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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?

Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina

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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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