, 5).

432


6 5 4 3 2 1

O

y

Amy’s House Park School Library 1 2 3 4 5 6 x

Name Points Using Ordered Pairs Name the point for the ordered pair (2, 3). Step 1

Start at the origin (0, 0). Move right along the x-axis until you reach 2, the x-coordinate.

Step 2 Move up until you reach 3, the y-coordinate.

7 6 5 4 3 2 1

O

So, point D is named by (2, 3).

y

C A D B 1 2 3 4 5 6 7 x

SCIENCE An archeologist found a vase at a point that was two units right and one unit up from the necklace. Name the location of the vase.

Make sure to start at the point of the necklace rather than the origin.

Step 1 Start from the necklace. Move two units to the right and one unit up.

y

Step 2 Write your location in relation to the origin. (5, 6) So, the vase was located at (5, 6).

O

x

Locate and name the ordered pair pair. See Examples 1 and 2 1. A

2. C

3. D

Locate and name the point. See Example 3 4. (4, 3)

5. (1, 6)

6. (5, 2)

7. Refer to Example 4. Write the ordered pair that names the ring on the grid. See Example 4 8.

E

7 6 5 4 3 2 1

O

y

E C D

B H

A 1 2 3 4 5 6 7 x

TALK MATH Are the points at (3, 8) and (8, 3) in the same

location? Explain your reasoning.

Lesson 2A Identify and Plot Ordered Pairs

433

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Locate L t and d name the th ordered d d pair. i See Examples 1 and 2 9. A

10. J

11. Q

12. R

13. E

14. N

y 8 Q P 7 M 6 R B 5 J A G 4 H 3 L F 2 1 N E C O 1 2 3 4 5 6 7 8 x

Locate and name the point. See Example 3 15. (2, 2)

16. (1, 5)

17. (4, 8)

18. (0, 3)

19. (6, 7)

20. (7, 0)

Use the map of the playground at the right.

y

See Examples 2 and 4

21. What is located at (7, 3)? 22. Write the ordered pair for the sandbox. 23. Suppose the x-coordinate of the water fountain was moved to the right 1 unit. What would be the new ordered pair of the water fountain? 24. If the y-coordinate of the slide was moved up 2 units, what would be the ordered pair of the slide?

O

25. Cam identified a point that was 4 units above the origin and 8 units to the right of the origin. What was the ordered pair? 26. Suppose point (6, 5) was moved 3 units to the left and moved 2 units down. Write the new ordered pair.

27. OPEN ENDED Create a map of a zoo using a coordinate plane. Locate five animals on the map. Include the ordered pairs for the location of the five animals. 28. CHALLENGE Name the ordered pair whose x-coordinate and y-coordinate are each located on an axis. 29. CHALLENGE Give the coordinates of the point located halfway between (3, 3) and (3, 4). 30.

434

E

WRITE MATH Describe the steps to locate point (7, 4).


x

Where’s My Line?

You will need: graph paper

Naming and Locating Points

Get Ready! Players: 2 players

Get Set! Players should sit so they cannot see each others’ papers. Each player draws a coordinate plane on graph paper and labels each axis from 0 to 10. Then each player draws a straight line on the graph paper. The line should pass through at least three points that are named by ordered pairs of whole numbers.

Player 1 calls out a whole number ordered pair. Player 2 calls out “Hit!” if the ordered pair describes a point on his or her line, or “Miss!” if it does not. If Player 1 scores a hit, he or she takes another turn. If not, Player 2 takes a turn. The first player to locate two additional points on the other player’s line wins.

Players choose who will go first.

Go! Each player begins by naming the ordered pair of one point on his or her line.

Game Time Where’s My Line? 435

Multi-Part Lesson

2

PART

Identify and Plot Ordered Pairs A

Main Idea I will graph points on a coordinate plane.

Vocabulary V

B

C

Graph Functions To graph a point in mathematics means to place a dot at the point named by an ordered pair.

graph

Get ConnectED GLE 0506.4.3 Describe length, distance relationships using the first quadrant of the coordinate system. SPI 0506.4.5 Find the length of vertical or horizontal line segments in the first quadrant of the coordinate system, including problems that require the use of fractions and decimals. Also addresses GLE 0506.1.5.

Bailey was making a treasure map for a game he was playing with his friend. Graph and label where the treasure is found, point X(3, 6), on a coordinate plane. Step 1 Start at the origin (0, 0). Step 2 Move 3 units to the right on the x-axis. Step 3 Then move up 6 units to locate the point. Step 4 Draw a dot and label the point X.

7 6 5 4 3 2 1

O

y

X

1 2 3 4 5 6 7 x

Graph Ordered Pairs Graph and label point M(2, 4) on a coordinate plane. Step 1 Start at the origin (0, 0). Step 2 Move 2 units to the right on the x-axis.

7 6 5 4 3 2 1

O

Step 3 Then move up 4 units to locate the point. Step 4 Draw a dot and label the point M.

436


y

M

1 2 3 4 5 6 7 x

The input and output values from a function table can be written as ordered pairs and graphed.

Graphing Functions TRANSPORTATION The cost of a taxi ride is $3 plus $2 for each mile. Given the function rule 3 + 2x, find the total cost of traveling 1, 2, 3, and 4 miles. The function rule is 3 + 2x. To find each output, follow the order of operations. Multiply each input by 2. Then add 3. Use the order of operations to solve. 3 + 2(2) | | 4 3 + / \ 7

Input (x)

3 + 2x

Output (y)

Ordered pairs

1

3 + 2(1)

5

(1, 5)

2

3 + 2(2)

7

(2, 7)

3

3 + 2(3)

9

(3, 9)

4

3 + 2(4)

11

(4, 11)

Now graph the ordered pairs. 11 10 9 8 7 6 5 4 3 2 1

O

y

1 2 3 4 5 6 7 8 9 1011 x

Graph and label each point on a coordinate plane plane. See Examples 1 and 2 1. Z(2, 2)

2. D(4, 0)

3. Y(5, 6)

4. W(7, 6)

5. C(0, 4)

6. B(3, 7)

7. A bag of birdseed weighs 5 pounds. Given the function rule 5x, find the total weight for 0, 1, 2, and 3 bags of birdseed. Make a function table and then graph the ordered pairs. See Example 3 8.

