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Study Guides > ALGEBRA / TRIG I

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. ⓐ = [latex]2[/latex]
  2. ⓑ = [latex]-2[/latex]
  3. ⓒ = [latex]-5[/latex]

Answer: This figure is a number line. Negative 5 is labeled with c, two units to the left of 0 is labeled b, and two units to the right of 0 is labeled a.

Exercise 2

  1. ⓐ = [latex]5[/latex]
  2. ⓑ = [latex]-5[/latex]
  3. ⓒ = [latex]-2[/latex]

Exercise 3

  1. ⓐ = [latex]-8[/latex]
  2. ⓑ = [latex]8[/latex]
  3. ⓒ = [latex]-6[/latex]

Answer: This figure is a number line. Negative 8 is labeled a, negative 6 is labeled c, and 5 is labeled b.

Exercise 4

  1. ⓐ = [latex]-7[/latex]
  2. ⓑ = [latex]7[/latex]
  3. ⓒ = [latex]-1[/latex]

Order Positive and Negative Numbers on the Number Line

In the following exercises, order each of the following pairs of numbers, using < or >.
  1. [latex]9\text{__}4[/latex]

    Answer: >

  2. [latex]-3\text{__}6[/latex]

    Answer: <

  3. [latex]-8\text{__}-2[/latex]

    Answer: <

  4. [latex]1\text{__}-10[/latex]

    Answer: >

  5. [latex]6\text{__}2[/latex]
  6. [latex]-7\text{__}4[/latex]
  7. [latex]-9\text{__}-1[/latex]
  8. [latex]9\text{__}-3[/latex]
  9. [latex]-5\text{__}1[/latex]

    Answer: <

  10. [latex]-4\text{__}-9[/latex]

    Answer: >

  11. [latex]6\text{__}10[/latex]

    Answer: <

  12. [latex]3\text{__}-8[/latex]

    Answer: >

  13. [latex]-7\text{__}3[/latex]
  14. [latex]-10\text{__}-5[/latex]
  15. [latex]2\text{__}-6[/latex]
  16. [latex]8\text{__}9[/latex]

Find Opposites

In the following exercises, find the opposite of each number.
  1. [latex]2[/latex]

    Answer: −2

  2. [latex]-6[/latex]

    Answer: 6

  3. [latex]9[/latex]
  4. [latex]-4[/latex]
  5. [latex]-8[/latex]

    Answer: 8

  6. [latex]1[/latex]

    Answer: -1

  7. [latex]-2[/latex]
  8. [latex]6[/latex]

Simplify Negatives

In the following exercises, simplify.
  1. [latex]-\left(-4\right)[/latex]

    Answer: 4

  2. [latex]-\left(-8\right)[/latex]
  3. [latex]-\left(-15\right)[/latex]

    Answer: 15

  4. [latex]-\left(-11\right)[/latex]

Simplify Negatives

In the following exercises, evaluate.

Exercise 1

[latex]-m[/latex] when
  1.  [latex]m=3[/latex]

    Answer: −3

  2. [latex]m=-3[/latex]

    Answer: 3

Exercise 2

[latex]-p[/latex] when
  1.  [latex]p=6[/latex]
  2. [latex]p=-6[/latex]

Exercise 3

[latex]-c[/latex] when
  1.  [latex]c=12[/latex]

    Answer: −12

  2. [latex]c=-12[/latex]

    Answer: 12

Exercise 3

[latex]-d[/latex] when
  1. [latex]d=21[/latex]
  2. [latex]d=-21[/latex]

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

Exercise 1

  1. [latex]|7|[/latex]

    Answer: 7

  2. [latex]|-25|[/latex]

    Answer: 25

  3.  [latex]|0|[/latex]

    Answer: 0

  4.  [latex]|5|[/latex]
  5.  [latex]|20|[/latex]
  6.  [latex]|-19|[/latex]
  7.  [latex]|-32|[/latex]

    Answer: 32

  8. [latex]|-18|[/latex]

    Answer: 18

  9.  [latex]|16|[/latex]

    Answer: 16

  10. [latex]|-41|[/latex]
  11.  [latex]|-40|[/latex]
  12. [latex]|22|[/latex]

