English
2
Spanish
Rational Numbers
and Equations
2.1
2.2
2.3
2.4
Rational Numbers
Adding and Subtracting Rational Numbers
Multiplying and Dividing Rational Numbers
Solving Equations Using Addition
or Subtraction
2.5 Solving Equations Using Multiplication
or Division
2.6
2
.6 Solving Two
Two-Step
Step Equations
m
tiles, so I a
”
y algebra
s.
m
it
d
u
n
c
fi
is
b
’t
dog
“I can
me of my
painting so
e
to solve th
l be able
2.”
=
)
“Now I wil
2
(2x
“On the count of 5, I’m goi
ng to give you
half of my dog biscuits.”
equation
“1, 2, 3, 4, 4 1 , 4 3 , 4 7 ,...”
2
4
8
English
Spanish
What You
Learned Before
“Let’s play
a game. Th
positive ra
tional num e goal is to say a
ber
the other p
et’s numb that is less than
er... You go
first.”
Example 1 Write 0.37 as a fraction.
2
5
Example 2 Write — as a decimal.
37
100
⋅
⋅
2
5
0.37 = —
2 2
5 2
4
10
— = — = — = 0.4
Write the decimal as a fraction or the fraction as a decimal.
1. 0.51
2. 0.731
1
3
1
5
Example 3 Find — + —.
1
3
⋅
⋅
1
5
1 5
3 5
3
5
⋅
⋅
1 3
5 3
3
15
4. —
1
4
9
36
⋅
⋅
2 4
9 4
8
36
1
36
=—
5
6
⋅ 34
Example 5 Find — —.
2
1 9
4 9
=—−—
8
15
⋅
⋅
⋅
2
9
=—
⋅
⋅
2
9
—−—=—−—
=—+—
5 3
5 3
— —=—
6 4 6 4
1
4
Find — − —.
Example 4
—+—=—+—
5
15
7
8
3. —
1
2
3
2
3
9
10
2
3
—÷—=—
10
⋅—
9
⋅
⋅
2 10
3 9
=—
5
8
=—
9
10
Find — ÷ —.
Example 6
Multiply by the
reciprocal of
the divisor.
20
27
=—
Evaluate the expression.
1
4
13
20
5. — + —
14
15
1
3
6. — − —
3
7
⋅ 109
7. — —
4
5
16
17
8. — ÷ —
English
2.1
Spanish
Rational Numbers
How can
canyou
youuse
useaanumber
numberline
lineto
toorder
order
rational numbers?
Rational
The word rational comes from the word ratio.
8h
.
24 h
If you sleep for 8 hours in a day, then
the ratio of your sleeping time to the
total hours in a day can be written as
—
A rational number is a number that can be written as the ratio of two integers.
−3
1
2
1
1
1
2
−3 = —
2=—
−1
2
1
4
−— = —
0.25 = —
ACTIVITY: Ordering Rational Numbers
Work in groups of five. Order the numbers from least to greatest.
1
3
5
3
a. Sample: −0.5, 1.25, −—, 0.5, −—
●
Make a number line on the floor using masking tape and a marker.
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
●
Write the numbers on pieces of paper. Then each person should
choose one.
●
Stand on the location of your number on the number line.
−2
●
−1.5
−1
−0.5
0
0.5
1
1.5
2
Use your positions to order the numbers from least to greatest.
5
3
1
3
So, the numbers from least to greatest are −—, −0.5, −—, 0.5, and 1.25.
7
4
1
2
1
10
b. −—, 1.1, —, −—, −1.3
3 9 1
5 2 4
d. −1.4, −—, —, —, 0.9
50
Chapter 2
Rational Numbers and Equations
1
4
3
4
c. −—, 2.5, —, −1.7, −0.3
9
4
5
4
e. —, 0.75, −—, −0.8, −1.1
Spanish
2
ACTIVITY: The Game of Math Card War
Preparation:
●
Cut index cards to make 40 playing cards.
●
Write each number in the table on a card.
-0.6
English
To Play:
3
4
●
Play with a partner.
●
Deal 20 cards to each player face-down.
●
Each player turns one card face-up. The player with the greater
number wins. The winner collects both cards and places them at
the bottom of his or her cards.
●
Suppose there is a tie. Each player lays three cards face-down, then
a new card face-up. The player with the greater of these new cards
wins. The winner collects all ten cards and places them at the
bottom of his or her cards.
●
Continue playing until one player has all the cards. This player
wins the game.
−—
3
2
—
3
10
−—
−0.6
1.25
−0.15
—
—
3
20
—
8
5
−1.2
—
19
10
0.75
−1.5
−—
1.5
1.9
−0.75
−0.4
—
3
4
−—
−1.9
6
5
−—
1.6
−—
0.6
0.15
—
3
2
—
3
4
3
10
2
5
5
4
5
4
3
5
−1.6
−0.3
−—
1.2
0.3
—
2
5
−—
3
20
−—
−1.25
0.4
−—
—
6
5
3
5
19
10
8
5
3. IN YOUR OWN WORDS How can you use a number line to order
rational numbers? Give an example.
The numbers are in order from least to greatest. Fill in the blank spaces
with rational numbers.
1
2
, —,
1
3
, −0.1,
4. −—,
6. −—,
1
3
7
5
5
2
5. −—,
, —,
4
5
, —,
7. −3.4,
, −1.9,
, −1.5,
2
3
, −—,
, 2.2,
Use what you learned about ordering rational numbers to
complete Exercises 28 –30 on page 54.
Section 2.1
Rational Numbers
51
English
2.1
Spanish
Lesson
Lesson Tutorials
A terminating decimal is a decimal that ends.
Key Vocabulary
1.5, –0.25, 10.625
terminating decimal,
p. 52
repeating decimal,
p. 52
rational number,
p. 52
A repeating decimal is a decimal that has a pattern that repeats.
—
−1.333 . . . = −1.3
—
0.151515 . . . = 0.15
Use bar notation to show
which of the digits repeat.
Terminating and repeating decimals are examples of rational numbers.
Rational Numbers
Rational Numbers
A rational number is a number that
−1.2
a
can be written as — where a and b are
b
Integers
−10
integers and b ≠ 0.
−
2
3
−2
0
Whole Numbers
3
EXAMPLE
1
4
1
2
5.8
Writing Rational Numbers as Decimals
1
4
5
11
a. Write −2 — as a decimal.
1
4
b. Write — as a decimal.
9
4
Notice that −2— = −—.
2.25
Divide 9 by 4.
0.4545
Divide 5 by 11.
5.0000
11 )‾
−44
60
− 55
50
− 44
60
The remainder repeats. So,
− 55
it is a repeating decimal.
5
9.00
4 )‾
−8
10
−8
20
The remainder is 0. So, it
− 20
is a terminating decimal.
0
5
—
So, — = 0.45.
1
4
So, −2— = −2.25.
11
Write the rational number as a decimal.
Exercises 11–18
52
Chapter 2
6
5
1. −—
3
8
2. −7 —
Rational Numbers and Equations
3
11
3. −—
5
27
4. 1—
English
Spanish
Writing a Decimal as a Fraction
2
EXAMPLE
Write −0.26 as a fraction in simplest form.
Write the digits after the decimal
point in the numerator.
26
100
−0.26 = −—
The last digit is in the hundredths
place. So, use 100 in the denominator.
13
50
= −—
Simplify.
Write the decimal as a fraction or mixed number in simplest form.
5. −0.7
Exercises 20–27
Creature
Elevations (km)
Anglerfish
−—
Squid
−2—
Shark
2
−—
11
The table shows the elevations of four sea creatures relative
to sea level. Which of the sea creatures are deeper than the
whale? Explain.
13
10
Write each rational number as a decimal.
1
5
Whale
8. −10.25
Ordering Rational Numbers
3
EXAMPLE
7. −3.1
6. 0.125
13
10
−— = −1.3
1
5
−2— = −2.2
−0.8
2
—
−— = −0.18
11
Then graph each decimal on a number line.
Squid
−2.2
−2.4
−2.0
Anglerfish
−1.3
−1.6
−1.2
Whale
−0.8
Shark
−0.18
−0.8
−0.4
0
Both −2.2 and −1.3 are less than −0.8. So, the squid and the
anglerfish are deeper than the whale.
1
10
9. WHAT IF? The elevation of a dolphin is −— kilometer. Which of the
Exercises 28– 33
sea creatures in Example 3 are deeper than the dolphin? Explain.
Section 2.1
Rational Numbers
53
English
Spanish
Exercises
2.1
Help with Homework
1. VOCABULARY How can you tell that a number is rational?
2. WRITING You have to write 0.63 as a fraction. How do you choose
the denominator?
Tell whether the number belongs to each of the following number sets:
rational numbers, integers, whole numbers.
—
3. −5
4. −2.16
5. 12
6. 0
Tell whether the decimal is terminating or repeating.
7. −0.4848 . . .
8. −0.151
—
10. −5.236
9. 72.72
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Write the rational number as a decimal.
7
8
5
11
1 11. —
7
9
5
6
17
18
14. −—
7
12
16. −2—
15. 1—
17
40
13. −—
12. —
15
22
17. −5—
19. ERROR ANALYSIS Describe and correct the error
in writing the rational number as a decimal.
18. 8—
✗
7
−— = −0.6—
3
11
Write the decimal as a fraction or mixed number in simplest form.