E

TALK MATH Explain how you would graph the point S(10, 7).

Lesson 2B Identify and Plot Ordered Pairs 437

EXTRA

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Begins on page EP2.

Graph h and d llabel b l each h point i on a coordinate di plane. l See Examples 1 and 2 9. J(1, 1)

10. K(7, 0)

11. L(2, 5)

12. M(0, 6)

13. N(4, 1)

14. P(8, 2)

15. Q(3, 4)

16. R(6, 3)

For Exercises 17–20, make a function table. Then graph the ordered pairs on a coordinate plane. See Example 3 17. Miss Henderson has a coupon for $2 off any item at Sport Inc. Find the new cost of items if they originally cost $4, $6, $8, and $10, given the function rule x - 2. 18. Measurement Amado’s book bag weighs 1 pound. Each book that he puts in his book bag weighs 3 pounds. Given the function rule 3x + 1, find the total weight of the book bag if Amado has 0, 1, 2, and 3 books in his book bag. What do you notice about the graph? 19. Reiko works in an electronics store. Every day he earns a flat rate of $10 plus $5 per hour. Given the function rule 5x + 10, find how much Reiko would earn if he worked 2, 3, 4, and 5 hours. 20. Gaspar and Joyce agreed to split the cost of a DVD. Given the

x function rule _ , find how much each friend would pay if the DVD 2

cost $8, $10, $12, and $14.

Science

The growth rate of a baby blue whale is one of the fastest in the animal kingdom. The table shows the age in months and length in feet of a baby blue whale. 21. Use the table to write the ordered pairs.

Growth of Blue Whale Age (months) Length (ft)

22. Graph the ordered pairs.

0

23

1

27

23. What is the length of a baby blue whale when it is 2 months old?

2

31

3

35

4

39

24. How old is a baby blue whale that is 37 feet long? 25. Estimate the length of a baby blue 1 whale that is 2_ months old. 2

438


26. OPEN ENDED Write an ordered pair for a point that would be graphed on the y-axis. 27.

E

WRITE MATH Write a real-world problem about a situation

that would be represented by the function 15x.

Test Practice 28. Ashley made the grid below to show the location of objects in her backyard. What are the coordinates of the object closest to the bird feeder?

29. The teacher asks you to place a point exactly halfway between points Q and R. Which ordered pair best describes this location? y 6 5 Q 4 3 2 1

y 6 tree house 5 swings shed 4 3 bird feeder 2 1 flowers O 1 2 3 4 5 6 x

O

R S

T 1 2 3 4 5 6 x

A. (5, 1)

C. (3, 6)

F. (0, 4)

H. (2, 2)

B. (7, 5)

D. (1, 4)

G. (2, 4)

I. (2, 5)

Graphing You have already learned how to find the distance between two points on a number line. You can also find the distance, or length, between two points on a coordinate plane.

Find the distance between the points A(3, 5) and B(3, 2) shown on the coordinate plane. If two points have the same x- or y-coordinates, find the distance by counting units on the grid or by subtracting. There are 3 units between points A and B. Note that 5 - 2 = 3.

8 y 7 6 5 A(3, 5) 4 3 2 B(3, 2) 1

O

3 units x

1 2 3 4 5 6 7 8

Find the distance between each pair of points. 31. M(4_, 9) and N(1, 9)

30. X(2, 8) and Y(2, 2) 1 1 32. W(14, 6_) and V(14, 20_) 2

2

1 2

33. A(15, 0) and B(2, 0)

Lesson 2B Identify and Plot Ordered Pairs 439

Multi-Part Lesson

2

Identify and Plot Ordered Pairs

PART

A

B

C

Compare Graphs of Functions Main Idea

The graphs of some functions are curves, not lines.

I will compare functions on a graph.

Get ConnectED GLE 0506.4.3 Describe length, distance relationships using the first quadrant of the coordinate system. SPI 0506.4.5 Find the length of vertical or horizontal line segments in the first quadrant of the coordinate system, including problems that require the use of fractions and decimals. Also addresses GLE 0506.1.5.

The square shown has side lengths of x units. Make function tables that show the perimeters and areas of squares of various lengths.

Step 1

Step 2

x x

Make a function table that shows perimeters. The perimeter is found by adding the lengths of all sides. Side Lengths (x)

Perimeter (y)

1 2 3 4

Make a function table that shows areas. The area is found by multiplying length by width. Side Lengths (x)

Area (y)

1

2 3 4

About It 1. Graph each function on the same coordinate plane. Connect the points with a line or smooth curve. 2. Compare the graphs of the two functions. 3. At what ordered pair do the two graphs intersect? What does this mean?

440


Mid-Chapter Check 1. MULTIPLE CHOICE What is the value of the expression (42 + 5) - 17? (Lesson 1A)

A. 2

C. 23

B. 4

D. 35

Find the value of each expression.

12. Reese is planting flowers in five rows. The first row has 28 flowers. Each additional row has 6 fewer flowers than the previous row. How many flowers will Reese plant? (Lesson 1C) 13. Heath ate half of his pretzels. Copy and complete the function table. (Lesson 1D)

(Lesson 1A)

2. 14 - 3 × 4

Input (x)

x÷2

Output

12

14

3. (23 - 17) × 9

16

4. (14 - 7) × (12 + 13) For Exercises 14–19, use the map shown. 5. (5 +

3)2

+7 7 6 5 4 3 2 1

6. Aimee has t tickets. Dale has 7 more tickets than Aimee. Write an expression for the number of tickets Dale has. Then evaluate the expression if t = 2. (Lesson 1B)

O

y

park

library pizza place grocery store school music store 1 2 3 4 5 6 7 x

Evaluate each expression if n = 3. (Lesson 1B)

7. n + 7 9. 12 + n

8. (n + 9) - 5 30 10. _ + (6 + n) 5

11. MULTIPLE CHOICE Alonzo waited x minutes to ride the bumper cars. Pearl waited 3 times as long plus an additional 5 minutes. Which expression could be used to find the number of minutes Pearl waited? (Lesson 1B) F. 3 + x + 5

H. 5x + 3

G. 3x + 5

I.