Simplify Expressions with Absolute Value

In the following exercises, evaluate each absolute value expression.
  1.  [latex]|x|\text{ when }x=-28[/latex]

    Answer: 28

  2.  [latex]|-u|\text{ when }u=-15[/latex]

    Answer: 15

  3. [latex]|y|\text{ when }y=-37[/latex]
  4. [latex]|-z|\text{ when }z=-24[/latex]
  5. [latex]-|p|\text{ when }p=19[/latex]

    Answer: −19

  6. [latex]-|q|\text{ when }q=-33[/latex]

    Answer: −33

  7. [latex]-|a|\text{ when }a=60[/latex]
  8. [latex]-|b|\text{ when }b=-12[/latex]

Simplify Expressions with Absolute Value

In the following exercises, fill in [latex]\text{<},\text{>},\text{or}=[/latex] to compare each expression.
  1. [latex]-6\text{__}|-6|[/latex]

    Answer: <

  2. [latex]-|-3|\text{__}-3[/latex]

    Answer: =

  3. [latex]-8\text{__}|-8|[/latex]
  4. [latex]-|-2|\text{__}-2[/latex]
  5. [latex]|-3|\text{__}-|-3|[/latex]

    Answer: >

  6. [latex]4\text{__}-|-4|[/latex]

    Answer: >

  7. [latex]|-5|\text{__}-|-5|[/latex]
  8. [latex]9\text{__}-|-9|[/latex]

Simplify Expressions with Absolute Value

In the following exercises, simplify each expression.
  1. [latex]|8 - 4|[/latex]

    Answer: 4

  2. [latex]|9 - 6|[/latex]
  3. [latex]8|-7|[/latex]

    Answer: 56

  4. [latex]5|-5|[/latex]
  5. [latex]|15 - 7|-|14 - 6|[/latex]

    Answer: 0

  6. [latex]|17 - 8|-|13 - 4|[/latex]
  7. [latex]18-|2\left(8 - 3\right)|[/latex]

    Answer: 8

  8. [latex]15-|3\left(8 - 5\right)|[/latex]
  9. [latex]8\left(14 - 2|-2|\right)[/latex]

    Answer: 80

  10. [latex]6\left(13 - 4|-2|\right)[/latex]

Translate Word Phrases into Expressions with Integers

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

Exercise 1

  1. the opposite of [latex]8[/latex]

    Answer: −8

  2. the opposite of [latex]-6[/latex]

    Answer: −(−6), or 6

  3. negative three

    Answer: −3

  4. [latex]4[/latex] minus negative [latex]3[/latex]

    Answer: 4−(−3)

  5. the opposite of [latex]11[/latex]
  6. the opposite of [latex]-4[/latex]
  7. negative nine
  8. [latex]8[/latex] minus negative [latex]2[/latex]
  9. the opposite of [latex]20[/latex]

    Answer: −20

  10. the opposite of [latex]-5[/latex]

    Answer: −(−5), or 5

  11. the opposite of [latex]12[/latex]

    Answer: −12

  12. [latex]18[/latex] minus negative [latex]7[/latex]

    Answer: 18−(−7)

  13. the opposite of [latex]15[/latex]
  14. the opposite of [latex]-9[/latex]
  15. negative sixty
  16. [latex]12[/latex] minus [latex]5[/latex]
  17. a temperature of [latex]6\text{degrees}[/latex] below zero

    Answer: −6 degrees

  18. a temperature of [latex]14\text{degrees}[/latex] below zero
  19. an elevation of [latex]40\text{ feet }[/latex] below sea level

    Answer: −40 feet

  20. an elevation of [latex]65\text{ feet }[/latex] below sea level
  21. a football play loss of [latex]12\text{ yards }[/latex]

    Answer: −12 yards

  22. a football play gain of [latex]4\text{ yards }[/latex]
  23. a stock gain of [latex]\$3[/latex]

    Answer: $3

  24. a stock loss of [latex]\$5[/latex]
  25. a golf score one above par

    Answer: +1

  26. a golf score of [latex]3[/latex] below par

Everyday Math

Elevation

The highest elevation in the United States is Mount McKinley, Alaska, at [latex]20,320\text{ feet}[/latex] above sea level. The lowest elevation is Death Valley, California, at [latex]282\text{ feet}[/latex] below sea level. Use integers to write the elevation of:
  1. Mount McKinley

