Problem Set: Integers

Identifying and Writing Integers

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the given points on a number line.

Exercise 1

  1. ⓐ = 22
  2. ⓑ = −2−2
  3. ⓒ = −5−5

Exercise 2

  1. ⓐ = 55
  2. ⓑ = −5−5
  3. ⓒ = −2−2

Exercise 3

  1. ⓐ = −8−8
  2. ⓑ = 88
  3. ⓒ = −6−6

Exercise 4

  1. ⓐ = −7−7
  2. ⓑ = 77
  3. ⓒ = −1−1

Order Positive and Negative Numbers on the Number Line

In the following exercises, order each of the following pairs of numbers, using < or >.

  1. 9__49__4

  2. −3__6−3__6

  3. −8__−2−8__−2

  4. 1__−101__−10

  5. 6__26__2
  6. −7__4−7__4
  7. −9__−1−9__−1
  8. 9__−39__−3
  9. −5__1−5__1

  10. −4__−9−4__−9

  11. 6__106__10

  12. 3__−83__−8

  13. −7__3−7__3
  14. −10__−5−10__−5
  15. 2__−62__−6
  16. 8__98__9

Find Opposites

In the following exercises, find the opposite of each number.

  1. 22

  2. −6−6

  3. 99
  4. −4−4
  5. −8−8

  6. 11

  7. −2−2
  8. 66

Simplify Negatives

In the following exercises, simplify.

  1. −(−4)−(−4)

  2. −(−8)−(−8)
  3. −(−15)−(−15)

  4. −(−11)−(−11)

Simplify Negatives

In the following exercises, evaluate.

Exercise 1

−m−m when

  1.  m=3m=3

  2. m=−3m=−3

Exercise 2

−p−p when

  1.  p=6p=6
  2. p=−6p=−6

Exercise 3

−c−c when

  1.  c=12c=12

  2. c=−12c=−12

Exercise 3

−d−d when

  1. d=21d=21
  2. d=−21d=−21

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

Exercise 1

  1. |7||7|

  2. |−25||−25|

  3.  |0||0|

  4.  |5||5|
  5.  |20||20|
  6.  |−19||−19|
  7.  |−32||−32|

  8. |−18||−18|

  9.  |16||16|

  10. |−41||−41|
  11.  |−40||−40|
  12. |22||22|

Simplify Expressions with Absolute Value

In the following exercises, evaluate each absolute value expression.

  1.  |x| when x=−28|x| when x=−28

  2.  |−u| when u=−15|−u| when u=−15

  3. |y| when y=−37|y| when y=−37
  4. |−z| when z=−24|−z| when z=−24
  5. −|p| when p=19−|p| when p=19

  6. −|q| when q=−33−|q| when q=−33

  7. −|a| when a=60−|a| when a=60
  8. −|b| when b=−12−|b| when b=−12

Simplify Expressions with Absolute Value

In the following exercises, fill in <,>,or=<,>,or= to compare each expression.

  1. −6__|−6|−6__|−6|

  2. −|−3|__−3−|−3|__−3

  3. −8__|−8|−8__|−8|
  4. −|−2|__−2−|−2|__−2
  5. |−3|__−|−3||−3|__−|−3|

  6. 4__−|−4|4__−|−4|

  7. |−5|__−|−5|
  8. 9__−|−9|

Simplify Expressions with Absolute Value

In the following exercises, simplify each expression.

  1. |8−4|

  2. |9−6|
  3. 8|−7|

  4. 5|−5|
  5. |15−7|−|14−6|

  6. |17−8|−|13−4|
  7. 18−|2(8−3)|

  8. 15−|3(8−5)|
  9. 8(14−2|−2|)

  10. 6(13−4|−2|)

Translate Word Phrases into Expressions with Integers

Translate each phrase into an expression with integers. Do not simplify.