3x - 5

Locate and name the ordered pair for each place on the map. (Lesson 2A) 14. park

15. school

16. library

Locate and name the place on the map for each ordered pair. (Lesson 2A) 17. (4, 6) 20.

E

18. (6, 1)

19. (3, 4)

WRITE MATH Write two different

expressions using n and 2, one with division and one with subtraction. Explain how to evaluate them if n = 6. (Lesson 1B)

Mid-Chapter Check 441

Multi-Part Lesson

3

Equations

PART

A

B

C

D

E

Model Addition Equations Main Idea I will explore writing and solving addition equations using models.

An equation is a number sentence that contains an equals sign, (=), showing that two expressions are equal. To solve an equation means to find the value of the variable so the sentence is true. You can use cups and counters to represent problem situations. The cup represents the variable.

Vocabulary V equation solve solution

Materials algebra mat

Collin had some goldfish, and then he bought 2 more. Now he has 8 goldfish. Solve the equation x + 2 = 8 to find how many goldfish Collin had at first.

Step 1

Place the cup on the left side to show x and two counters to show 2. Place 8 counters on the right side to show 8.

balance

Step 2 connecting cubes counters cups

Get ConnectED GLE 0506.3.4 Solve single-step linear equations and inequalities. Also addresses GLE 0506.1.4, GLE 0506.1.8.

Model the equation.

=

x +2

8

Solve the equation. THINK How many counters need to be in the cup so there is an equal number of counters on each side of the mat?

=

x =6

When you found the number of counters in the cup, you found the solution of the equation. A solution is the value of the variable that makes the sentence true. So, 6 is the solution of the equation x + 2 = 8. Collin had 6 goldfish at first.

442

=


You can also use a balance, cup, and connecting cubes to model equations. The cup represents the variable.

How many cubes are in the cup? x + 2 = 6 Write the equation. THINK If you take 2 cubes off each side, it would still be balanced.

x+2= 6 -2=-2 −−−−−−−− x 4 So, x = 4.

About It 1. Refer to Activity 1. What does x represent? 2. How did you know that 6 was the solution to the equation? 3. Describe how you would model and solve x + 4 = 6.

and Apply It Write an equation for each model. Then solve. 5.

4.

=

Write an equation for each model. Then, find the weight of the item. 7.

6. =1 =5

8.

E

=1 =5 = 10

WRITE MATH How could you solve x + 3 = 8 without

using models?

Lesson 3A Equations 443

Multi-Part Lesson

3

PART

Equations A

Main Idea I will write and solve addition and subtraction equations.

Vocabulary V defining the variable

B

C

D

E

Addition and Subtraction Equations Addition and subtraction are opposite operations. You can undo addition using subtraction.


Subtraction Property of Equality

Words

If you subtract the same number from each side of an equation, the two sides remain equal.

Examples

6= 6 2=-2 −−−−−−− 4= 4

Symbols

7 + x = 10 7 =-7 −−−−−−−−−− x= 3

On Monday and Tuesday, Aubrey walked a total of 7 blocks. If she walked 3 blocks on Tuesday, how many blocks did she walk on Monday?

One Way:

Use a model.

Let × represent the number of blocks walked on Monday. You know that 4 plus 3 is equal to 7. So, x = 4.

Another Way:

x

Use symbols.

x+3= 7

Write an equation for the situation.

- 3 =-3 x = 4

Subtract 3 from each side.

So, Aubrey walked 4 blocks on Monday.

444

7


3

Addition Property of Equality

Words

If you add the same number to each side of an equation, the two sides remain equal.

Examples

6= 6 + 2=+2 −−−−−−− 8= 8

Symbols

x-3= 5 +3=+3 −−−−−−−−− x = 8

Addition Property of Equality Solve x - 6 = 8.

One Way:

Use a model.

You know that 14 minus 6 is equal to 8. So, x = 14. You can check your answer by replacing x n with 14 in the equatio and evaluating. 14 - 6 = 8

Another Way: x-6=

8 +6=+6 x = 14

x 6

8

Use symbols.

Write the equation. Add 6 to each side. The solution is 14.

Choosing a variable to represent an unknown value is called defining the variable .

You can use the first letter of the word you are defining as a variable. For example: b is for the number of home runs Barry hit.

SPORTS Ricky, Manny, and Barry hit a total of 78 home runs. If Ricky hit 34 home runs and Manny hit 21 home runs, how many home runs did Barry hit? 34 + 21 + b =

78

55 + b = 78 - 55 = - 55 −−−−−−−−−−−−−−−−− b = 23

Write the equation. Let b represent Barry’s home runs. Add 34 and 21. Subtract 55 from each side.

So, Barry hit 23 home runs.

Lesson 3B Equations 445

Solve each equation equation. Check your solution. solution See Examples 1 and 2 1. h + 4 = 8

2. 2 + x = 9

3. p - 7 = 1

4. r - 5 = 4

5. During Wednesday’s game, Mona stole 2 bases, giving her a total of 8 stolen bases. Write and solve an equation to find how many stolen bases Mona had before Wednesday’s game. See Example 3 6.

E

TALK MATH Explain what it means to solve an equation.

EXTRA

% )# E # T4 IC !C 2A PR 0

Begins on page EP2.

Solve S l each h equation. ti Ch Check k your solution. l ti See Examples 1 and 2 7. x + 1 = 5

8. n - 4 = 2

10. g + 18 = 39

11. (25 - 5) + c = 50

9. d - 36 = 40 12. 57 + 12 + t = 75

Write an equation and then solve. Check your solution. See Example 3 13. A box contained some snack bars. Tyrese ate 4 snack bars. Now there are 8 snack bars left. How many snack bars were there at the beginning? 14. Molly spent $2.50 on comic books. She now has $7.00. How much money did she have to start with? 15. Leroy used 1_ cups of butter for a batch of cookies he made. Then 4 1 he baked a cake. He used a total of 2_ cups of butter for the cookies 4 and cake. How many cups of butter did he use for the cake? 3

Use the information to solve the problem.

16. Write an equation that can be used to find the cost of the video game. Then solve.

446


17. OPEN ENDED Write two different equations that have 12 as the solution. Sample answer: x + 2 = 14, x + 3 = 15 18. WHICH ONE DOESN’T BELONG? Identify the equation that does not belong with the other three. Explain your reasoning.