    Answer: 20,320 feet

  2. Death Valley

    Answer: −282 feet

Extreme temperatures

The highest recorded temperature on Earth is [latex]58^{\circ}\text{ Celsius}[/latex], recorded in the Sahara Desert in 1922. The lowest recorded temperature is [latex]90^{\circ}[/latex] below [latex]0^{\circ}\text{ Celsius}[/latex], 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 [latex]\$540\text{ million}[/latex]. That same month, Texas estimated it would have a budget deficit of [latex]\$27\text{ billion}[/latex]. Use integers to write the budget:
  1. surplus

    Answer: $540 million

  2. deficit

    Answer: −$27 billion

College enrollments

Across the United States, community college enrollment grew by [latex]1,400,000[/latex] students from [latex]2007[/latex] to [latex]2010[/latex]. In California, community college enrollment declined by [latex]110,171[/latex] students from [latex]2009[/latex] to [latex]2010[/latex]. 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. [practice-area rows="4"][/practice-area]

Answer: Sample answer: I have experienced negative temperatures.

What are the three uses of the "−" sign in algebra? Explain how they differ. [practice-area rows="4"][/practice-area]

Adding Integers

Model Addition of Integers

In the following exercises, model the expression to simplify.
  1. [latex]7+4[/latex]

    Answer: This figure shows a row of 11 light pink circles, representing positive counters. They are separated into a group of seven and a group of four. 11

  2. [latex]8+5[/latex]
  3. [latex]-6+\left(-3\right)[/latex]

    Answer: This figure shows a row of 9 dark pink circles, representing negative counters. They are separated into a group of six and a group of three. −9 

  4. [latex]-5+\left(-5\right)[/latex]
  5. [latex]-7+5[/latex]

    Answer: This figure shows two rows of circles. The top row shows 7 dark pink circles, representing negative counters. The bottom row shows 5 light pink circles, representing positive counters. −2

  6. [latex]-9+6[/latex]
  7. [latex]8+\left(-7\right)[/latex]

    Answer: This figure shows two rows of circles. The top row shows 8 light pink circles, representing positive counters. The bottom row shows 7 light pink circles, representing negative counters. 1

  8. [latex]9+\left(-4\right)[/latex]

Simplify Expressions with Integers

In the following exercises, simplify each expression.
  1. [latex]-21+\left(-59\right)[/latex]

    Answer: −80

  2. [latex]-35+\left(-47\right)[/latex]
  3. [latex]48+\left(-16\right)[/latex]

    Answer: 32

  4. [latex]34+\left(-19\right)[/latex]
  5. [latex]-200+65[/latex]

    Answer: −135

  6. [latex]-150+45[/latex]
  7. [latex]2+\left(-8\right)+6[/latex]

    Answer: 0

  8. [latex]4+\left(-9\right)+7[/latex]
  9. [latex]-14+\left(-12\right)+4[/latex]

    Answer: −22

  10. [latex]-17+\left(-18\right)+6[/latex]
  11. [latex]135+\left(-110\right)+83[/latex]

    Answer: 108

  12. [latex]140+\left(-75\right)+67[/latex]
  13. [latex]-32+24+\left(-6\right)+10[/latex]

    Answer: −4

  14. [latex]-38+27+\left(-8\right)+12[/latex]
  15. [latex]19+2\left(-3+8\right)[/latex]

    Answer: 29

  16. [latex]24+3\left(-5+9\right)[/latex]

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Exercise 1

[latex]x+8[/latex] when
  1.  [latex]x=-26[/latex]

    Answer: −18

  2. [latex]x=-95[/latex]

    Answer: −87

Exercise 2

[latex]y+9[/latex] when
  1.  [latex]y=-29[/latex]
  2. [latex]y=-84[/latex]

Exercise 3

[latex]y+\left(-14\right)[/latex] when
  1. [latex]y=-33[/latex]

    Answer: −47

  2. [latex]y=30[/latex]