2 20. −0.9
24. −2.32
21. 0.45
22. −0.258
23. −0.312
25. −1.64
26. 6.012
27. −12.405
Order the numbers from least to greatest.
3
4
2
3
9
5
7
3
3 28. −—, 0.5, —, −—, 1.2
6
10
9
4
4
5
29. —, −2.5, −1.1, −—, 0.8
5
3
31. 2.1, −—, −—, −0.75, —
7
2
5 4
4 3
32. −—, −2.8, −—, —, 1.3
8
5
11
5
a. Write the amount as a decimal.
b. Write the amount as a fraction in simplest form.
35. HIBERNATION A box turtle hibernates in sand at −1— feet. A spotted turtle
16
25
hibernates at −1— feet. Which turtle is deeper?
54
Chapter 2
Rational Numbers and Equations
15
10
33. −—, −2.4, 1.6, —, −2.25
34. COINS You lose one quarter, two dimes and two nickels.
5
8
1
4
30. −1.4, −—, 0.6, −0.9, —
English
Spanish
Copy and complete the statement using <, >, or =.
36. −2.2
6
10
39. −4 —
−2.42
37. −1.82
3
11
−4.65
40. −5 —
15
8
7
8
−1.81
38. —
—
−5.2
41. −2 —
1—
13
16
11
14
−2 —
42. OPEN-ENDED Find one terminating decimal and one repeating decimal
1
2
1
3
between −— and −—.
Player
Hits
At Bats
Eva
42
90
Michelle
38
80
43. SOFTBALL In softball, a batting average is the number
of hits divided by the number of times at bat. Does Eva
or Michelle have the higher batting average?
44. QUIZ You miss 3 out of 10 questions on a science quiz and 4 out of 15 questions
on a math quiz. Which quiz has a higher percent of correct answers?
45. SKATING Is the half pipe deeper than the skating pool? Explain.
Lip
Lip
Skating pool
Half pipe
−9
−10 ft
Base
Base
46. ENVIRONMENT The table shows the
changes from the average water level of
a pond over several weeks. Order the
numbers from least to greatest.
47.
5
ft
6
Week
Change
(inches)
1
7
5
−—
2
5
11
−1—
3
4
−1.45
−1—
91
200
Given: a and b are integers.
1
a
1
ab
a. When is −— positive?
b. When is — positive ?
SKILLSReview
REVIEW
HANDBOOK
Add or subtract. (Skills
Handbook)
3
5
2
7
48. — + —
9
10
2
3
49. — − —
50. 8.79 − 4.07
51. 11.81 + 9.34
52. MULTIPLE CHOICE In one year, a company has a profit of −$2 million. In the
next year, the company has a profit of $7 million. How much more money
SECTION
1.3
did the company make the second year? (Section
1.3)
A $2 million
○
B $5 million
○
C $7 million
○
Section 2.1
D $9 million
○
Rational Numbers
55
English
Spanish
2.2
Adding and Subtracting
Rational Numbers
How does adding and subtracting rational
numbers compare with adding and subtracting integers?
1
ACTIVITY: Adding and Subtracting Rational Numbers
Work with a partner. Use a number line to find the sum or difference.
a. Sample: 2.7 + (−3.4)
Start at 0. Move 2.7
units to the right.
−3
−2
Then move 3.4 units
left to end at −0.7.
Add −3.4.
2.7
−1
0
1
2
3
So, 2.7 + (−3.4) = −0.7.
3
10
( )
9
10
6
10
3
10
b. — + −—
c. −— − 1—
d. 1.3 + (−3.4)
e. −1.9 − 0.8
2
ACTIVITY: Adding and Subtracting Rational Numbers
Work with a partner. Write the numerical expression shown on the number
line. Then find the sum or difference.
a.
Start at 0. Move 1.5
units to the right.
−3
b.
−2
Then move
1.5
−1
1
unit
2
0
Subtract
left to end at −2.
−3
56
Chapter 2
Then move 2.3 units
left to end at −0.8.
Add −2.3.
1
Rational Numbers and Equations
3
Start at 0. Move
1
.
2
−1
−2
2
−1
1
2
1 units to the left.
1
2
0
1
2
3
English
Spanish
3
ACTIVITY: Financial Literacy
Work with a partner. The table shows the balance in
a checkbook.
➡
●
Black numbers are amounts added to the account.
●
Red numbers are amounts taken from the account.
Date
Check #
––
––
Previous balance
1/02/2009
124
Groceries
Amount
Balance
––
100.00
34.57
1/06/2009
Check deposit
1/11/2009
ATM withdrawal
40.00
Electric company
78.43
Music store
10.55
Shoes
47.21
1/14/2009
125
1/17/2009
1/18/2009
➡
Transaction
126
1/20/2009
Check deposit
1/21/2009
Interest
1/22/2009
127
875.50
125.00
2.12
Cell phone
59.99
You can find the balance in the second row two different ways.
100.00 − 34.57 = 65.43
Subtract 34.57 from 100.00.
100.00 + (−34.57) = 65.43
Add −34.57 to 100.00.
a. Copy the table. Then complete the balance column.
b. How did you find the balance in the tenth row?
c. Use a different way to find the balance in part (b).
4. IN YOUR OWN WORDS How does adding and subtracting rational numbers
compare with adding and subtracting integers? Give an example.
PUZZLE Find a path through the table so that the numbers add up to the sum.
You can move horizontally or vertically.
3
4
6. Sum: −0.07
5. Sum: —
Start
1
2
—
1
8
−—
2
3
—
5
7
−—
3
4
−—
1
3
—
Start
End
2.43
1.75
−0.98
−1.09
3.47
−4.88
End
Use what you learned about adding and subtracting rational
numbers to complete Exercises 7–9 and 16 –18 on page 60.
Section 2.2
Adding and Subtracting Rational Numbers
57
English
Spanish
Lesson
2.2
Lesson Tutorials
Adding and Subtracting Rational Numbers
To add or subtract rational numbers, use the same rules for
signs as you used for integers.
Words
Numbers
4 1 4−1 3
—−—=—=—
5 5
5
5
1
3
1
6
−2
6
1
6
−2 + 1
6
−1
6
1
6
−— + — = — + — = — = — = −—
EXAMPLE
1
Adding Rational Numbers
8
3
5
6
8
3
5
6
Find − — + —.
Study Tip
8
3
5
6
Rewrite using the LCD (least common denominator).
−16 + 5
6
how −— is written as
−8
3
−16
6
−— + — = — + —
In Example 1, notice
8
3
Estimate −3 + 1 = −2
Write the sum of the numerators
over the like denominator.
=—
−16
6
− — = — = —.
−11
6
5
6
= —, or −1 —
5
6
The sum is −1 —.
EXAMPLE
2
Simplify.
5
6
Reasonable? −1 — ≈ −2
✓
Adding Rational Numbers
Find −4.05 + 7.62.
∣ 7.62 ∣ > ∣ − 4.05 ∣. So, subtract ∣ − 4.05 ∣ from ∣ 7.62 ∣.
−4.05 + 7.62 = 3.57
Use the sign of 7.62.
The sum is 3.57.
Add.
Exercises 4 –12
58
Chapter 2
7
8
1
4
1
3
20
3
( )
7
2
1. −— + —
2.
−6 — + —
3.
2 + −—
4. −12.5 + 15.3
5.
−8.15 + (−4.3)
6.
0.65 + (−2.75)
Rational Numbers and Equations
English
Spanish
EXAMPLE
3
Subtracting Rational Numbers
1
7
( )
− ( − ) = −4
6
7
Find −4 — − − — .
1
7
−4 —
6
7
—
Estimate −4 − (−1) = −3
1
7
6
7
Add the opposite of −—.
29
7
6
7
Write the mixed number
as an improper fraction.
6
7
—+—
= −— + —
−23
7
2
7
= —, or −3 —
2
7
Simplify.
Reasonable? −3 — ≈ −3
✓
1
3
1
4
2
7
The difference is −3 —.
Subtract.
1
3
( )
1
3
7. — − −—
EXAMPLE
4
8.
5
6
−3 — − —
9.
1
2
4— − 5—
Real-Life Application
In the water, the bottom of a boat is 2.1 feet below the surface and the
top of the boat is 8.7 feet above it. Towed on a trailer, the bottom of the
boat is 1.3 feet above the ground. Can the boat and trailer pass under
the bridge?
Step 1: Find the height h of the boat.
Clearance: 11 ft 8 in.
h = 8.7 − (−2.1)
Subtract the lowest point from the highest point.
= 8.7 + 2.1
Add the opposite of −2.1.
= 10.8
Add.
Step 2: Find the height t of the boat and trailer.
t = 10.8 + 1.3
= 12.1
Add the trailer height to the boat height.
Add.
Because 12.1 feet is greater than 11 feet 8 inches, the boat and
trailer cannot pass under the bridge.
Exercises 13–21
10. WHAT IF? In Example 4, the clearance is 12 feet 1 inch.
Can the boat and trailer pass under the bridge?
Section 2.2
Adding and Subtracting Rational Numbers
59
English
Spanish
Exercises
2.2
Help with Homework
1. WRITING Explain how to find the sum −8.46 + 5.31.
1
2
2. OPEN-ENDED Write an addition expression using fractions that equals −—.
3. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Add −4.8 and 3.9.