14 - x = 8 19.

E

x + 4 = 10

15 - x = 9

x + 2 = 14; This is the only equation with a solution that is not 6

x + 2 = 14

WRITE MATH Explain why the equation n + 7 = 15 has

the same solution as 15 - n = 7. Sample answer: fact family; 8 + 7 = 15; 15 - 8 = 7

Test Practice 20. Becka and Dina scored a total of 18 points. If Becka scored 7 points, which equation should you use to find the number of points Dina scored? B A. 18 + 7 = 25 B. 7 + d = 18 C. 18 + d = 25 D. d + 18 = 7 21. The area of Lake Palmer is 3,200 square yards. The area of Lake Palmer is 1,850 square yards more than Lake Hunter. Which equation can be used to find the area of Lake Hunter? F

22. Sample answer: 5.5 + 5 + e = 15; 4.5 22.

SHORT RESPONSE Three students were working together on a 15-page report. The table lists the number of pages each student wrote. Write an equation to determine how many pages Estella wrote. Then solve. Student

Pages Written

Mike Amon Estella

5.5 5

23. Which of the following can be used to solve the equation shown? D x - 15 = 39 A. Multiply each side of the equation by 15. B. Divide each side of the equation by 15.

F. x + 1,850 = 3,200 G. 1,850x = 3,200 H. x - 1,850 = 3,200

C. Subtract 15 from each side of the equation. D. Add 15 to each side of the equation.

I. 1,850 + 3,200 = x


447

Multi-Part Lesson

3

PART

Equations A

Main Idea I will explore writing and solving multiplication equations.

B

C

D

E

Model Multiplication Equations You can use cups and counters to represent problem situations involving multiplication. Recall the cup represents the variable.

Materials algebra mat

Two friends split the cost of a pizza evenly. If the cost of the pizza is $8, how much did each friend spend? Solve the equation 2x = 8 to find how much each friend spent. balance

Step 1

Place 2 cups on the left side to show 2x. Place 8 counters on the right side to show 8.

connecting cubes counters


Model the equation.

Step 2

=

2x

8

THINK How many counters need to be in each cup so there is an equal number of counters in each cup and an equal number of counters on each side of the mat?

=

x =4

So, x = 4. Each friend paid $4. Check 2x = 8 2·48 8=8

448

=

Solve the equation.


Write the equation. Replace x with 4. Simplify.

Solve the equation 3x = 15.

Step 1

Model the equation. Place 3 cups on one side of the balance and 15 cubes on the other.

Step 2

Solve the equation. THINK How many cubes need to be in each cup so there is an equal number of cubes in each cup and the balance is equal?

So, x = 5. There are 5 cubes in each cup.

and Apply It Write an equation for each model and then solve. Check your solution. 2.

1. =

=

3.

4.

5.

6.

7.

E

WRITE MATH How could you solve 4x = 32 without

using models?

Lesson 3C Equations 449

Multi-Part Lesson

3

Equations

PART

A

Main Idea I will write and solve multiplication and division equations.

Get ConnectED

B

D

C

E

Multiplication and Division Equations Since multiplication and division are opposite operations, you can undo division using multiplication. 8 For example, _ means 8 ÷ 4. To undo dividing by 4, multiply by 4.

GLE 0506.3.4 Solve single-step linear equations and inequalities. Also addresses GLE 0506.1.4.

4

8 You can write 8 ÷ 4 × 4 as _(4). 4

Multiplication Property of Equality

Words

If you multiply each side of an equation by the same nonzero number, the two sides remain equal.

Examples

4

= 4

4(2) = 4(2) 8

Solve

_x = 7 2

Symbols

= 8

_x (2) = 7(2) 2 x

Multiplication Property of Equality

_y = 4. 3

One Way:

y

Use a model.

THINK What number divided by 3 is equal to 4?

4

You know that 12 divided by 3 is equal to 4. So, y = 12.

Another Way:

Use symbols.

_y

Write the equation.

3

450

= 14

= 4

_y (3) = 4(3) 3

Multiply each side by 3.

y

The solution is 12.

= 12


4

4

Division Property of Equality

4x _ Think of 4 as 4 times x divided by 4.

Words

If you divide each side of an equation by the same nonzero number, the sides remain equal.

Examples

8

= 8

8 _ 2

=

8 _ 2

4

= 4

Symbols

4x = 12 4x 12 _ = _ 4 4

x

=

3

Division Property of Equality Coach Carlota needs to purchase 18 gold medals for her soccer team. The medals are sold in packages of 3. How many packages should Coach Carlota buy?

Check the answer by replacing p with 6 in the equation. 3p = 18 3(6) = 18 18 = 18

3p = 18

Write an equation for the situation.

3p _ 18 _ =

Divide each side by 3.

3

3

p=6 So, Coach Carlota needs to buy 6 packages.

FOOTBALL The Jaguars scored 3 times as many points as their opponent. If the Jaguars scored 21 points, how many points did their opponent score? Words

Jaguar’s points equals 3 times the opponent’s points.

Variable

Let p represent the opponent’s points.

Equation

21 =

21 = 3p

3p

3p 21 _ = _ 3 3

7 =

Write the equation. Divide each side by 3.

p

So, the opponents scored 7 points.

Lesson 3D Equations 451

Solve each equation equation. Check your solution. solution See Examples 1 and 2 t 2. _ = 9

1. 2b = 8

3. 21 = 7x

3

4. 6x = 24

Write an equation and then solve. Check your solution. See Example 3 5. Irena is half as old as Yoko. If Irena is 20, how old is Yoko? 6. To paint a classroom, you need 3 gallons of paint. If you have 27 gallons of paint, how many classrooms can you paint if they are all identical? 7.

E

TALK MATH Describe how to find the solution of 8x = 72. EXTRA

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Begins on page EP2.