    Answer: 16

Exercise 4

[latex]x+\left(-21\right)[/latex] when
  1.  [latex]x=-27[/latex]
  2. [latex]x=44[/latex]

Exercise 5

When [latex]a=-7[/latex], evaluate:
  1.  [latex]a+3[/latex]

    Answer: −4

  2. [latex]-a+3[/latex]

    Answer: 10

Exercise 6

When [latex]b=-11[/latex], evaluate:
  1. [latex]b+6[/latex]
  2. [latex]-b+6[/latex]

Exercise 7

When [latex]c=-9[/latex], evaluate:
  1. [latex]c+\left(-4\right)[/latex]

    Answer: −13

  2. [latex]-c+\left(-4\right)[/latex]

    Answer: 5

Exercise 8

When [latex]d=-8[/latex], evaluate:
  1. [latex]d+\left(-9\right)[/latex]
  2. [latex]-d+\left(-9\right)[/latex]

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.
  1. [latex]m+n[/latex] when, [latex]m=-15[/latex] , [latex]n=7[/latex]

    Answer: −8

  2. [latex]p+q[/latex] when, [latex]p=-9[/latex] , [latex]q=17[/latex]
  3. [latex]r - 3s[/latex] when, [latex]r=16[/latex] , [latex]s=2[/latex]

    Answer: 10

  4. [latex]2t+u[/latex] when, [latex]t=-6[/latex] , [latex]u=-5[/latex]
  5. [latex]{\left(a+b\right)}^{2}[/latex] when, [latex]a=-7[/latex] , [latex]b=15[/latex]

    Answer: 64

  6. [latex]{\left(c+d\right)}^{2}[/latex] when, [latex]c=-5[/latex] , [latex]d=14[/latex]
  7. [latex]{\left(x+y\right)}^{2}[/latex] when, [latex]x=-3[/latex] , [latex]y=14[/latex]

    Answer: 121

  8. [latex]{\left(y+z\right)}^{2}[/latex] when, [latex]y=-3[/latex] , [latex]z=15[/latex]

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.
  1. The sum of [latex]-14[/latex] and [latex]5[/latex]

    Answer: −14 + 5 = −9

  2. The sum of [latex]-22[/latex] and [latex]9[/latex]
  3. [latex]8[/latex] more than [latex]-2[/latex]

    Answer: −2 + 8 = 6

  4. [latex]5[/latex] more than [latex]-1[/latex]
  5. [latex]-10[/latex] added to [latex]-15[/latex]

    Answer: −15 + (−10) = −25

  6. [latex]-6[/latex] added to [latex]-20[/latex]
  7. [latex]6[/latex] more than the sum of [latex]-1[/latex] and [latex]-12[/latex]

    Answer: [−1 + (−12)] + 6 = −7

  8. [latex]3[/latex] more than the sum of [latex]-2[/latex] and [latex]-8[/latex]
  9. the sum of [latex]10[/latex] and [latex]-19[/latex], increased by [latex]4[/latex]

    Answer: [10 + (−19)] + 4 = −5

  10. the sum of [latex]12[/latex] and [latex]-15[/latex], increased by [latex]1[/latex]

Add Integers in Applications

In the following exercises, solve.

Temperature

The temperature in St. Paul, Minnesota was [latex]-19^{\circ}\text{ F}[/latex] at sunrise. By noon the temperature had risen [latex]26^{\circ}\text{ F.}[/latex] What was the temperature at noon?

Answer: 7°F

Temperature

The temperature in Chicago was [latex]-15^{\circ}\text{ F}[/latex] at 6 am. By afternoon the temperature had risen [latex]28^{\circ}\text{ F}[/latex]. What was the afternoon temperature?

Credit Cards

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

Answer: −$118

Credit Cards

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

Weight Loss

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

Weight Loss

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

Football

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

Answer: 25-yard line

Football

The Cowboys began with the ball on their own [latex]\text{20-yard line.}[/latex] They gained [latex]\text{15 yards,}[/latex] lost [latex]\text{3 yards}[/latex] and then gained [latex]\text{6 yards}[/latex] 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 [latex]\text{20 minutes,}[/latex] she burned [latex]\text{90 calories.}[/latex] The frozen yogurt she ate was [latex]\text{110 calories.}[/latex] What was her total calorie gain or loss?