What is 3.9 less than −4.8?
What is −4.8 increased by 3.9?
Find the sum of −4.8 and 3.9.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Add. Write fractions in simplest form.
1
2
( )
( )
11
12
9
14
5
6
15
4
7
12
5. −— + —
8
15
8. 4 + −1 —
4. — + −—
2
7
( )
2
3
7. 2 — + −—
10. −3.1 + (−0.35)
( )
1
3
6. — + −4 —
9. −4.2 + 3.3
11. 12.48 + (−10.636)
12. 20.25 + (−15.711)
Subtract. Write fractions in simplest form.
3
( )
5
8
1
4
7
8
4 13. — − −—
11
16
1
2
14. — − —
5
3
3
8
( )
5
9
15. −— − −—
1
6
16. −5 − —
17. −8 — − 10 —
18. −1 − 2.5
19. 5.5 − 8.1
20. −7.34 − (−5.51)
21. 6.673 − (−8.29)
22. ERROR ANALYSIS Describe and correct
the error in finding the difference.
✗
3
4
9
2
3−9
4−2
−6
2
— − — = — = — = −3
5
6
23. SPORTS DRINK Your sports drink bottle is — full. After practice the bottle is
3
8
— full. Write the difference of the amounts after practice and before practice.
24. BANKING Your bank account balance is −$20.85. You deposit $15.50.
What is your new balance?
Evaluate.
1
6
( ) ( )
8
3
7
9
25. 2 — − −— + −4 —
60
Chapter 2
26. 6.3 + (−7.8) − (−2.41)
Rational Numbers and Equations
12
5
∣
13
6
∣ ( )
2
3
27. −— + −— + −3 —
English
Spanish
28. REASONING When is the difference of two decimals an integer? Explain.
2
3
3
4
29. RECIPE A cook has 2 — cups of flour. A recipe calls for 2 — cups of flour. Does
the cook have enough flour? If not, how much more flour is needed?
Springville
30. ROADWAY A new road that connects Uniontown to
1
3
Springville is 4 — miles long. What is the change in
new road
3
mi.
8
distance when using the new road instead of the
dirt roads?
Uniontown
3
5
mi.
6
RAINFALL In Exercises 31– 33, the bar graph shows the differences in a city’s rainfall
from the historical average.
Monthly Rainfall
31. What is the difference in
rainfall between the wettest
and driest months?
4.0
Rainfall (inches)
2
32. Find the sum of the differences
for the year.
33. What does the sum in Exercise 32
tell you about the rainfall for
the year?
3.0
Historical Average
2.36
2.0
0.94
1.0
1.39
0.83
0.35
0
Ź1.0 Ź0.45
Ź0.88
Ź3.0
Ź0.90
Ź1.35
Ź1.39
Ź0.96
Ź1.67
Ź2.0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
ALGEBRA Add or subtract. Write the answer in simplest form.
3n
8
5b
8
a
3
36. −4a − —
38.
2n
8
n
8
35. −— + — − —
34. −4x + 8x − 6x
( )
2b
3
37. — + −—
Fill in the blanks to make the solution correct.
5.
4−(
.8
) = −3.61
SKILLSReview
REVIEW
HANDBOOK
Evaluate. (Skills
Handbook)
39. 5.2 × 6.9
40. 7.2 ÷ 2.4
2
3
1
4
41. 2 — × 3 —
4
5
1
2
42. 9 — ÷ 3 —
43. MULTIPLE CHOICE A sports store has 116 soccer balls. Over 6 months, it sells
eight soccer balls per month. How many soccer balls are in inventory at the
end of the 6 months? (Section
1.31.3
and Section
1.4) 1.4
SECTION
SECTION
A −48
○
B 48
○
Section 2.2
C 68
○
D 108
○
Adding and Subtracting Rational Numbers
61
English
2.3
Spanish
Multiplying and
Dividing Rational Numbers
How can you use
operations with rational numbers in a story?
1
EXAMPLE: Writing a Story
Write a story that uses addition, subtraction
multiplication, or division of rational
numbers. Draw pictures
for your story.
There are many possible
stories. Here is an example.
Lauryn decides to earn
some extra money. She sets up a lemonade stand. To get
customers, she uses big plastic glasses and makes a sign
saying “All you can drink for 50¢!”
Lauryn can see that her daily profit is negative. But, she decides to keep
trying. After one week, she has the same profit each day.
Lauryn is frustrated. Her profit for the first week is
7(−5.75) = (−5.75) + (−5.75) + (−5.75) + (−5.75) + (−5.75) + (−5.75) + (−5.75)
= −40.25.
She realizes that she has too many customers who are drinking a second
and even a third glass of lemonade. So, she decides to try a new strategy.
Soon, she has a customer. He buys a glass of lemonade and drinks it.
He hands the empty glass to Lauryn and says “That was great. I’ll have
another glass.” Today, Lauryn says “That will be 50¢ more, please.” The
man says “But, you only gave me one glass and the sign says ‘All you can
drink for 50¢!’” Lauryn replies, “One glass IS all you can drink for 50¢.”
With her new sales strategy, Lauryn starts making a profit of $8.25 per
day. Her profit for the second week is
7(8.25) = (8.25) + (8.25) + (8.25) + (8.25) + (8.25) + (8.25) + (8.25) = 57.75.
Her profit for the two weeks is −40.25 + 57.75 = $17.50. So, Lauryn has
made some money. She decides that she is on the right track.
62
Chapter 2
Rational Numbers and Equations
English
Spanish
2
ACTIVITY: Writing a Story
Work with a partner. Write a story that uses addition, subtraction,
multiplication, or division of rational numbers.
●
At least one of the numbers in the story has to be negative and not
an integer.
●
Draw pictures to help illustrate what is happening in the story.
●
Include the solution of the problem in the story.
If you are having trouble thinking of a story, here are some common uses
of negative numbers.
●
A profit of −$15 is a loss of $15.
●
An elevation of −100 feet is a depth of 100 feet below sea level.
●
A gain of −5 yards in football is a loss of 5 yards.
●
A score of −4 in golf is 4 strokes under par.
●
A balance of −$25 in your checking account means the account is
overdrawn by $25.
3. IN YOUR OWN WORDS How can you use operations with rational
numbers in a story? You already used rational numbers in your story.
Describe another use of a negative rational number in a story.
PUZZLE Read the cartoon. Fill in the blanks using 4s or 8s to make the
equation true.
4.
5.
( )( )
( ) ( )
1
1
“Dear Mom, I’m in a hurry. To save
time I won’t be typing any 4’s or 8’s.”
Section 2.3
1
1
−—
(
8. −4.
6
) = −1.
( ) ( )
3
−—
1
× −— = —
× −0.
6. 1.
7.
1
−— + −— = −—
3
1
2
÷ — = −—
÷ 2 = −2.
Multiplying and Dividing Rational Numbers
63
English
2.3
Spanish
Lesson
Lesson Tutorials
Multiplying and Dividing Rational Numbers
Words
To multiply or divide rational numbers, use the same rules for
signs as you used for integers.
Remember
a
b
2
7
b
a
⋅ 13
⋅
⋅
−2 1
7 3
−2
21
2
21
−— — = — = — = −—
Numbers
The reciprocal of — is —.
−1
2
4
9
1
2
⋅ 94
⋅
⋅
−1 9
2 4
−9
8
9
8
−— ÷ — = — — = — = — = −—
EXAMPLE
1
Dividing Rational Numbers
1
5
1
3
1
2
Find −5 — ÷ 2 —.
1
5
1
3
Estimate −5 ÷ 2 = −2 —
26
5
7
3
−5 — ÷ 2— = −— ÷ —
−26
5
Write mixed numbers as improper fractions.
⋅ 37
7
3
=— —
Multiply by the reciprocal of —.
⋅
−26 3
5 7
=—
⋅
−78
35
Multiply the numerators and the denominators.
8
35
= —, or −2 —
8
35
The quotient is −2 —.
EXAMPLE
2
Simplify.
Multiplying Rational Numbers
⋅
Find −2.5 3.6.
−2.5
× 3.6
The decimals have different signs.
150
750
−9.0 0
The product is negative.
The product is −9.
64
Chapter 2
Rational Numbers and Equations
8
35
1
2
Reasonable? −2 — ≈ −2 —
✓
English
Spanish
EXAMPLE
Standardized Test Practice
3
5
Which number, when multiplied by − —, gives a product
3
between 5 and 6?
1
4
A −6
○
1
4
B −3 —
○
C −—
○
D 3
○
Use the guess, check, and revise method.
Guess 1: Because the product is positive and the known factor is
C .
negative, choose a number that is negative. Try Choice ○
1
4
⋅
⋅
−1 (−5)
4 3
( )
5
3
5
12
−— −— = — = —
Guess 2: The result of Choice ○
C is not between 5 and 6. So,
B.
choose another number that is negative. Try Choice ○
1
4
( )
5
3
13
4
( )
5
3
⋅
⋅
−13 (−5)
4 3
65
12
5
12
−3 — −— = −— −— = — = — = 5 —
5
12
5 — is between 5 and 6. So, the correct answer is ○
B .
Multiply or divide.
EXAMPLE
4
( )
1
2
2.
— ÷ −2 —
3.
( )
4. 1.8(−5.1)
5.