Solve Check S l each h equation. ti Ch k your solution. l ti See Examples l 1 and d2 d 9. _ = 6

8. 4b = 16

4

12. 33 = 11t

13. 25 = r ÷ 3

10. 30 = 5h

z 11. _ = 4

14. r ÷ 8 = 14 - 4

15. 30 = 5m

8

Write an equation and then solve. Check your solution. See Example 3 16. Seven students spent a total of 35 hours cleaning and 28 hours painting. Each student spent an equal amount of time. Find how many hours each student spent cleaning and painting. 17. A Girl Scout troop collected 54 cans for the food drive. There are 6 members in the troop, and each member collected the same number of cans. How many cans did each member collect? Solve each equation. Check your solution. 18. 3x = 12.4 - 0.4

19. 10x = 45.3 + 4.7

20. 7x = 53.8 - 4.8

21. 5x = 21.9 + 8.1

2 1 22. 5x = 14_ + _

4 4 23. 3x = 6_ - _

3 1 24. 6x = 5_ + _

8 1 25. 8x = 31_ + _

3

3

5

5

Write an equation and then solve. Check your solution. 26. Greg bought three admissions to the aquarium. Each admission cost the same amount. He spent $48. How much did each admission cost?

452


4

4

9

9

The Naples Zoo opened in 1919. Today, animals and d plants fill the 52 acres of land. Write an equation. Then solve. Check your solution. on. 27. Mr. Graban bought some adult tickets. The total cost was $38. How many adult tickets did Mr. Graban buy?

Naples Zoo Admission Prices Ticket

28. The Solomon family bought some children’s tickets. The cost of the tickets was $70. How many children’s tickets did the family buy?

Price ($)

Adults

19

Seniors

17

Children

10

29. OPEN ENDED Write two different multiplication equations that each have a solution of 9. 30. WHICH ONE DOESN’T BELONG? Identify which equation does not belong with the other three. Explain your reasoning.

35 - n = 28 31.

E

21 = 3n

n + 49 = 56

7n = 63

WRITE MATH Write a real-world problem that can be solved

using a multiplication equation.

Test Practice 32. Vito and Marisol are making party favors. Vito made 18 party favors. If he had made 2 more party favors, he would have made exactly 2 times as many party favors as Marisol did. How many party favors did Marisol make? A. 10

C. 25

B. 20

D. 32

33. A full basket contained 27 apples. There are 9 apples left in the basket now. Which equation could be used to find how many apples were taken from the basket? F. 27 + x = 9

H. 27 + 9 = x

G. 27 - x = 9

I. x - 9 = 27

Lesson 3D Equations 453

Multi-Part Lesson

3

PART

Equations A

B

C

E

D

Problem-Solving Investigation Main Idea I will choose the best strategy to solve a problem.

STEPHANIE: I sold brownies at a bake sale. I sold large brownies for $2 and small brownies for $1. After one hour, I sold 11 brownies and earned $14. How many of each size did I sell, if I sold at least one of each size? YOUR MISSION: Find the number of each size sold.

Understand You know that Stephanie earned $14. She sold large brownies for $2 and small brownies $1. You need to know how many of each size Stephanie sold. She sold 11 brownies.

plan

You need to think about the different combinations of large and small brownies she could have sold. So, the guess, check, and revise strategy is a good choice.

solve

Use a calculator. Since 7 × $2 = $14, less than 7 large brownies were sold. Let’s try 6. Large Earnings Brownie

Small Brownie

Earnings

Total

1st guess

6

$12

5

$5

$17

Revise.

2nd guess

4

$8

7

$7

$15

Revise.

3rd guess

3

$6

8

$8

$14

So, Stephanie sold 3 large brownies and 8 small brownies.

check

Since (3 × $2) + (8 × $1) = $14, the answer is correct.

GLE 0506.3.4 Solve single-step linear equations and inequalities. GLE 0506.1.2 Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.

454


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• • • • •

Guess, check, and revise. Work backward. Make a table. Solve a simpler problem. Choose an operation.

6. Lupe bought sandwiches for herself and 4 friends. She spent a total of $12. How many of each type of sandwich did she buy?

Use any strategy shown to solve each problem. 1. The number on Jarvis’s basketball jersey is between 30 and 50. The sum of the digits is 4. What is the number on Jarvis’s jersey? 2. Algebra Jasmine is making bookmarks. Each day she makes twice as many bookmarks as the day before. On the fifth day she makes 32 bookmarks. How many bookmarks did she make on the first day? 3. Measurement A plumber needs to cut the pipe shown below into 2-inch pieces. If one cut takes 3 minutes, how many minutes will it take the plumber to make the cuts?

7. Geometry Square tables are put together end-to-end to make one long table for a birthday party. A total of 18 people attend the party. How many tables are needed if only one person can sit on each side of the square tables?

8. At the school bake sale, Kenji’s mom bought 3 cookies, 1 brownie, and 1 cupcake. She gave the cashier $2 and received $1.05 in change. Find the cost of a cupcake. Bake Sale Prices Item

in.

4. Some students are standing in the lunch line. Tammie is fourth. Clara is two places in front of Tammie. Eight places behind Clara is Mike. What place is Mike?

Cookie

0.15

Brownie

0.20

Cupcake

9. An electronics store is selling handheld games at 3 for $27. How much will 5 handheld games cost?

10. 5. Measurement A sandbox is made of two squares side-by-side. Each square is 9 feet by 9 feet. What is the total distance around the sandbox?

Price ($)

E

WRITE MATH Write a real-world

problem that you can solve using any problem-solving strategy. What strategy would you use to solve the problem? Explain your reasoning.

Lesson 3E Equations 455

Oceans cover _ of Earth’s surface. 10 Just as there are maps of land, there are maps of the ocean’s floor. Oceanographers use sonar to measure the depths of different places in the ocean. The average depth of Earth’s oceans is 12,200 feet. 7

The deepest part of the ocean floor has been recorded at 36,198 feet. This distance is 7,163 feet greater than the height of Mount Everest, the highest mountain on Earth.

456


Oceanographers send out sound waves from a sonar device on a ship. They measure how long it takes for sound waves to reach the bottom of the ocean and return. They use the fact that sound waves travel through sea water at about 5,000 feet per second. That’s about 3,400 miles per hour!

Use the information from the previous page and the table below to solve each problem. Use a calculator.

1.

Make a function table to find the number of feet sound travels through sea water for 1, 2, 3, 4, and 5 seconds.

2.

Use your table to estimate how long it will take a sound wave to reach the average depth of Earth’s oceans.

3.

Suppose your ship sends out a sound wave and it returns to the ship in 8 seconds. About how deep is the ocean where you are?