Answer: 20 calories

Calories

Ozzie rode his bike for [latex]\text{30 minutes,}[/latex] burning [latex]\text{168 calories.}[/latex] Then he had a [latex]\text{140-calorie}[/latex] 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: [latex-display]\begin{array}{cccccc}\text{Monday}\hfill & -504\hfill & \text{Tuesday}\hfill & +142\hfill & \text{Wednesday}\hfill & -449\hfill \\ \text{Thursday}\hfill & +410\hfill & \text{Friday}\hfill & +369\hfill & \end{array}[/latex-display] What was the overall change for the week?

Answer: −32

Stock Market

During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was: [latex-display]\begin{array}{cccccc}\text{Monday}\hfill & -201\hfill & \text{Tuesday}\hfill & -16\hfill & \text{Wednesday}\hfill & -23\hfill \\ \text{Thursday}\hfill & +172\hfill & \text{Friday}\hfill & -34\hfill & \end{array}[/latex-display] What was the overall change for the week?

Writing Exercises

Explain why the sum of [latex]-8[/latex] and [latex]\text{2}[/latex] is negative, but the sum of [latex]\text{8}[/latex] and [latex]-2[/latex] and is positive. [practice-area rows="4"][/practice-area]

Answer: In the first case, there are more negatives so the sum is negative. In the second case, there are more positives so the sum is positive.

Give an example from your life experience of adding two negative numbers. [practice-area rows="4"][/practice-area]

Subtracting Integers

Model Subtraction of Integers

In the following exercises, model each expression and simplify.
  1. [latex]8 - 2[/latex]

    Answer: This figure shows a row of 8 light pink circles, representing positive counters. The first 2 are circles and are separated from the last 6. 6

  2. [latex]9 - 3[/latex]
  3. [latex]-5-\left(-1\right)[/latex]

    Answer: This figure ishows a row of 5 dark pink circles. The first one is circled. –4

  4. [latex]-6-\left(-4\right)[/latex]
  5. [latex]-5 - 4[/latex]

    Answer: This figure has a row of 9 dark pink circles representing negative counters. The first 5 are separated from the last 4. Below the last 4 is a row of 4 light pink circles, representing positive counters. These four positive counters are circled. –9

  6. [latex]-7 - 2[/latex]
  7. [latex]8-\left(-4\right)[/latex]

    Answer: This figure has a row of 12 light pink circles, representing positive counters. The first 8 are separated from the last 4. Below the last 4 is a row of 4 dark pink circles, representing negative counters. These four negative counters are circled. 12 

  8. [latex]7-\left(-3\right)[/latex]

Simplify Expressions with Integers

In the following exercises, simplify each expression.
  1.  [latex]15 - 6[/latex]

    Answer: 9

  2. [latex]15+\left(-6\right)[/latex]

    Answer: 9

  3. [latex]12 - 9[/latex]
  4. [latex]12+\left(-9\right)[/latex]
  5. [latex]44 - 28[/latex]

    Answer: 16

  6. [latex]44+\left(-28\right)[/latex]

    Answer: 16

  7.  [latex]35 - 16[/latex]
  8. [latex]35+\left(-16\right)[/latex]
  9. [latex]8-\left(-9\right)[/latex]

    Answer: 17

  10. [latex]8+9[/latex]

    Answer: 17

  11. [latex]4-\left(-4\right)[/latex]
  12. [latex]4+4[/latex]
  13. [latex]27-\left(-18\right)[/latex]

    Answer: 45

  14. [latex]27+18[/latex]

    Answer: 45

  15. [latex]46-\left(-37\right)[/latex]
  16. [latex]46+37[/latex]

Simplify Expressions with Integers

In the following exercises, simplify each expression.
  1. [latex]15-\left(-12\right)[/latex]

    Answer: 27

  2. [latex]14-\left(-11\right)[/latex]
  3. [latex]10-\left(-19\right)[/latex]

    Answer: 29

  4. [latex]11-\left(-18\right)[/latex]
  5. [latex]48 - 87[/latex]

    Answer: −39

  6. [latex]45 - 69[/latex]
  7. [latex]31 - 79[/latex]