−6.3(−0.6)
6.
(−1.3)2
1
3
2
3
1 3
2
−—
Real-Life Application
Account Positions
Stock
( )
1. −— ÷ −—
6
5
Exercises 10 – 33
Original Value
Current Value
Change
A
600.54
420.15
−180.39
B
391.10
518.38
127.28
C
380.22
99.70
−280.52
An investor owns stocks A, B, and C.
What is the mean change in value of
the stocks?
−180.39 + 127.28 + (−280.52)
3
−333.63
3
mean = ——— = — = −111.21
The mean change in value of the stocks is −$111.21.
7. In Example 4, the change in value of stock D is $568.23.
What is the mean change in value of the four stocks?
Section 2.3
Multiplying and Dividing Rational Numbers
65
English
Spanish
Exercises
2.3
Help with Homework
1. WRITING How is multiplying and dividing rational numbers similar
to multiplying and dividing integers?
Find the reciprocal.
2
5
2. −—
16
9
3. −3
1
3
5. −2—
4. —
Tell whether the expression is positive or negative without evaluating.
( )
3
10
8
15
6. −— × −—
( )
1
2
1
4
7. 1— ÷ −—
−8.16
−2.72
8. −6.2 × 8.18
9. —
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Divide. Write fractions in simplest form.
7
10
( )
(
1
4
2
5
1 10. −— ÷ —
3
8
4
5
2
7
( )
8
9
11. — ÷ −—
4
11
14. −2— ÷ (−7)
15. −10 — ÷ −4—
18. −3.45 ÷ (−15)
19. −0.18 ÷ 0.03
)
8
9
1
5
12. −— ÷ −—
13. −— ÷ 20
16. −9 ÷ 7.2
17. 8 ÷ 2.2
20. 8.722 ÷ (−3.56)
21. 12.42 ÷ (−4.8)
Multiply. Write fractions in simplest form.
2
2
3
2
9
1
3
26. −3 —
4
3
23. −— × −—
7
⋅ ( −2 —
10 )
30. −8(0.09)
( )
( )
1
4
3 22. −— × —
27.
2 3
3
−1—
⋅
31. −9.3 (−5.1)
5
6
( )
( )
8
15
1
4
24. — −—
25. −2 −1—
28. 0.4 × (−0.03)
29. −0.05 × (−0.5)
⋅
32. −95.2 (−0.12)
33. (−0.4)3
ERROR ANALYSIS Describe and correct the error.
34.
✗
35.
−2.2 × 3.7 = 8.14
✗
1
4
3
2
4
1
3
2
12
2
− — ÷ — = −— × — = −— = −6
36. HOUR HAND The hour hand of a clock moves −30° every
1
5
hour. How many degrees does it move in 2— hours?
37. SUNFLOWER SEEDS How many 0.75-pound packages
can be made with 6 pounds of sunflower seeds?
66
Chapter 2
Rational Numbers and Equations
−30°
English
Spanish
Evaluate.
39. 2.85 − 6.2 ÷ 22
38. −4.2 + 8.1 × (−1.9)
5
9
( ) ( )
2
3
3
5
3
4
41. 1— ÷ −— + −2—
5
6
40. −3.64
⋅ ∣ −5.3 ∣ − 1.5
3
( ) ( )
2 2 3 1
− — 2—
3
4 3
1
3
42. −3 — × — − 2 —
43. −—
3
5
44. OPEN-ENDED Write two fractions whose product is −—.
30
50
45. FENCING A farmer needs to
enclose two adjacent rectangular
pastures. How much fencing does
the farmer need?
2
yd
9
5
yd
8
3
4
46. GASOLINE A 14.5-gallon gasoline tank is — full. How many gallons will it take
to fill the tank?
47. BOARDWALK A section of a boardwalk is made using 15 boards. Each board is
1
4
9 — inches wide. The total width of the section is 144 inches. The spacing between
each board is equal. What is the width of the spacing between each board?
48. RUNNING The table shows the changes in the times
(in seconds) of four teammates. What is the mean change?
1
49.
2
3
4
Teammate
Change
1
−2.43
2
−1.85
3
0.61
4
−1.45
5
Consider (−2) , (−2) , (−2) , (−2) , (−2) ,
and (−2)6.
a. Evaluate each expression.
b. What pattern do you notice?
c. What is the sign of (−2)49?
Add or subtract.
50. −6.2 + 4.7
SECTION 2.2
51. −8.1 − (−2.7)
9
5
(
7
10
52. — − −2—
)
54. MULTIPLE CHOICE What are the coordinates of the
point in quadrant IV?
SECTION 1.6
A (−4, 1)
○
B (−3, −3)
○
C (0, −2)
○
D (3, −3)
○
( )
5
6
4
9
53. −4— + −3—
y
4
A
3
2
C
1
−4 −3 −2
B
D
O
−2
−3
1
2
4 x
3
F
E
−4
Section 2.3
Multiplying and Dividing Rational Numbers
67
English
2
Spanish
Study Help
Graphic Organizer
You can use a process diagram to show the steps involved in a procedure. Here is an
example of a process diagram for adding rational numbers.
Adding rational
numbers
with
the same sign
with
different signs
Add the absolute values
of the rational numbers.
Subtract the lesser
absolute value from the
greater absolute value.
Write the sum using the
common sign.
Write the sum using the sign
of the rational number with
the greater absolute value.
Example
−5.5 + (−6.9)
Because the numbers have the
same sign, add −5.5 and −6.9 .
Example
−5.5 + 6.9
Because the numbers have different
signs, subtract −5.5 from 6.9 .
−5.5 + (−6.9) = −12.4
Use the common sign.
−5.5 + 6.9 = 1.4
Use the sign of 6.9.
Make a process diagram with examples to
help you study these topics. Your process
diagram can have one or more branches.
1. writing rational numbers as decimals
2. subtracting rational numbers
3. dividing rational numbers
After you complete this chapter, make process
diagrams with examples for the following topics.
4. solving equations using addition
or subtraction
5. solving equations using multiplication
or division
6. solving two-step equations
68
Chapter 2
Rational Numbers and Equations
“Does this process diagram accurately
show how a cat claws furniture?”
English
Spanish
Quiz
2.1–2.3
Progress Check
Write the rational number as a decimal. (Section 2.1)
3
20
11
6
2. −—
1. −—
Write the decimal as a fraction or mixed number in simplest form. (Section 2.1)
3. −0.325
4. −1.28
Add or subtract. Write fractions in simplest form. (Section 2.2)
( )
( )
4
5
3
8
5. −— + −—
12
7
6. −5.8 + 2.6
2
9
7. — − −—
8. 9.1 − 12.9
Multiply or divide. Write fractions in simplest form. (Section 2.3)
3
8
8
5
5
9
( )
9. −2 — × —
10. −9.4 × (−4.7)
4
7
11. −8 — ÷ −1 —
12. −8.4 ÷ 2.1
13. STOCK The value of stock A changes −$3.68 and the value of stock B
changes −$3.72. Which stock has the greater loss? Explain. (Section 2.1)
14. PARASAILING A parasail is
at 200.6 feet above the water.
After five minutes, the parasail
is at 120.8 feet above the water.
What is the change in height
of the parasail? (Section 2.2)
15. FOOTBALL The table shows the
statistics of a running back in a
football game. How many total
yards did he gain? (Section 2.2)
Quarter
Yards
1
1
−8 —
2
2
3
4
Total
23
1
42 —
2
1
−2 —
4
?
16. LATE FEES You were overcharged $4.52 on your cell phone bill three months in
a row. The cell phone company will add −$4.52 to your next bill for each month
you were overcharged. How much will be added to your next bill? (Section 2.3)
Sections 2.1–2.3
Quiz
69
English
2.4
Spanish
Solving Equations Using
Addition or Subtraction
How can you use inverse operations to solve
an equation?
Key:
1
= Variable
=1
= Zero Pair
= −1
EXAMPLE: Using Addition to Solve an Equation
Use algebra tiles to model and solve x − 3 = −4.
Model the equation x − 3 = −4.
=
To get the green tile by itself, remove the red tiles on
the left side by adding three yellow tiles to each side.
=
Remove the three “zero pairs” from each side.
=
The remaining tile shows the value of x.
=
So, x = −1.
2
EXAMPLE: Using Addition to Solve an Equation
Use algebra tiles to model and solve −5 = n + 2.
Model the equation −5 = n + 2.
Remove the yellow tiles on the right side
by adding two red tiles to each side.
=
=
Remove the two “zero pairs” from the right side.
=
The remaining tiles show the value of n.
So, −7 = n or n = −7.
70
Chapter 2
Rational Numbers and Equations
=
English
Spanish
3
ACTIVITY: Solving Equations Using Algebra Tiles
Work with a partner. Use algebra tiles to model and solve the equation.
a. y + 10 = −5
b. p − 7 = −3
c. −15 = t − 5
d. 8 = 12 + z
4
ACTIVITY: Writing and Solving Equations
Work with a partner. Write an equation shown by the algebra tiles. Then solve.
a.
c.
b.
=
d.
=
=
=
5. Decide whether the statement is true or false. Explain your reasoning.
a. In an equation, any letter can be used as a variable.
b. The goal in solving an equation is to get the variable by itself.
c. In the solution, the variable always has to be on the left side of
the equal sign.
d. If you add a number to one side, you should add it to the other side.