4.

Suppose a thunderstorm is about 1 mile away from where you are standing. About how long will it take before you hear the thunder? (Hint: 1 mile = 5,280 feet)

5.

Suppose you know the depth of the ocean is 25,060 feet. Write and solve an equation to find about how long it will take a sound wave to travel this distance.

Speed of Sound Through Different Media (feet per second) Air (2 Ai (20°C) C)

1,,12 1 21

P re Wa Pu atter

4,88 4, 882 82

S a Wa Se W te er

5,01 012 2

Sttee eell

16 6,,0 000 0

Diam Di a on nd

39,2 39 ,2 240

Problem Solving in Science 457

Multi-Part Lesson

4

Inequalities

PART

A

B

Inequalities Main Idea Use models to represent and solve simple addition and subtraction inequalities.

An inequality is a mathematical sentence stating that two quantities are not equal. The expression x < 2 means that the value of x is less than 2. To solve an inequality, find the values of the variables so the inequality is true.

Solve Inequalities Get ConnectED GLE 0506.3.4 Solve single-step linear equations and inequalities. SPI 0506.3.4 Given a set of values, identify those that make an inequality a true statement. Also addresses GLE 0506.1.4, GLE 0506.1.8.

Solve x < 2 using a model. The balance below contains a cup and two counters. Note that the left side weighs less than the right side. Copy and complete the table. x

Is x < 2?

True or False

0

02

true

1

12

2

22

3

32

4

42

x5 8>5 This is true.

The model shows one cup and two counters on the left side and five counters on the rightt side. Remove two counters from om each side of the balance.. There are three counters remaining on the right side of the balance. x+2>5 The solution is any number ber greater than 3. So, x > 3.

Solve a Subtraction Inequality Solve x - 3 < 7 using a table. Check possible values of x. The table suggests that the solution is any number less than 10. So, x < 10.

x

Is x - 3 7?

True or False

8

8-37

true

9

9-37

10

10 - 3 7

11

11 - 3 7

Check the solution with other values of x.

and Apply It Solve each inequality by using a model or mental math. 1. x + 1 < 6

2. x - 3 > 1

3. x + 4 > 8

4. Write two different inequalities, one involving addition and one involving subtraction, both with the solution x < 3. 5.

E

WRITE MATH Explain how you could solve the inequality

x + 13 < 18.

Lesson 4A Inequalities 459

Multi-Part Lesson

4

Inequalities

PART

A

Main Idea I will solve one-step inequalities.

Vocabulary V iinequality

B

Inequalities You have already learned how to compare whole numbers and decimals using inequalities. Inequalities contain the symbols , ≤, and ≥. Inequalities

Get ConnectED Symbols GLE 0506.3.4 Solve single-step linear equations and inequalities. SPI 0506.3.4 Given a set of values, identify those that make an inequality a true statement. Also addresses GLE 0506.1.4.

≤

≥

Words

is less than

is greater than

is less than or equal to

is greater than or equal to

Examples

41

6 ≤ 10

9≥3

Inequalities can be solved by finding the values of the variables that make the inequality true.

Determine the Solution of an Inequality Of the numbers 4, 5, or 6, which is a solution of the inequality x - 1 < 4?

Value of x

x-1 20; 6, 7, 8

13. x - 13 ≥ 30; 41, 42, 43

14. 8 + x ≤ 32; 26, 25, 24

15. 5x > 45; 8, 9, 10

16. 7x ≤ 42; 8, 7, 6

x 17. _ ≥ 3; 36, 18, 9

x 18. _ < 4; 55, 44, 33 11

9

Solve each inequality. See Example 2 19. x - 7 > 12

20. x + 8 < 22

21. 4x > 64

x 22. _ > 8

23. x + 20 ≤ 35

24. 5x > 55

6

Write an inequality. Then solve. See Example 3 25. Callie needs to save at least $100 for her ski trip. She has already saved $45. What is the least amount she still needs to save?

26. Rodrigo has $60 to buy pizzas for the Art Club. Each pizza is $12. What is the greatest number of pizzas he can buy?

27. The table shows the number of online concert tickets that were sold for different concerts. If the number of tickets sold is greater than 1,000, the concert is considered sold out. What is the least number of additional online tickets that need to be sold in order for the symphony concert to be considered sold out? 28. Pasha would like to run at least 15 miles to train for a race. He would like to divide the total number of miles over 5 days. What is the least number of miles he needs to run each day?

462


29. OPEN ENDED Write a real-word problem that would have the solution x ≤ 10. 30. NUMBER SENSE Is it possible for x > 8 and x ≤ 7 to have the same solutions? Explain. 31. CHALLENGE Solve the inequality 5x + 7 ≤ 52. 32.

E

WRITE MATH Is it possible to list all values that are solutions of

the inequality x ≥ 10? Explain.

Test Practice 33. Which of the following is a solution of the inequality x + 9 < 24? A. 14

C. 16

B. 15

D. 17

34. Which of the following inequalities can be used to represent the phrase no more than 35 miles per hour? F. x > 35 G. x < 35

35. Which of the following values is NOT a solution of x > 12 and x ≤ 15? A. 12

C. 14

B. 13

D. 15

36. Ben needs to save at least $250 for a new electric guitar. He would like to save the same amount each week for 10 weeks. Which inequality can be used to find the least amount he should save each week?

H. x ≥ 35

F. x < 25

H. x ≤ 25

I. x ≤ 35

G. x > 25

I. x ≥ 25

37. Brandon’s dog weighs 10 more pounds than Sarah’s dog. The two dogs weigh a total of 80 pounds. What is the weight of Sarah’s dog? (Lesson 3E) Solve each equation. Check your solution. (Lesson 3D) 38. 4x = 44

39. 7x = 49

x 40. _ = 3 8


463

Chapter Study Guide and Review Be sure the following Key Concepts are noted in your C Foldable. F

Vocabulary coefficient equation

Order of Operations

Ordered Pairs

Equa tions

expression Ine qua litie s

function inequality ordered pair order of operations variable

Key Concepts Order of Operations (Lesson 1) • The order of operations is a set of rules that tell you which operation to do first when evaluating an expression. Evaluating Expressions (Lesson 1)

Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence.