Exercise 1

  1. the opposite of 8

  2. the opposite of −6

  3. negative three

  4. 4 minus negative 3

  5. the opposite of 11
  6. the opposite of −4
  7. negative nine
  8. 8 minus negative 2
  9. the opposite of 20

  10. the opposite of −5

  11. the opposite of 12

  12. 18 minus negative 7

  13. the opposite of 15
  14. the opposite of −9
  15. negative sixty
  16. 12 minus 5
  17. a temperature of 6degrees below zero

  18. a temperature of 14degrees below zero
  19. an elevation of 40 feet  below sea level

  20. an elevation of 65 feet  below sea level
  21. a football play loss of 12 yards 

  22. a football play gain of 4 yards 
  23. a stock gain of $3

  24. a stock loss of $5
  25. a golf score one above par

  26. a golf score of 3 below par

Everyday Math

Elevation

The highest elevation in the United States is Mount McKinley, Alaska, at 20,320 feet above sea level. The lowest elevation is Death Valley, California, at 282 feet below sea level. Use integers to write the elevation of:

  1. Mount McKinley

  2. Death Valley

Extreme temperatures

The highest recorded temperature on Earth is 58∘ Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature is 90∘ below 0∘ Celsius, recorded in Antarctica in 1983. Use integers to write the:

  1.  highest recorded temperature
  2. lowest recorded temperature

State budgets

In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of $540 million. That same month, Texas estimated it would have a budget deficit of $27 billion. Use integers to write the budget:

  1. surplus

  2. deficit

College enrollments

Across the United States, community college enrollment grew by 1,400,000 students from 2007 to 2010. In California, community college enrollment declined by 110,171 students from 2009 to 2010. Use integers to write the change in enrollment:

  1. growth
  2. decline

Writing Exercises

Give an example of a negative number from your life experience.

What are the three uses of the “−” sign in algebra? Explain how they differ.

Adding Integers

Model Addition of Integers

In the following exercises, model the expression to simplify.

  1. 7+4

  2. 8+5
  3. −6+(−3)

  4. −5+(−5)
  5. −7+5

  6. −9+6
  7. 8+(−7)

  8. 9+(−4)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

  1. −21+(−59)

  2. −35+(−47)
  3. 48+(−16)

  4. 34+(−19)
  5. −200+65

  6. −150+45
  7. 2+(−8)+6

  8. 4+(−9)+7
  9. −14+(−12)+4

  10. −17+(−18)+6
  11. 135+(−110)+83

  12. 140+(−75)+67
  13. −32+24+(−6)+10

  14. −38+27+(−8)+12
  15. 19+2(−3+8)

  16. 24+3(−5+9)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Exercise 1

x+8 when

  1.  x=−26

  2. x=−95

Exercise 2

y+9 when

  1.  y=−29
  2. y=−84

Exercise 3

y+(−14) when

  1. y=−33

  2. y=30

Exercise 4

x+(−21) when

  1.  x=−27
  2. x=44

Exercise 5

When a=−7, evaluate:

  1.  a+3

  2. −a+3

Exercise 6

When b=−11, evaluate:

  1. b+6
  2. −b+6

Exercise 7

When c=−9, evaluate:

  1. c+(−4)

  2. −c+(−4)

Exercise 8

When d=−8, evaluate:

  1. d+(−9)
  2. −d+(−9)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

  1. m+n when, m=−15 , n=7

  2. p+q when, p=−9 , q=17
  3. r−3s when, r=16 , s=2

  4. 2t+u when, t=−6 , u=−5
  5. (a+b)2 when, a=−7 , b=15

  6. (c+d)2 when, c=−5 , d=14
  7. (x+y)2 when, x=−3 , y=14

  8. (y+z)2 when, y=−3 , z=15

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

  1. The sum of −14 and 5

  2. The sum of −22 and 9
  3. 8 more than −2

  4. 5 more than −1
  5. −10 added to −15

  6. −6 added to −20
  7. 6 more than the sum of −1 and −12

  8. 3 more than the sum of −2 and −8
  9. the sum of 10 and −19, increased by 4

  10. the sum of 12 and −15, increased by 1

Add Integers in Applications

In the following exercises, solve.

Temperature

The temperature in St. Paul, Minnesota was −19∘ F at sunrise. By noon the temperature had risen 26∘ F. What was the temperature at noon?

Temperature

The temperature in Chicago was −15∘ F at 6 am. By afternoon the temperature had risen 28∘ F. What was the afternoon temperature?

Credit Cards

Lupe owes $73 on her credit card. Then she charges $45 more. What is the new balance?

Credit Cards

Frank owes $212 on his credit card. Then he charges $105 more. What is the new balance?

Weight Loss

Angie lost 3 pounds the first week of her diet. Over the next three weeks, she lost 2 pounds, gained 1 pound, and then lost 4 pounds. What was the change in her weight over the four weeks?

Weight Loss

April lost 5 pounds the first week of her diet. Over the next three weeks, she lost 3 pounds, gained 2 pounds, and then lost 1 pound. What was the change in her weight over the four weeks?