6. IN YOUR OWN WORDS How can
you use inverse operations to
solve an equation without
algebra tiles? Give two examples.
7. What makes the cartoon funny?
8. The word variable comes from
the word vary. For example,
the temperature in Maine
varies a lot from winter
to summer.
Write two other English
sentences that use the
word vary.
“Dear Sir: Yesterday you said x = 2.
Today you are saying x = 3.
Please make up your mind.”
Use what you learned about solving equations using inverse
operations to complete Exercises 5 – 8 on page 74.
Section 2.4
Solving Equations Using Addition or Subtraction
71
English
2.4
Spanish
Lesson
Lesson Tutorials
Key Vocabulary
equivalent equations,
p. 72
Addition Property of Equality
Two equations are equivalent equations if they have the
same solutions. Adding the same number to each side of an
equation produces an equivalent equation.
Words
If a = b, then a + c = b + c.
Algebra
Subtraction Property of Equality
Subtracting the same number from each side of an equation
produces an equivalent equation.
Words
Algebra
EXAMPLE
1
If a = b, then a − c = b − c.
Solving Equations
a. Solve x − 5 = −1.
Remember
x − 5 = −1
To solve equations, use
inverse operations that
“undo” each other. For
example, use addition
to solve an equation
with subtraction.
+5
Write the equation.
+5
x=4
Check
Add 5 to each side.
x − 5 = −1
Simplify.
?
4 − 5 = −1
3
2
✓
−1 = −1
So, the solution is x = 4.
1
2
b. Solve z + — = —.
3
2
—
1
2
Write the equation.
3
2
−—
3
2
Subtract — from each side.
z+—=
−—
z = −1
3
2
Simplify.
So, the solution is z = −1.
Solve the equation. Check your solution.
Exercises 5 –20
72
Chapter 2
1. p − 5 = −2
Rational Numbers and Equations
2.
w + 13.2 = 10.4
3.
5
6
1
6
x − — = −—
English
Spanish
Standardized Test Practice
2
EXAMPLE
A company has a profit of $750 this week. This profit is $900 more
than the profit P last week. Which equation can be used to find P?
A
750 = 900 − P
○
C
900 = P − 750
○
B
750 = P + 900
○
D
900 = P + 750
○
The profit this week is $900 more than the profit last week.
Words
=
750
Equation
+
P
900
The equation is 750 = P + 900. The correct answer is ○
B .
4. A company has a profit of $120.50 today. This profit is $145.25 less
than the profit P yesterday. Write an equation that can be used
to find P.
Exercises 22–25
3
EXAMPLE
Real-Life Application
The line graph shows the scoring while you and your friend
played a video game. Write and solve an equation to find your
score after Level 4.
Score (points)
Video Game Scoring
30
You
Your Friend
You can determine the following from the graph.
20
10
0
−10
33 points
1
2
Words
Your friend’s score is 33 points less than your score.
3
(4, −8)
−20
Variable Let s be your score after Level 4.
−8
Equation
Level
−8 = s − 33
+ 33
25 = s
+ 33
=
s
−
33
Write equation.
Add 33 to each side.
Simplify.
Your score after Level 4 is 25 points.
Reasonable? From the graph, your score after Level 4 is between
20 points and 30 points. So, 25 points is a reasonable answer.
5. WHAT IF? In Example 3, you have −12 points after Level 1. Your
score is 27 points less than your friend’s score. What is your
friend’s score?
Section 2.4
Solving Equations Using Addition or Subtraction
73
English
Spanish
Exercises
2.4
Help with Homework
1. VOCABULARY What property would you use to solve m + 6 = −4?
2. VOCABULARY Name two inverse operations.
3. WRITING Are the equations m + 3 = −5 and m = −2 equivalent? Explain.
4. WHICH ONE DOESN’T BELONG? Which equation does not belong with the
other three? Explain your reasoning.
x + 3 = −1
x + 1 = −5
x − 2 = −6
x − 9 = −13
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Solve the equation. Check your solution.
1
5. a − 6 = 13
9. c − 7.6 = −4
13. g − 9 = −19
5
9
1
6
17. q + — = —
6. −3 = z − 8
7. −14 = k + 6
1
2
2
3
8. x + 4 = −14
1
6
11. — = q + —
12. p − 3 — = −2 —
14. −9.3 = d − 3.4
15. 4.58 + y = 2.5
16. x − 5.2 = −18.73
1
4
4
5
18. −2 — = r − —
3
8
5
6
19. w + 3 — = 1 —
21. ERROR ANALYSIS Describe and correct the error in
finding the solution.
✗
2
5
2 22. 4 less than a number n is −15.
x+8 =
10
+8
+8
x =
18
23. 10 more than a number c is 3.
24. The sum of a number y and −3 is −8.
25. The difference between a number p and 6 is −14.
In Exercises 26 –28, write an equation. Then solve.
26. DRY ICE The temperature of dry ice is −109.3°F. This is 184.9°F less than
the outside temperature. What is the outside temperature?
27. PROFIT A company makes a profit of $1.38 million. This is $2.54 million more
than last year. What was the profit last year?
28. PIER The difference between the lengths of a paddle boat and a pier is
1
2
−7 — feet. The pier is 18 — feet long. How long is the paddle boat?
74
Chapter 2
Rational Numbers and Equations
2
11
20. 4 — + k = −3 —
Write the verbal sentence as an equation. Then solve.
3
4
1
2
10. −10.1 = w + 5.3
English
Spanish
GEOMETRY Write and solve an equation to find the unknown side length.
29. Perimeter = 12 cm
30. Perimeter = 24.2 in.
31. Perimeter = 34.6 ft
?
8.3 in.
?
3 cm
?
3.8 in.
5 cm
6.4 ft
5.2 ft
8.3 in.
11.1 ft
In Exercises 32−36, write an equation. Then solve.
32. STATUE OF LIBERTY The total height of the Statue of Liberty and
its pedestal is 153 feet more than the height of the statue. What is
the height of the statue?
1
6
33. BUNGEE JUMPING Your first jump is 50 — feet higher than your second
2
5
jump. Your first jump reaches −200 — feet. What is the height of your
305 ft
second jump?
3
5
34. TRAVEL Boatesville is 65 — kilometers from Stanton. A bus traveling from
1
3
Stanton is 24 — kilometers from Boatesville. How far has the bus traveled?
m°
35. GEOMETRY The sum of the measures of the angles
of a triangle equals 180°. What is the measure of
the missing angle?
36. SKATEBOARDING The table shows your
scores in a skateboarding competition.
The leader has 311.62 points. What score
do you need in the fourth round to win?
30.3°
40.8°
Round
1
2
3
4
Points
63.43
87.15
81.96
?
37. CRITICAL THINKING Find the value of 2x − 1 when x + 6 = 2.
Find the values of x.
38.
∣x∣=2
39.
∣x∣−2=4
40.
∣ x ∣ + 5 = 18
SECTION
SECTION
1.5
Multiply or divide. (Section
1.4 1.4
and Section
1.5)
41. −7 × 8
42. 6 × (−12)
43. 18 ÷ (−2)
44. −26 ÷ 4
45. MULTIPLE CHOICE A class of 144 students voted for a class president. Three-fourths
5
9
of the students voted for you. Of the students who voted for you, — are female. How
many female students voted for you?
A 50
○
B 60
○
Section 2.4
SECTION
(Section
2.3)2.3
C 80
○
D 108
○
Solving Equations Using Addition or Subtraction
75
English
Spanish
2.5
Solving Equations Using
Multiplication or Division
How can you use multiplication or division
to solve an equation?
1
ACTIVITY: Using Division to Solve an Equation
Work with a partner. Use algebra tiles to model and solve the equation.
a. Sample: 3x = −12
Model the equation 3x = −12.
=
Your goal is to get one green tile by itself. Because
there are three green tiles, divide the red tiles into
three equal groups.
=
=
=
Keep one of the groups. This shows the value of x.
=
So, x = −4.
b. 2k = −8
c. −15 = 3t
d. −20 = 5m
e. 4h = −16
2
ACTIVITY: Writing and Solving Equations
Work with a partner. Write an equation shown by the algebra tiles. Then solve.
a.
b.
=
c.
d.
=
76
Chapter 2
=
Rational Numbers and Equations
=
English
Spanish
3
ACTIVITY: The Game of Math Card War
Preparation:
●
Cut index cards to make 40 playing cards.
●
Write each equation in the table on a card.
To Play:
●
Play with a partner. Deal 20 cards to each player face-down.
●
Each player turns one card face-up. The player with the greater
solution wins. The winner collects both cards and places them at
the bottom of his or her cards.
●
Suppose there is a tie. Each player lays three cards face-down, then
a new card face-up. The player with the greater solution of these
new cards wins. The winner collects all ten cards, and places them
at the bottom of his or her cards.
●
Continue playing until one player has all the cards. This player
wins the game.
−4x = −12
x−1=1
x−3=1
2x = −10
−9 = 9x
3 + x = −2
x = −2
−3x = −3
— = −2
x
−2
x = −6
6x = −36
−3x = −9
−7x = −14
x −2 = 1
−1 = x + 5
x = −1
9x = −27
— = −1
−8 = −2x
x=3
−7 = −1 + x
x = −5
−10 = 10x
x = −4
−2 = −3 + x
−20 = 10x
x+9=8
−16 = 8x
x=2
x + 13 = 11
x = −3
−8 = 2x
x=1
— = −2
−4 + x = −2
−6 = x − 3
x=4
x+6=2
x − 5 = −4
x
5
— = −1
x
3
x
2
4. IN YOUR OWN WORDS How can you use multiplication or division to solve
an equation without using algebra tiles? Give two examples.
