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The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

The difference in temperatures was 45 degrees.

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The temperature in Denver was −6 degrees at lunchtime. By sunset the temperature had dropped to −15 degrees. What was the difference in the lunchtime and sunset temperatures?

The difference in temperatures was 9 degrees.

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Apply a strategy to solve applications with integers.

  1. Read the problem. Make sure all the words and ideas are understood
  2. Identify what we are asked to find.
  3. Write a phrase that gives the information to find it.
  4. Translate the phrase to an expression.
  5. Simplify the expression.
  6. Answer the question with a complete sentence.

The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

Solution

Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are asked to find. the number of yards lost Step 3. Write a phrase that gives the information to find it. three times a 15-yard penalty Step 4. Translate the phrase to an expression. 3 ( −15 ) Step 5. Simplify the expression. −45 Step 6. Answer the question with a complete sentence. The team lost 45 yards.

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The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

The Bears lost 105 yards.

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Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM?

A $16 fee was deducted from his checking account.

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Key concepts

  • Multiplication and Division of Two Signed Numbers
    • Same signs—Product is positive
    • Different signs—Product is negative
  • Strategy for Applications
    1. Identify what you are asked to find.
    2. Write a phrase that gives the information to find it.
    3. Translate the phrase to an expression.
    4. Simplify the expression.
    5. Answer the question with a complete sentence.

Practice makes perfect

Multiply Integers

In the following exercises, multiply.

Divide Integers

In the following exercises, divide.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

5 ( −6 ) + 7 ( −2 ) 3

−47

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8 ( −4 ) + 5 ( −4 ) 6

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( 8 11 ) ( 9 12 )

9

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( 6 11 ) ( 8 13 )

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65 ÷ ( −5 ) + ( −28 ) ÷ ( −7 )

−9

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52 ÷ ( −4 ) + ( −32 ) ÷ ( −8 )

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9 2 [ 3 8 ( −2 ) ]

−29

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11 3 [ 7 4 ( −2 ) ]

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( −3 ) 2 24 ÷ ( 8 2 )

5

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( −4 ) 2 32 ÷ ( 12 4 )

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Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

y + ( −14 ) when
y = −33
y = 30

−47 16

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x + ( −21 ) when
x = −27
x = 44

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a + 3 when a = −7
a + 3 when a = −7

−4 10

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d + ( −9 ) when d = −8
d + ( −9 ) when d = −8

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m + n when
m = −15 , n = 7

−8

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p + q when
p = −9 , q = 17

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r + s when r = −9 , s = −7

−16

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t + u when t = −6 , u = −5

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( x + y ) 2 when
x = −3 , y = 14

121

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( y + z ) 2 when
y = −3 , z = 15

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−2 x + 17 when
x = 8
x = −8

1 33

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−5 y + 14 when
y = 9
y = −9

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10 3 m when
m = 5
m = −5

−5 25

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18 4 n when
n = 3
n = −3

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2 w 2 3 w + 7 when
w = −2

21

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3 u 2 4 u + 5 when u = −3

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9 a 2 b 8 when
a = −6 and b = −3

−56

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7 m 4 n 2 when
m = −4 and n = −9

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Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

the sum of 3 and −15 , increased by 7

( 3 + ( −15 ) ) + 7 ; 5

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the sum of −8 and −9 , increased by 23

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the difference of 10 and −18

10 ( −18 ) ; 28

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the difference of −5 and −30

−5 ( −30 ) ; 25

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subtract −6 from −13

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the product of −3 and 15

−3 · 15 ; 45

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the product of −4 and 16

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the quotient of −60 and −20

−60 ÷ ( −20 ) ; 3

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the quotient of −40 and −20

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the quotient of −6 and the sum of a and b

−6 a + b

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the quotient of −7 and the sum of m and n

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the product of −10 and the difference of p and q

−10 ( p q )

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the product of −13 and the difference of c and d

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Use Integers in Applications

In the following exercises, solve.

Temperature On January 15 , the high temperature in Anaheim, California, was 84 ° . That same day, the high temperature in Embarrass, Minnesota was −12 ° . What was the difference between the temperature in Anaheim and the temperature in Embarrass?

96 °

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Temperature On January 21 , the high temperature in Palm Springs, California, was 89 ° , and the high temperature in Whitefield, New Hampshire was −31 ° . What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

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Football At the first down, the Chargers had the ball on their 25 yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down?

21

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Football At the first down, the Steelers had the ball on their 30 yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down?

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Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?

$28

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Checking Account Selina has $165 in her checking account. She writes a check for $207. What is the new balance in her checking account?

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Checking Account Diontre has a balance of $38 in his checking account. He deposits $225 to the account. What is the new balance?

$187

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Checking Account Reymonte has a balance of $49 in his checking account. He deposits $281 to the account. What is the new balance?

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Everyday math

Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio?

$3600

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Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women?

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Writing exercises

In your own words, state the rules for multiplying integers.

Answers may vary

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In your own words, state the rules for dividing integers.

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Why is 2 4 ( −2 ) 4 ?

Answers may vary

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Why is 4 3 = ( −4 ) 3 ?

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A table is shown that is composed of four columns and seven rows. The titles of the columns are “I can …”, “Confidently”, “With some help” and “No – I don’t get it!”. The first column reads “multiple integers.”, “divide integers.”, “simplify expressions with integers.”, “evaluate variable expressions with integers.”, “translate English phrases to algebraic expressions.” and “use integers in applications.”

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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