Ordered Pairs (Lesson 2)

1. A function is a relationship in which one input quantity is paired with exactly one output quantity.

• A location on a coordinate plane can be described by using an ordered pair.

2. The solution of the inequality x + 5 > 7 is x < 2.

• To evaluate an expression means to find the value of the expression.

• An example of an ordered pair is (7, 4). 3. A variable represents the known value.

(7, 4) x-coordinate

y-coordinate

Solving Equations (Lesson 3)

4. An equation is a number sentence that contains an equals sign.

• An equation is a number sentence that contains an equals sign.

5. Points on a coordinate plane are found using variables.

Inequalities (Lesson 4) • Inequalities can be solved in the same way as equations.

464


6. When evaluating 15 - 3 × 4, the order of operations tells you to perform subtraction first.

Multi-Part Lesson Review Lesson 1

Order of Operations

Order of Operations

(Lesson 1A)

Find the value of each expression. 7. (12 + 6) × 2

8. 26 - 3 × 6

9. Elena bought 3 pairs of running shorts that cost $15 each from an internet store. Shipping costs $5 more. Write an expression to find the total cost. Then evaluate.

Evaluate Expressions

EXAMPLE 1

Find the value of 3 × (4 + 5). 3 × (4 + 5) Add. 3 × 9

Multiply.

27 So, 3 × (4 + 5) = 27.

(Lesson 1B)

Evaluate each expression if b = 2 and c = 5. 10. 3c

11. 5 - b

12. 12 + b

10 13. _ + (4 - b) 2

EXAMPLE 2

Evaluate the expression 4w if w = 7. 4w


4×7

Replace w with 7.

28

Multiply 4 and 7.

So, 4w = 28, if w = 7.

Problem-Solving Strategy: Make a Table

(Lesson 1C)

Solve. Use the make a table strategy.

EXAMPLE 3

14. Sabrina starts by saving $2.50 in one week. She plans to increase her savings by $3.50 each week until she is saving $20 each week. How many weeks will it take Sabrina until she is saving $20 each week?

A marching band has 5 rows with 9 band members in the front row. Each row has 3 more band members than the row in front of it. How many band members are there?

15. Hayden is arranging 10 rows of chairs in the gymnasium. In the last row, there are 54 chairs. Each row has 4 fewer chairs than the previous row. How many chairs will Hayden have to arrange?

Row Members

1 9

2 12

+3 +3

3 15

4 18

+3

5 21

+3

Add the band members in each row. 9 + 12 + 15 + 18 + 21 = 75. So, there are 75 band members.

Chapter Study Guide and Review 465

Chapter Study Guide and Review

Lesson 1

Order of Operations

Function Tables

(continued)

(Lesson 1D)

16. Dino has 3 more pets than his friend. Copy and complete the table. x+3

Input (x)

Output

6 8

EXAMPLE 4

Sofia jumped rope for 2 minutes less than her sister. Find the function rule. Then make a function table to find the number of minutes Sofia jumped if her sister jumped for 6, 8, and 10 minutes. Subtract 2 from each input. The function rule is x - 2.

10

17. Paperback books cost $5 each at Book Smart. Find the function rule. Then make a function table to find the cost of buying 3, 4, and 5 books at Book Smart. What is the cost of 5 books?

Lesson 2

Input (x)

x-2

Output

6

6 -2

4

8

8-2

6

10

10 - 2

8


Ordered Pairs

(Lesson 2A)

Use the coordinate plane below. y 6 P S 5 4 O 3 Q 2 1

O

Locate and write the ordered pair that names point T.

T

5 4 3 2 1

R

1 2 3 4 5 6 x

Locate and write the point for each ordered pair. 18. (3, 5)

19. (5, 6)

EXAMPLE 5

20. (1, 4)

21. Ria identified a point that was 5 units above the origin and 2 units to the right of the origin. What was the ordered pair?

O

y

T

1 2 3 4 5 x

Step 1 Start at the origin (0, 0). Move right along the x-axis until you are under point T. The x-coordinate of the ordered pair is 4. Step 2 Move up until you reach point T. The y-coordinate is 2. So, the point T is located at (4, 2).

466


Lesson 2


Graph Functions

(Lesson 2B)

Graph and label each point on a coordinate plane. 22. A(4, 1) 25. D(4, 4)

(continued)

23. B(2, 1) 26. E(2, 7)

EXAMPLE 6

Graph and label point Q(2, 4).

24. C(6, 5)

6 5 4 3 2 1

27. F(2, 3)

O

Lesson 3

y

Q

1 2 3 4 5 6 x

Equations

Addition and Subtraction Equations

(Lesson 3B)

Solve each equation.

EXAMPLE 7

28. p + 5 = 13

29. 14 = x - 6

Solve x + 4 = 10. x + 4 = 10

30. r - 5 = 3

- 4 =- 4

31. 22 = 11 + n x

32. Celeste gave 4 baseball cards to her brother. She now has 16 baseball cards left. Write an equation and solve it to find how many baseball cards Celeste had at first.

6

=

Subtract 4 from each side. The solution is 6.

EXAMPLE 8

Solve m - 2 = 8. m-2=

8

+ 2 = + 2 Add 2 to each side. m = 10 The solution is 10.

Multiplication and Division Equations Solve each equation. Check your solution. 33. 40 = 10c

34. 2z = 16

35. k ÷ 5 = 6

36. v ÷ 7 = 8

(Lesson 3D)

EXAMPLE 9

Solve 4s = 32. 4s 4

=

32 4

s = 8

Divide each side by 4. The solution is 8.

Chapter Study Guide and Review 467

Chapter Study Guide and Review

Lesson 3

Equations

(continued)

Problem-Solving Investigation: Choose the Best Strategy Use any strategy to solve each problem.

(Lesson 3E)

EXAMPLE 10

37. At a snack bar, Urick was served before Mirna, but after Mallory. Trent was the first of the four friends to be served. Which friend was served last?

Last week Dee gave half of her student council buttons away. This week she gave 7 buttons away. There are 16 buttons remaining. How many buttons did Dee have to begin with?

38. Selena has $150 to spend on skateboarding equipment. Does she have enough money to buy all the equipment listed below? Explain.

Use the work backward strategy.