Use what you learned about solving equations to complete
Exercises 7–10 on page 80.
Section 2.5
Solving Equations Using Multiplication or Division
77
English
2.5
Spanish
Lesson
Lesson Tutorials
Multiplication Property of Equality
Multiplying each side of an equation by the same number
produces an equivalent equation.
Words
Algebra
⋅
⋅
If a = b, then a c = b c.
Division Property of Equality
Dividing each side of an equation by the same number
produces an equivalent equation.
Words
Algebra
EXAMPLE
1
If a = b, then a ÷ c = b ÷ c, c ≠ 0.
Solving Equations
x
3
a. Solve — = −6.
x
3
— = −6
⋅ 3x
Write the equation.
⋅
3 — = 3 (−6)
Multiply each side by 3.
x = −18
Simplify.
So, the solution is x = −18.
b. Solve 18 = −4y.
18 = −4y
18
−4
Write the equation.
−4y
−4
—=—
Check
18 = −4y
Divide each side by −4.
−4.5 = y
?
18 = −4(−4.5)
Simplify.
18 = 18
✓
So, the solution is y = −4.5.
Solve the equation. Check your solution.
Exercises 7–18
78
Chapter 2
x
5
1. — = −2
Rational Numbers and Equations
2.
−a = −24
3.
3 = −1.5n
English
Spanish
EXAMPLE
2
Solving an Equation Using a Reciprocal
4
5
Solve −— x = −8.
4
5
−— x = −8
5
4
Write the equation.
⋅ ( −—45x ) = −—54⋅ (−8)
5
4
−—
4
5
Multiply each side by −—, the reciprocal of −—.
x = 10
Simplify.
So, the solution is x = 10.
Solve the equation. Check your solution.
2
3
Exercises 19–22
EXAMPLE
Ź40í F
4. −14 = — x
3
8
5
−— b = 5
6.
3
8
— h = −9
Real-Life Application
The record low temperature in Arizona is 1.6 times the record
low temperature in Rhode Island. What is the record low temperature
in Rhode Island?
Words
Record low temperature
in Arizona
5.
The record low in Arizona is 1.6 times the record low
in Rhode Island.
Variable Let t be the record low in Rhode Island.
−40
Equation
−40 = 1.6t
40
1.6
1.6t
1.6
= 1.6
×
t
Write equation.
−— = —
Divide each side by 1.6.
−25 = t
Simplify.
The record low temperature in Rhode Island is −25°F.
7. The record low temperature in Hawaii is –0.15 times the record
low temperature in Alaska. The record low temperature in Hawaii
is 12°F. What is the record low temperature in Alaska?
Section 2.5
Solving Equations Using Multiplication or Division
79
English
Spanish
Exercises
2.5
Help with Homework
1. WRITING Explain why multiplication can be used to solve equations
involving division.
2. OPEN-ENDED Turning a light on and then turning the light off are considered
to be inverse operations. Describe two other real-life situations that can be
thought of as inverse operations.
Describe the inverse operation that will undo the given operation.
3. Multiplying by 5
5. Dividing by −8
4. Subtracting 12
6. Adding −6
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Solve the equation. Check your solution.
1
7. 3h = 15
n
2
k
−3
9. — = −7
8. −5t = −45
10. — = 9
11. 5m = −10
12. 8t = −32
13. −0.2x = 1.6
15. −6p = 48
16. −72 = 8d
17. — = 5
3
4
2 19. — g = −12
2
5
20. 8 = −— c
b
4
14. −10 = −—
n
1.6
18. −14.4 = −0.6p
4
9
8
5
21. −— f = −3
✗
23. ERROR ANALYSIS Describe and correct the
error in finding the solution.
22. 26 = −— y
−4.2x = 21
−4.2x
4.2
21
4.2
—=—
x=5
Write the verbal sentence as an equation. Then solve.
24. A number divided by −9 is −16.
2
5
26. The product of 15 and a number is −75.
27. The quotient of a number and −1.5 is 21.
In Exercises 28 and 29, write an equation. Then solve.
28. NEWSPAPERS You make $0.75 for every newspaper
you sell. How many newspapers do you have to sell
to buy the soccer cleats?
3
5
29. ROCK CLIMBING A rock climber averages 12 — feet per minute.
How many feet does the rock climber climb in 30 minutes?
80
Chapter 2
Rational Numbers and Equations
3
20
25. A number multiplied by — is —.
English
Spanish
OPEN-ENDED (a) Write a multiplication equation that has the given solution.
(b) Write a division equation that has the same solution.
30. −3
1
2
31. −2.2
1
4
33. −1—
32. −—
2
3
34. REASONING Which of the methods can you use to solve −—c = 16?
2
3
Multiply each side by −—.
3
2
Multiply each side by −—.
Multiply each side by 3, then
divide each side by −2.
2
3
Divide each side by −—.
35. STOCK A stock has a return of −$1.26 per day. Write and solve an equation to
find the number of days until the total return is −$10.08.
3
4
36. ELECTION In a school election, — of the students vote. There are 1464 ballots.
Write and solve an equation to find the number of students.
37. OCEANOGRAPHY Aquarius is an underwater ocean
laboratory located in the Florida Keys National Marine
31
Sanctuary. Solve the equation — x = −62 to find the
25
value of x.
x
5
38. SHOPPING The price of a bike at store A is — the price at
6
−62 ft
store B. The price at store A is $150.60. Write and solve an
equation to find how much you save by buying the bike
at store A.
39. CRITICAL THINKING Solve −2∣ m ∣ = −10.
5
7
In four days, your family drives — of a trip.
40.
Your rate of travel is the same throughout the trip. The total trip
is 1250 miles. How many more days until you reach your destination?
Subtract.
SECTION 1.3
41. 5 − 12
42. −7 − 2
43. 4 − (−8)
44. −14 − (−5)
45. MULTIPLE CHOICE Of the 120 apartments in a building, 75 have been
scheduled to receive new carpet. What fraction of the apartments have
not been scheduled to receive new carpet?
SECTION 2.1
A
○
1
4
—
B
○
3
8
—
Section 2.5
C
○
5
8
—
D
○
3
4
—
Solving Equations Using Multiplication or Division
81
English
Spanish
2.6
Solving Two-Step Equations
In a two-step equation, which step should
you do first?
1
EXAMPLE: Solving a Two-Step Equation
Use algebra tiles to model and solve 2x − 3 = −5.
Model the equation 2x − 3 = −5.
=
Remove the three red tiles on the left side
by adding three yellow tiles to each side.
=
Remove the three “zero pairs” from each side.
=
Because there are two green tiles, divide the
red tiles into two equal groups.
=
Keep one of the groups. This shows the value of x.
=
=
So, x = −1.
2
EXAMPLE: The Math Behind the Tiles
Solve 2x − 3 = −5 without using algebra tiles. Describe each step.
Which step is first, adding 3 to each side or dividing each side by 2?
Use the steps in Example 1 as a guide.
2x − 3 = −5
Write the equation.
2x − 3 + 3 = −5 + 3
Add 3 to each side.
2x = −2
2x
2
−2
2
—=—
x = −1
Simplify.
Divide each side by 2.
Simplify.
So, x = −1. Adding 3 to each side is the first step.
82
Chapter 2
Rational Numbers and Equations
English
Spanish
3
ACTIVITY: Solving Equations Using Algebra Tiles
Work with a partner.
●
Write an equation shown by the algebra tiles.
●
Use algebra tiles to model and solve the equation.
●
Check your answer by solving the equation without using algebra tiles.
b.
a.
=
4
=
ACTIVITY: Working Backwards
Work with a partner.
a. Sample: Your friend pauses a video game to get a drink. You continue
the game. You double the score by saving a princess. Then you lose
75 points because you do not collect the treasure. You finish the game
with −25 points. How many points did you start with?
One way to solve the problem is to work backwards. To do this, start
with the end result and retrace the events.
You have −25 points at the end of the game.
−25
You lost 75 points for not collecting the treasure,
so add 75 to −25.
−25 + 75 = 50
You doubled your score for saving the princess,
so find half of 50.
50 ÷ 2 = 25
So, you started the game with 25 points.
b. You triple your account balance by making a deposit. Then you
withdraw $127.32 to buy groceries. Your account is now overdrawn
by $10.56. By working backwards, find your account balance before
you made the deposit.
5. IN YOUR OWN WORDS In a two-step equation, which step should you do
first? Give four examples.
6. Solve the equation 2x − 75 = −25. How do your steps compare with the
strategy of working backwards in Activity 4?
Use what you learned about solving two-step equations to
complete Exercises 6 –11 on page 86.
Section 2.6
Solving Two-Step Equations
83
English
2.6
Spanish
Lesson
Lesson Tutorials
EXAMPLE
1
Solving a Two-Step Equation
Solve −3x + 5 = 2. Check your solution.
−3x + 5 =
−5
2
Write the equation.
−5
−3x + 5 = 2
?