    Answer: −48

  8. [latex]39 - 81[/latex]
  9. [latex]-31 - 11[/latex]

    Answer: −42

  10. [latex]-32 - 18[/latex]
  11. [latex]-17 - 42[/latex]

    Answer: −59

  12. [latex]-19 - 46[/latex]
  13. [latex]-103-\left(-52\right)[/latex]

    Answer: −51

  14. [latex]-105-\left(-68\right)[/latex]
  15. [latex]-45-\left(-54\right)[/latex]

    Answer: 9

  16. [latex]-58-\left(-67\right)[/latex]
  17. [latex]8 - 3 - 7[/latex]

    Answer: −2

  18. [latex]9 - 6 - 5[/latex]
  19. [latex]-5 - 4+7[/latex]

    Answer: −2

  20. [latex]-3 - 8+4[/latex]
  21. [latex]-14-\left(-27\right)+9[/latex]

    Answer: 22

  22. [latex]-15-\left(-28\right)+5[/latex]
  23. [latex]71+\left(-10\right)-8[/latex]

    Answer: 53

  24. [latex]64+\left(-17\right)-9[/latex]
  25. [latex]-16-\left(-4+1\right)-7[/latex]

    Answer: −20

  26. [latex]-15-\left(-6+4\right)-3[/latex]
  27. [latex]\left(2 - 7\right)-\left(3 - 8\right)[/latex]

    Answer: 0

  28. [latex]\left(1 - 8\right)-\left(2 - 9\right)[/latex]
  29. [latex]-\left(6 - 8\right)-\left(2 - 4\right)[/latex]

    Answer: 4

  30. [latex]-\left(4 - 5\right)-\left(7 - 8\right)[/latex]
  31. [latex]25-\left[10-\left(3 - 12\right)\right][/latex]

    Answer: 6

  32. [latex]32-\left[5-\left(15 - 20\right)\right][/latex]
  33. [latex]6\cdot 3 - 4\cdot 3 - 7\cdot 2[/latex]

    Answer: −8

  34. [latex]5\cdot 7 - 8\cdot 2 - 4\cdot 9[/latex]
  35. [latex]{5}^{2}-{6}^{2}[/latex]

    Answer: −11

  36. [latex]{6}^{2}-{7}^{2}[/latex]

Evaluate Variable Expressions with Integers

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

Exercise 1

[latex-display]x - 6\text{ when }[/latex-display]
  1.  [latex]x=3[/latex]

    Answer: -3

  2. [latex]x=-3[/latex]

    Answer: −9

Exercise 2

[latex-display]x - 4\text{ when }[/latex-display]
  1.  [latex]x=5[/latex]
  2. [latex]x=-5[/latex]

Exercise 3

[latex-display]5-y\text{ when }[/latex-display]
  1. [latex]y=2[/latex]

    Answer: 3

  2. [latex]y=-2[/latex]

    Answer: 7

Exercise 4

[latex-display]8-y\text{ when }[/latex-display]
  1.  [latex]y=3[/latex]
  2. [latex]y=-3[/latex]

Exercise 5

  1. [latex]4{x}^{2}-15x+1\text{ when }x=3[/latex]

    Answer: −8

  2. [latex]5{x}^{2}-14x+7\text{ when }x=2[/latex]
  3. [latex]-12 - 5{x}^{2}\text{ when }x=6[/latex]

    Answer: −192

  4. [latex]-19 - 4{x}^{2}\text{ when }x=5[/latex]

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.
  1. The difference of [latex]3[/latex] and [latex]-10[/latex]

    Answer: −3 − (−10) = 13

  2. Subtract [latex]-20[/latex] from [latex]45[/latex]

    Answer: 45 − (−20) = 65

  3. The difference of [latex]8[/latex] and [latex]-12[/latex]
  4. Subtract [latex]-13[/latex] from [latex]50[/latex]
  5. The difference of [latex]-6[/latex] and [latex]9[/latex]

    Answer: −6 − 9 = −15

  6.  Subtract [latex]-12[/latex] from [latex]-16[/latex]

    Answer: −16 − (−12) = −4

  7. The difference of [latex]-8[/latex] and [latex]9[/latex]
  8. Subtract [latex]-15[/latex] from [latex]-19[/latex]
  9. [latex]8[/latex] less than [latex]-17[/latex]