Football

The Rams took possession of the football on their own 35-yard line. In the next three plays, they lost 12 yards, gained 8 yards, then lost 6 yards. On what yard line was the ball at the end of those three plays?

Football

The Cowboys began with the ball on their own 20-yard line. They gained 15 yards, lost 3 yards and then gained 6 yards on the next three plays. Where was the ball at the end of these plays?

Calories

Lisbeth walked from her house to get a frozen yogurt, and then she walked home. By walking for a total of 20 minutes, she burned 90 calories. The frozen yogurt she ate was 110 calories. What was her total calorie gain or loss?

Calories

Ozzie rode his bike for 30 minutes, burning 168 calories. Then he had a 140-calorie iced blended mocha. Represent the change in calories as an integer.

Everyday Math

Stock Market

The week of September 15, 2008, was one of the most volatile weeks ever for the U.S. stock market. The change in the Dow Jones Industrial Average each day was:

Monday−504Tuesday+142Wednesday−449Thursday+410Friday+369

What was the overall change for the week?

Stock Market

During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:

Monday−201Tuesday−16Wednesday−23Thursday+172Friday−34

What was the overall change for the week?

Writing Exercises

Explain why the sum of −8 and 2 is negative, but the sum of 8 and −2 and is positive.

Give an example from your life experience of adding two negative numbers.

Subtracting Integers

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

  1. 8−2

  2. 9−3
  3. −5−(−1)

  4. −6−(−4)
  5. −5−4

  6. −7−2
  7. 8−(−4)

  8. 7−(−3)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

  1.  15−6

  2. 15+(−6)

  3. 12−9
  4. 12+(−9)
  5. 44−28

  6. 44+(−28)

  7.  35−16
  8. 35+(−16)
  9. 8−(−9)

  10. 8+9

  11. 4−(−4)
  12. 4+4
  13. 27−(−18)

  14. 27+18

  15. 46−(−37)
  16. 46+37

Simplify Expressions with Integers

In the following exercises, simplify each expression.

  1. 15−(−12)

  2. 14−(−11)
  3. 10−(−19)

  4. 11−(−18)
  5. 48−87

  6. 45−69
  7. 31−79

  8. 39−81
  9. −31−11

  10. −32−18
  11. −17−42

  12. −19−46
  13. −103−(−52)

  14. −105−(−68)
  15. −45−(−54)

  16. −58−(−67)
  17. 8−3−7

  18. 9−6−5
  19. −5−4+7

  20. −3−8+4
  21. −14−(−27)+9

  22. −15−(−28)+5
  23. 71+(−10)−8

  24. 64+(−17)−9
  25. −16−(−4+1)−7

  26. −15−(−6+4)−3
  27. (2−7)−(3−8)

  28. (1−8)−(2−9)
  29. −(6−8)−(2−4)

  30. −(4−5)−(7−8)
  31. 25−[10−(3−12)]

  32. 32−[5−(15−20)]
  33. 6⋅3−4⋅3−7⋅2

  34. 5⋅7−8⋅2−4⋅9
  35. 52−62

  36. 62−72

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression for the given values.

Exercise 1

x−6 when 

  1.  x=3

  2. x=−3

Exercise 2

x−4 when 

  1.  x=5
  2. x=−5

Exercise 3

5−y when 

  1. y=2

  2. y=−2

Exercise 4

8−y when 

  1.  y=3
  2. y=−3

Exercise 5

  1. 4x2−15x+1 when x=3

  2. 5x2−14x+7 when x=2
  3. −12−5x2 when x=6

  4. −19−4x2 when x=5

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

  1. The difference of 3 and −10

  2. Subtract −20 from 45

  3. The difference of 8 and −12
  4. Subtract −13 from 50
  5. The difference of −6 and 9

  6.  Subtract −12 from −16

  7. The difference of −8 and 9
  8. Subtract −15 from −19
  9. 8 less than −17

  10. −24 minus 37

  11. 5 less than −14
  12. −13 minus 42
  13. 21 less than 6

  14. 31 subtracted from −19

  15. 34 less than 7
  16. 29 subtracted from −50

Subtract Integers in Applications

In the following exercises, solve the following applications.

Temperature

One morning, the temperature in Urbana, Illinois, was 28∘ Fahrenheit. By evening, the temperature had dropped 38∘ Fahrenheit. What was the temperature that evening?

Temperature

On Thursday, the temperature in Spincich Lake, Michigan, was 22∘ Fahrenheit. By Friday, the temperature had dropped 35∘ Fahrenheit. What was the temperature on Friday?