−3(1) + 5 = 2
Subtract 5 from each side.
−3x = −3
−3x
−3
Check
?
−3 + 5 = 2
Simplify.
−3
−3
—=—
Divide each side by −3.
x=1
2=2
Simplify.
So, the solution is x = 1.
Solve the equation. Check your solution.
1. 2x + 12 = 4
Exercises 6 –17
EXAMPLE
2
2.
−5c + 9 = −16
3.
3(x − 4) = 9
6.
— + 4a = −—
Solving a Two-Step Equation
x
8
1
2
7
2
Solve — − — = −—.
Study Tip
You can simplify the
equation in Example 2
before solving. Multiply
each side by the LCD of
the fractions, 8.
x
8
1
2
7
2
Write the equation.
1
2
+—
1
2
Add — to each side.
— − — = −—
+—
1
2
x
8
— = −3
x
1
7
— − — = −—
8
2
2
x − 4 = −28
x = −24
⋅ 8x
Simplify.
⋅
8 — = 8 (−3)
Multiply each side by 8.
x = −24
Simplify.
So, the solution is x = −24.
Solve the equation. Check your solution.
Exercises 20–25
84
Chapter 2
m
2
4. — + 6 = 10
Rational Numbers and Equations
5.
z
3
−— + 5 = 9
2
5
6
5
✓
English
Spanish
EXAMPLE
3
Combining Like Terms Before Solving
Solve 3y − 8y = 25.
3y − 8y = 25
Write the equation.
−5y = 25
Combine like terms.
y = −5
Divide each side by −5.
So, the solution is y = −5.
EXAMPLE
4
Real-Life Application
Top
10h
−100
The height at the top of a roller coaster hill
is 10 times the height h of the starting point.
The height decreases 100 feet from the top
to the bottom of the hill. The height at the
bottom of the hill is −10 feet. Find h.
Start
h
−10
Bottom
Location
Verbal Description
Start
The height at the start is h.
Top of hill
The height at the top of the hill is
10 times the starting height h.
Bottom of hill
Height decreases by 100 feet.
So, subtract 100.
Expression
h
10h
10h − 100
The height at the bottom of the hill is −10 feet. Solve
10h − 100 = −10 to find h.
10h − 100 = −10
Write equation.
10h = 90
Add 100 to each side.
h=9
Divide each side by 10.
The height at the start is 9 feet.
Solve the equation. Check your solution.
Exercises 29–34
7. 4 − 2y + 3 = −9
8.
7x − 10x = 15
9.
−8 = 1.3m − 2.1m
10. WHAT IF? In Example 4, the height at the bottom of the hill is
−5 feet. Find the height h.
Section 2.6
Solving Two-Step Equations
85
English
Spanish
Exercises
2.6
Help with Homework
1. WRITING How do you solve two-step equations?
Match the equation with the first step to solve it.
n
4
n
4
2. 4 + 4n = −12
3. 4n = −12
4. — = −12
5. — − 4 = −12
A. Add 4.
B. Subtract 4.
C. Multiply by 4.
D. Divide by 4.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Solve the equation. Check your solution.
1
6. 2v + 7 = 3
7. 4b + 3 = −9
8. 17 = 5k − 2
9. −6t − 7 = 17
10. 8n + 16.2 = 1.6
11. −5g + 2.3 = −18.8
12. 2t − 5 = −10
13. −4p + 9 = −5
14. 11 = −5x − 2
15. 4 + 2.2h = −3.7
16. −4.8f + 6.4 = −8.48
17. 7.3y − 5.18 = −51.9
ERROR ANALYSIS Describe and correct the error in finding the solution.
18.
✗
19.
−6 + 2x = −10
2x
2
10
2
−6 + — = −—
✗
−3x + 2 = −7
−3x = −9
3x
3
−9
3
−— = —
−6 + x = −5
x = −3
x=1
Solve the equation. Check your solution.
3
5
1
3
10
3
2 20. — g − — = −—
b
3
5
2
23. 2 − — = −—
a
4
5
6
1
2
1
3
21. — − — = −—
2
3
3
7
5
6
22. −— + 2z = −—
1
2
24. −— x + — = —
In Exercises 26 –28, write an equation. Then solve.
26. WEATHER Starting at 1:00 p.m., the temperature changes
−4 degrees per hour. How long will it take to reach −1°?
9
4
4
5
Temperature
at 1:00 P.M.
35°F
27. BOWLING It costs $2.50 to rent bowling shoes. Each game
costs $2.25. You have $9.25. How many games can you bowl?
28. CELL PHONES A cell phone company charges a monthly fee
plus $0.25 for each text message. The monthly fee is $30.00
and you owe $59.50. How many text messages did you have?
86
Chapter 2
Rational Numbers and Equations
7
8
25. −— v + — = —
°F
English
Spanish
Solve the equation. Check your solution.
3 29. 3v − 9v = 30
32. 6(x − 2) = −18
30. 12t − 8t = −52
31. −8d − 5d + 7d = 72
33. −4(m + 3) = 24
34. −8(y + 9) = −40
1
35. WRITING Write a real-world problem that can be modeled by — x − 2 = 8.
2
Then solve the equation.
3m
36. GEOMETRY The perimeter of the parallelogram
is 102 feet. Find m.
m
REASONING Exercises 37 and 38 are missing information. Tell what information
is needed to solve the problem.
37. TAXI A taxi service charges an initial fee plus $1.80 per mile. How far can you
travel for $12?
38. EARTH The coldest surface temperature on the moon is 57 degrees colder
than twice the coldest surface temperature on Earth. What is the coldest
surface temperature on Earth?
39. SCIENCE On Saturday, you catch insects for your science class. Five of the insects
escape. The remaining insects are divided into three groups to share in class.
Each group has nine insects. How many insects did you catch on Saturday?
a. Solve the problem by working backwards.
x−5
b. Solve the equation — = 9. How does the answer compare with the
3
answer to part (a)?
−15 ft
40. UNDERWATER HOTEL You must scuba dive to the entrance
of your room at Jule’s Undersea Lodge in Key Largo, Florida.
2
3
The diver is 1 foot deeper than — of the elevation of the entrance.
What is the elevation of the entrance?
Entrance
41.
How much should you change the length of
the rectangle so that the perimeter is 54 centimeters? Write
an equation that shows how you found your answer.
Multiply or divide.
42. −6.2 × 5.6
15
20
—
25 cm
SECTION 2.3
8
3
( )
1
2
43. — × −2 —
5
2
( )
4
5
44. — ÷ −—
45. −18.6 ÷ (−3)
46. MULTIPLE CHOICE Which fraction is not equivalent to 0.75?
A
○
12 cm
B
○
9
12
—
C
○
Section 2.6
6
9
—
SKILLS REVIEW HANDBOOK
D
○
3
4
—
Solving Two-Step Equations
87
English
Spanish
Quiz
2.4 –2.6
Progress Check
Solve the equation. Check your solution. (Section 2.4 and Section 2.5)
.5)
1
2
3
4
2. 4 — + p = −5 —
1. −6.5 + x = −4.12
b
7
3. −— = 4
4. 2h = −57
Write the verbal sentence as an equation. Then solve. (Section 2.4 and Section 2.5)
5. The difference between a number b and 7.4 is −6.8.
2
5
1
2
6. 5 — more than a number a is 7 —.
3
8
15
32
7. A number x multiplied by — is −—.
8. The quotient of two times a number k and −2.6 is 12.
Write and solve an equation to find the value of x. (Section 2.4 and Section 2.6)
9. Perimeter = 26
10. Perimeter = 23.59
x
10.5
2.8
11. Perimeter = 33
x
13
3x
5.62
5.62
9
12
3.65
12. BANKING You withdraw $29.79 from your bank account. Now, your balance
is −$20.51. Write and solve an equation to find the amount of money in your
bank account before you withdrew the money. (Section 2.4)
1
5
13. WATER LEVEL During a drought, the water level of a lake changes −3 — feet
per day. Write and solve an equation to find how long it takes for the water
level to change −16 feet. (Section 2.5)
14. BASKETBALL A basketball game has four quarters. The length of a game is
1
2
32 minutes. You play the entire game except 4 — minutes. Write and solve an
equation to find the mean time you play per quarter.
(Section 2.6)
15. SCRAPBOOKING The mat needs to be cut to have
a 0.5-inch border on all four sides. (Section 2.6)
6 in.
a. How much should you cut from the left
and right sides?
b. How much should you cut from the
top and bottom?
9.6 in.
4 in.
7.8 in.
88
Chapter 2
Rational Numbers and Equations
English
2
Spanish
Chapter Review
Vocabulary Help
Review Key Vocabulary
terminating decimal, p. 52
repeating decimal, p. 52
rational number, p. 52
equivalent equations, p. 72
Review Examples and Exercises
2.1
Rational Numbers
(pp. 50–55)
Write −0.14 as a fraction in simplest form.
Write the digits after the
decimal point in the numerator.
14
−0.14 = −—
100
The last digit is in the hundredths
place. So, use 100 in the denominator.
7
50
= −—
Simplify.
Write the rational number as a decimal.
5
8
8
15
1. −—
13
6
7
16
3. −—
2. —
4. 1 —
Write the decimal as a fraction or mixed number in simplest form.