Gloves Helmet Skateboard Mouth guard

Lesson 4

$14.95 $34.50 $84.50 $9.95

There were 23 buttons before Dee handed out any this week. Last week she handed out half of the original amount. So, multiply 23 by 2. 23 × 2 = 46 Dee had 46 buttons at the beginning.

Check Work forward to check.

Inequalities

Inequalities

(Lesson 4B)

Write an inequality and then solve. 39. Water boils at a temperature of at least 212°F. The temperature of a pot of water is 180°F. What is the number of degrees the water needs to rise in order to boil? 40. Ebony is saving her money for a school trip. She needs to save at least $50. If she has 10 weeks to save, how much should she save each week?

468

Dee gave out 7 buttons. Add 7 to the remaining buttons. 16 + 7 = 23


EXAMPLE 11

Solve x + 9 ≤ 20. x + 9 ≤ 20 - 9 -9 x ≤ 11

Write the inequality. Subtract 9 from each side.

The solution is x ≤ 11. This means that any number less than or equal to 11 is a solution.

Practice Chapter Test Evaluate each expression if x = 7 and y = 5. 1. x + 7 - 5

Find the value of each expression.

2. 12y + 8

3. y² + 3x

For Exercises 14–19, use the coordinate plane below.

4. x² - 6y

5. Mr. Gomez is buying gumballs. The table shows the price of different amounts of gumballs. Number of Gumballs Price

13. 26 + (7 × 2)

12. 5 × 6 + 2 × 3

20

40

60

80

100

$2

$4

$6

$8

$10

What is the relationship between the number of gumballs and the price? 6. Martina’s fish tank has 5 less goldfish than Bill’s fish tank, x. Copy and complete the function table. Input (x)

x-5

Output

6

18

Solve each equation. Check your solution.

9. 6z = 42

O

y

B D

F G

C

E 1 2 3 4 5 6 7 x

Name the ordered pair for each point. 15. C

14. B

16. D

Name the point for each ordered pair. 17. (3, 1)

18. (4, 3)

19. (6, 6)

20. The sum of two whole numbers between 20 and 40 is 58. The difference of the two numbers is 2. What are the two numbers? Tell what strategy you used to find the numbers.

12

7. x + 5 = 8

7 6 5 4 3 2 1

8. y - 2 = 11 d 10. _ = 4 9

11. MULTIPLE CHOICE The daily cost of renting a boat is $50. Which equation represents c, the cost in dollars for boating for d days? A. c = 50 + d

C. 50 = c + d

B. c = 50d

D. 50 = c - d

Solve each inequality. 21. x - 6 > 12 22. 4x ≥ 44 x 23. _ < 3 12

24.

E

WRITE MATH Explain why the

variable x can have any value in x + 3, but in x + 3 = 7, the variable x can have only one value.

Practice Chapter Test

469

Test Practice

A store parking lot has 30 rows with 15 spaces in each row. In addition, there are 8 spaces near the front of the store. Which expression can be used to find the total number of parking spaces?

A. (30 × 15) + 8

C. (30 + 8) × 15

Remember, when an expression contains parentheses, perform the operation(s) inside the parentheses first.

B. (30 × 15) + (30 × 8) D. (30 + 8) × (8 + 15) Read the Test Item You need to find which expression could be used to find the total number of parking spaces. Solve the Test Item You know the parking lot has 30 rows with 15 spaces in each row. The total number of spaces in these rows can be represented by 30 × 15. Plus, there are 8 spaces at the front. So, the total number of parking spaces can be represented by (30 × 15) + 8. The answer is A.

Read each question. Then fill in the correct answer on the answer sheet provided by your teacher or on a separate sheet of paper. 1. Mrs. Grey bought 5 boxes of fruit snacks. Each box contains 12 packages of fruit snacks. She also had 4 packages at home. Which expression could be used to find the total number of fruit snack packages? A. 5 × 12 + 12 × 4 B. 4 × 12 + 5 C. 5 × 4 + 12 D. 5 × 12 + 4

470


2. Otis records how much money he has saved. Which is not a way to find the amount he saves each week? Week

Total Saved

3

$45

4

$60

5

$75

6

$90

F. Divide $60 by 4. G. Divide $90 by 6. H. Subtract $45 from $75. I. Subtract $60 from $75.

3.

SHORT RESPONSE Marita ate _ of 2 1 the pie. Jonas ate _ of the pie. What 4 fraction of the pie is left? 1

?

8. What is the missing value in the table? 2

Output

0

4

6

8

10

4

6

8

Jonas

Marita

4. The gym teacher bought 8 dodge balls for $32. If each dodge ball costs the same amount, what is the cost of one dodge ball?

5.

Input

A. $4

C. $8

B. $6

D. $128

F. 2

H. 5

G. 3

I. 7

9. There are 640 cans of coffee at a distribution warehouse. The cans will be packed into boxes that hold 40 cans each. How many boxes are needed?

GRIDDED RESPONSE What is the next number in the pattern?

A. 12

C. 16

B. 15

D. 18

7, 15, 23, 31, 39, . . .

F. 5

H. 67

G. 55

I. 101

SHORT RESPONSE Tariq purchased 60 baseball cards this week and 15 baseball cards last week. If there are 5 cards in each pack, write a number sentence to show how many packs of cards he bought.

10.

6. Evaluate the expression 12x - 17 if x = 7.

7. Which point is located at (2, 4)? 5 4 3 2 1

O

y

A B

C

11. There are 120 players at a soccer camp. The players are divided into groups of 15 for warm-ups. How many groups of players are there?

D

1 2 3 4 5 x

A. Point A

C. Point C

F. 6

H. 10

B. Point B

D. Point D

G. 8

I. 15

NEED EXTRA HELP? If You Missed Question . . . Go to Chapter-Lesson . . . For help with . . .

1

2

3

4

5

6

7

8

9

10

11

9-1A

9-1C

8-2D

3-1A

7-3C

9-1B

9-2A

9-1D

4-2B

9-1A

4-2B

SPI 3.2

GLE 1.2

SPI 2.5

SPI 2.4

GLE 1.2

SPI 3.1

SPI 2.4

SPI 3.2

SPI 2.4

GLE 4.3 GLE 3.3

Test Practice 471