    Answer: −17 − 8 = −25

  10. [latex]-24[/latex] minus [latex]37[/latex]

    Answer: −24 − 37 = −61

  11. [latex]5[/latex] less than [latex]-14[/latex]
  12. [latex]-13[/latex] minus [latex]42[/latex]
  13. [latex]21[/latex] less than [latex]6[/latex]

    Answer: 6 − 21 = −15

  14. [latex]31[/latex] subtracted from [latex]-19[/latex]

    Answer: −19 − 31 = −50

  15. [latex]34[/latex] less than [latex]7[/latex]
  16. [latex]29[/latex] subtracted from [latex]-50[/latex]

Subtract Integers in Applications

In the following exercises, solve the following applications.

Temperature

One morning, the temperature in Urbana, Illinois, was [latex]28^{\circ}\text{ Fahrenheit}[/latex]. By evening, the temperature had dropped [latex]38^{\circ}\text{ Fahrenheit}[/latex]. What was the temperature that evening?

Answer: −10°

Temperature

On Thursday, the temperature in Spincich Lake, Michigan, was [latex]22^{\circ}\text{ Fahrenheit}[/latex]. By Friday, the temperature had dropped [latex]35^{\circ}\text{ Fahrenheit}[/latex]. What was the temperature on Friday?

Temperature

On January 15, the high temperature in Anaheim, California, was [latex]84^{\circ}\text{ Fahrenheit}[/latex]. That same day, the high temperature in Embarrass, Minnesota was [latex]-12^{\circ}\text{ Fahrenheit}[/latex]. What was the difference between the temperature in Anaheim and the temperature in Embarrass?

Answer: 96°

Temperature

On January 21, the high temperature in Palm Springs, California, was [latex]89^{\circ}[/latex], and the high temperature in Whitefield, New Hampshire was [latex]-31^{\circ}[/latex]. 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 [latex]30\text{-yard line}[/latex]. On the next three downs, they gained [latex]2\text{ yards}[/latex], lost [latex]7\text{ yards}[/latex], and lost [latex]4\text{ yards}[/latex]. What was the yard line at the end of the third down?

Answer: 21-yard line

Football

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

Checking Account

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

Answer: $65

Checking Account

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

Checking Account

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

Answer: −$40

Checking Account

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

Checking Account

Bill has a balance of [latex]-\$14[/latex] in his checking account. He deposits [latex]\$40[/latex] to the account. What is the new balance?

Answer: $26

Checking Account

Patty has a balance of [latex]-\$23[/latex] in her checking account. She deposits [latex]\$80[/latex] to the account. What is the new balance?

Everyday Math

Camping

Rene is on an Alpine hike. The temperature is [latex]-7^{\circ}[/latex]. Rene’s sleeping bag is rated "comfortable to [latex]-20^{\circ}[/latex]." How much can the temperature change before it is too cold for Rene’s sleeping bag?

Answer: 13°

Scuba Diving

Shelly’s scuba watch is guaranteed to be watertight to [latex]-100\text{ feet}[/latex]. She is diving at [latex]-45\text{ feet}[/latex] 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 [latex]9[/latex] and [latex]-6[/latex] is [latex]15[/latex]. [practice-area rows="4"][/practice-area]

Answer: Sample answer: On a number line, 9 is 15 units away from −6.

Why is the result of subtracting [latex]3-\left(-4\right)[/latex] the same as the result of adding [latex]3+4?[/latex] [practice-area rows="4"][/practice-area]

Multiplying and Dividing Integers

Multiply Integers

In the following exercises, multiply each pair of integers.
  1. [latex]-4\cdot 8[/latex]

    Answer: −32

  2. [latex]-3\cdot 9[/latex]
  3. [latex]-5\left(7\right)[/latex]

    Answer: −35

  4. [latex]-8\left(6\right)[/latex]
  5. [latex]-18\left(-2\right)[/latex]

    Answer: 36

  6. [latex]-10\left(-6\right)[/latex]
  7. [latex]9\left(-7\right)[/latex]