5. −0.6
2.2
6. −0.35
7. −5.8
Adding and Subtracting Rational Numbers
8. 24.23
(pp. 56–61)
Find −8.18 + 3.64.
∣ −8.18 ∣ > ∣ 3.64 ∣. So, subtract ∣ 3.64 ∣ from ∣ −8.18 ∣.
−8.18 + 3.64 = −4.54
Use the sign of −8.18.
Add or subtract. Write fractions in simplest form.
5
9
8
9
5
12
9. −4 — + —
3
10
10. −— − —
11. −2.53 + 4.75
12. 3.8 − (−7.45)
5
6
13. TURTLES A turtle is 20 — inches below the surface of a pond. It dives to a
1
4
depth of 32 — inches. How far did it dive?
Chapter Review
89
English
Spanish
2.3
Multiplying and Dividing Rational Numbers
1
6
(pp. 62–67)
1
3
Find −4 — ÷ 1—.
1
6
1
3
25
6
4
3
−4 — ÷ 1 — = −— ÷ —
−25
6
Write mixed numbers as improper fractions.
⋅ 34
4
3
=— —
Multiply by the reciprocal of —.
⋅
−25 3
6 4
=—
⋅
−25
8
Multiply the numerators and the denominators.
1
8
= —, or −3 —
Simplify.
Multiply or divide. Write fractions in simplest form.
4
9
( )
( )
9
10
7
9
6
5
8
15
( )
2
3
4
11
2
7
14. −— −—
15. — ÷ −—
16. — −—
17. −— ÷ —
18. −5.9(−9.7)
19. 6.4 ÷ (−3.2)
20. 4.5(−5.26)
21. −15.4 ÷ (−2.5)
22. SUNKEN SHIP The elevation of a sunken ship is −120 feet. Your elevation is
5
8
— of the ship’s elevation. What is your elevation?
2.4
Solving Equations Using Addition or Subtraction
(pp. 70 –75)
Solve x − 9 =−6.
x − 9 = −6
+9
Write the equation.
+9
Add 9 to each side.
x=3
Simplify.
Solve the equation. Check your solution.
23. p − 3 = −4
3
4
1
4
27. n + — = —
24. 6 + q = 1
5
6
7
8
28. v − — = −—
25. −2 + j = −22
26. b − 19 = −11
29. t − 3.7 = 1.2
30. ℓ + 15.2 = −4.5
31. GIFT CARD A shirt costs $24.99. After using a gift card as a partial
payment, you still owe $9.99. What is the value of the gift card?
90
Chapter 2
Rational Numbers and Equations
English
Spanish
2.5
Solving Equations Using Multiplication or Division
(pp. 76– 81)
x
5
Solve — = −7.
x
5
— = −7
⋅ 5x
Write the equation.
⋅
5 — = 5 (−7)
x = −35
Multiply each side by 5.
Simplify.
Solve the equation. Check your solution.
x
3
y
7
z
4
3
4
w
20
32. — = −8
33. −7 = —
34. −— = −—
35. −— = −2.5
36. 4x = −8
37. −10 = 2y
38. −5.4z = −32.4
39. −6.8w = 3.4
40. TEMPERATURE The mean temperature change is −3.2°F per day for five
days. What is the total change over the five-day period?
2.6
Solving Two-Step Equations
x
5
(pp. 82– 87)
3
10
7
10
Solve — + — = − —.
x
5
7
10
3
10
— + — = −—
Write the equation.
— = −1
x
5
Subtract — from each side.
x = −5
Multiply each side by 5.
7
10
Solve the equation. Check your solution.
41. −2c + 6 = −8
w
6
5
8
3
8
43. — + — = −1 —
42. 3(3w − 4) = −20
44. −3x − 4.6 = 5.9
45. EROSION The floor of a canyon has an elevation of −14.5 feet. Erosion causes
the elevation to change by −1.5 feet per year. How many years will it take for
the canyon floor to have an elevation of −31 feet?
Chapter Review
91
English
2
Spanish
Chapter Test
Test Practice
Write the rational number as a decimal.
7
40
1
9
36
5
21
16
2. −—
1. —
3. −—
4. —
Write the decimal as a fraction or mixed number in simplest form.
5. −0.122
7. −4.45
6. 0.33
8. −7.09
Add or subtract. Write fractions in simplest form.
4
9
( )
23
18
9. −— + −—
( )
17
12
1
8
10. — − −—
11. 9.2 + (−2.8)
12. 2.86 − 12.1
Multiply or divide. Write fractions in simplest form.
9
10
( )
8
3
13. 3 — × −—
5
6
1
6
14. −1 — ÷ 4 —
15. −4.4 × (−6.02)
16. −5 ÷ 1.5
Solve the equation. Check your solution.
2
9
17. 7x = −3
18. 2(x + 1) = −2
19. — g = −8
20. z + 14.5 = 5.4
21. −14 = 6c
22. — k − — = −—
2
7
3
8
19
8
23. MARATHON A marathon is a 26.2-mile race. You run three marathons in one
year. How many miles do you run?
24. RECORD A runner is compared with the world record
holder during a race. A negative number means the
runner is ahead of the time of the world record holder,
and a positive number means that the runner is behind
the time of the world record holder. The table
shows the time difference between the runner and
the world record holder for each lap. What time
difference does the runner need for the fourth lap
to match the world record?
Lap
Time Difference
1
−1.23
2
0.45
3
0.18
4
?
25. GYMNASTICS You lose 0.3 point for stepping out of bounds during a floor
routine. Your final score is 9.124. Write and solve an equation to find your
score before the penalty.
26. PERIMETER The perimeter of the
triangle is 45. Find the value of x.
17
5x
15
92
Chapter 2
Rational Numbers and Equations
English
2
Spanish
Standardized Test Practice
Test-Takin
g Strateg
y
Estimate
the Answ
er
1. When José and Sean were each 5 years old,
1
2
José was 1— inches taller than Sean. José grew
3
4
at an average rate of 2 — inches per year from
the time that he was 5 years old until the time
he was 13 years old. José was 63 inches tall
when he was 13 years old. How tall was Sean
when he was 5 years old?
1
2
1
B. 42 — in.
2
3
4
3
D. 47 — in.
4
A. 39 — in.
C. 44 — in.
“Using
e
there are stimation you
can see
about 1
th
30 are n 0 tabbies. So a at
bout
ot tabb
ies.”
2. A line is graphed in the coordinate
plane below.
y
(9, 3)
3
2
1
−5 −4 −3 −2
O
1
2
3
5
6
7
8
9 x
−2
(−3, −5)
−4
−5
−6
Which point is not on the line?
F. (−3, 0)
H. (3, −1)
G. (0, −3)
I. (6, 1)
3. What is the missing number in the sequence below?
9
16
9 9
8 4
9
2
—, −—, —, −—, 9,
4. What is the value of the expression below?
∣ −2 − (−2.5) ∣
A. −4.5
C. 0.5
B. −0.5
D. 4.5
Standardized Test Practice
93
English
Spanish
5. Which equation is equivalent to the equation shown below?
3
4
1
8
3
8
−—x + — = −—
3
3 1
4
8 8
3
3 1
G. −— x = −— + —
4
8 8
1
3
8
8
1
3
I. x + — = −—
8
8
F. −— x = −— − —
H. x + — = −—
⋅ ( −—43 )
⋅ ( −—34 )
6. What is the value of the expression below?
−5 ÷ 20
7. Karina was solving the equation in the box below.
−96 = −6(15 − 2x)
−96 = −90 − 12x
−96 + 90 = −90 + 90 − 12x
−6 = −12x
−6
−12
−12x
−12
—=—
1
2
—=x
What should Karina do to correct the error that she made?
A. First add 6 to both sides of the equation.
B. First add 2x to both sides of the equation.
C. Distribute the −6 to get 90 − 12x.
D. Distribute the −6 to get −90 + 12x.
8. Current, voltage, and resistance are related according to the formula below,
where I represents the current, in amperes, V represents the voltage, in volts,
and R represents the resistance, in ohms.
V
R
I=—
What is the voltage when the current is 0.5 ampere and the resistance
is 0.8 ohm?
94
F. 4.0 volts
H. 0.4 volt
G. 1.3 volts
I. 0.3 volt
Chapter 2
Rational Numbers and Equations
English
Spanish
1
2
9. What is the area of a triangle with a base length of 2 — inches and a
height of 3 inches?
3
4
3
B. 3 — in.2
4
1
2
1
D. 7 — in.2
2
A. 2 — in.2
C. 5 — in.2
10. What is the circumference of the circle below? (Use 3.14 for π .)
10.2 cm
F. 64.056 cm
H. 32.028 cm
G. 60.028 cm
I. 30.028 cm
11. Four points are graphed on the number line below.
R
−3
S
−2
−1
T
0
U
1
2
3
Part A
Choose the two points whose values have the greatest sum.
Approximate this sum. Explain your reasoning.
Part B
Choose the two points whose values have the greatest difference.
Approximate this difference. Explain your reasoning.
Part C
Choose the two points whose values have the greatest product.
Approximate this product. Explain your reasoning.
Part D
Choose the two points whose values have the greatest quotient.
Approximate this quotient. Explain your reasoning.
12. What number belongs in the box to make the equation true?
−0.4
— + 0.8 = −1.2
A. 1
C. −0.2
B. 0.2
D. −1
Standardized Test Practice
95