Temperature

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

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?

Football

At the first down, the Warriors football team had the ball on their 30-yard line. On the next three downs, they gained 2 yards, lost 7 yards, and lost 4 yards. What was the yard line at the end of the third down?

Football

At the first down, the Barons football team had the ball on their 20-yard line. On the next three downs, they lost 8 yards, gained 5 yards, and lost 6 yards. What was the yard line at the end of the third down?

Checking Account

John has $148 in his checking account. He writes a check for $83. What is the new balance in his checking account?

Checking Account

Ellie has $426 in her checking account. She writes a check for $152. What is the new balance in her checking account?

Checking Account

Gina has $210 in her checking account. She writes a check for $250. What is the new balance in her checking account?

Checking Account

Frank has $94 in his checking account. He writes a check for $110. What is the new balance in his checking account?

Checking Account

Bill has a balance of −$14 in his checking account. He deposits $40 to the account. What is the new balance?

Checking Account

Patty has a balance of −$23 in her checking account. She deposits $80 to the account. What is the new balance?

Everyday Math

Camping

Rene is on an Alpine hike. The temperature is −7∘. Rene’s sleeping bag is rated “comfortable to −20∘.” How much can the temperature change before it is too cold for Rene’s sleeping bag?

Scuba Diving

Shelly’s scuba watch is guaranteed to be watertight to −100 feet. She is diving at −45 feet on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed?

Writing Exercises

Explain why the difference of 9 and −6 is 15.

Why is the result of subtracting 3−(−4) the same as the result of adding 3+4?

Multiplying and Dividing Integers

Multiply Integers

In the following exercises, multiply each pair of integers.

  1. −4⋅8

  2. −3⋅9
  3. −5(7)

  4. −8(6)
  5. −18(−2)

  6. −10(−6)
  7. 9(−7)

  8. 13(−5)
  9. −1⋅6

  10. −1⋅3
  11. −1(−14)

  12. −1(−19)

Divide Integers

In the following exercises, divide.

  1. −24÷6

  2. −28÷7
  3. 56÷(−7)

  4. 35÷(−7)
  5. −52÷(−4)

  6. −84÷(−6)
  7. −180÷15

  8. −192÷12
  9. 49÷(−1)

  10. 62÷(−1)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

  1. 5(−6)+7(−2)−3

  2. 8(−4)+5(−4)−6
  3. −8(−2)−3(−9)

  4. −7(−4)−5(−3)
  5. (−5)3

  6. (−4)3
  7. (−2)6

  8. (−3)5
  9. −42

  10. −62
  11. −3(−5)(6)

  12. −4(−6)(3)
  13. −4⋅2⋅11

  14. −5⋅3⋅10
  15. (8−11)(9−12)

  16. (6−11)(8−13)
  17. 26−3(2−7)

  18. 23−2(4−6)
  19. −10(−4)÷(−8)

  20. −8(−6)÷(−4)
  21. 65÷(−5)+(−28)÷(−7)

  22. 52÷(−4)+(−32)÷(−8)
  23. 9−2[3−8(−2)]

  24. 11−3[7−4(−2)]
  25. (−3)2−24÷(8−2)

  26. (−4)2−32÷(12−4)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Exercise 1

−2x+17 when 

  1.  x=8

  2. x=−8

Exercise 2

−5y+14 when 

  1. y=9
  2. y=−9

Exercise 3

10−3m when 

  1. m=5

  2. m=−5

Exercise 4

  1. 18−4n when 
  2. n=3

Exercise 5

n=−3

  1. p2−5p+5 when p=−1

  2. q2−2q+9 when q=−2
  3. 2w2−3w+7 when w=−2

  4. 3u2−4u+5 when u=−3
  5. 6x−5y+15 when x=3 and y=−1

  6. 3p−2q+9 when p=8 and q=−2
  7. 9a−2b−8 when a=−6 and b=−3

  8. 7m−4n−2 when m=−4 and n=−9

Translate Word Phrases to Algebraic Expressions

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

  1. The product of −3 and 15

  2. The product of −4 and 16
  3. The quotient of −60 and −20

  4. The quotient of −40 and −20
  5. The quotient of −6 and the sum of a and b

  6. The quotient of −7 and the sum of m and n
  7. The product of −10 and the difference of p and q

  8. The product of −13 and the difference of c and d

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?

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?

Writing Exercises

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

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

Why is −24≠(−2)4?

Why is −42≠(−4)2?

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