    Answer: −63

  8. [latex]13\left(-5\right)[/latex]
  9. [latex]-1\cdot 6[/latex]

    Answer: −6

  10. [latex]-1\cdot 3[/latex]
  11. [latex]-1\left(-14\right)[/latex]

    Answer: 14

  12. [latex]-1\left(-19\right)[/latex]

Divide Integers

In the following exercises, divide.
  1. [latex]-24\div 6[/latex]

    Answer: −4

  2. [latex]-28\div 7[/latex]
  3. [latex]56\div \left(-7\right)[/latex]

    Answer: −8

  4. [latex]35\div \left(-7\right)[/latex]
  5. [latex]-52\div \left(-4\right)[/latex]

    Answer: 13

  6. [latex]-84\div \left(-6\right)[/latex]
  7. [latex]-180\div 15[/latex]

    Answer: −12

  8. [latex]-192\div 12[/latex]
  9. [latex]49\div \left(-1\right)[/latex]

    Answer: −49

  10. [latex]62\div \left(-1\right)[/latex]

Simplify Expressions with Integers

In the following exercises, simplify each expression.
  1. [latex]5\left(-6\right)+7\left(-2\right)-3[/latex]

    Answer: −47

  2. [latex]8\left(-4\right)+5\left(-4\right)-6[/latex]
  3. [latex]-8\left(-2\right)-3\left(-9\right)[/latex]

    Answer: 43

  4. [latex]-7\left(-4\right)-5\left(-3\right)[/latex]
  5. [latex]{\left(-5\right)}^{3}[/latex]

    Answer: −125

  6. [latex]{\left(-4\right)}^{3}[/latex]
  7. [latex]{\left(-2\right)}^{6}[/latex]

    Answer: 64

  8. [latex]{\left(-3\right)}^{5}[/latex]
  9. [latex]-{4}^{2}[/latex]

    Answer: −16

  10. [latex]-{6}^{2}[/latex]
  11. [latex]-3\left(-5\right)\left(6\right)[/latex]

    Answer: 90

  12. [latex]-4\left(-6\right)\left(3\right)[/latex]
  13. [latex]-4\cdot 2\cdot 11[/latex]

    Answer: −88

  14. [latex]-5\cdot 3\cdot 10[/latex]
  15. [latex]\left(8 - 11\right)\left(9 - 12\right)[/latex]

    Answer: 9

  16. [latex]\left(6 - 11\right)\left(8 - 13\right)[/latex]
  17. [latex]26 - 3\left(2 - 7\right)[/latex]

    Answer: 41

  18. [latex]23 - 2\left(4 - 6\right)[/latex]
  19. [latex]-10\left(-4\right)\div \left(-8\right)[/latex]

    Answer: −5

  20. [latex]-8\left(-6\right)\div \left(-4\right)[/latex]
  21. [latex]65\div \left(-5\right)+\left(-28\right)\div \left(-7\right)[/latex]

    Answer: −9

  22. [latex]52\div \left(-4\right)+\left(-32\right)\div \left(-8\right)[/latex]
  23. [latex]9 - 2\left[3 - 8\left(-2\right)\right][/latex]

    Answer: −29

  24. [latex]11 - 3\left[7 - 4\left(-2\right)\right][/latex]
  25. [latex]{\left(-3\right)}^{2}-24\div \left(8 - 2\right)[/latex]

    Answer: 5

  26. [latex]{\left(-4\right)}^{2}-32\div \left(12 - 4\right)[/latex]

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Exercise 1

[latex-display]-2x+17\text{ when }[/latex-display]
  1.  [latex]x=8[/latex]

    Answer: 1

  2. [latex]x=-8[/latex]

    Answer: 33

Exercise 2

[latex-display]-5y+14\text{ when }[/latex-display]
  1. [latex]y=9[/latex]
  2. [latex]y=-9[/latex]

Exercise 3

[latex-display]10 - 3m\text{ when }[/latex-display]
  1. [latex]m=5[/latex]

    Answer: −5

  2. [latex]m=-5[/latex]

    Answer: 25

Exercise 4

  1. [latex]18 - 4n\text{ when }[/latex]
  2. [latex]n=3[/latex]

Exercise 5

[latex-display]n=-3[/latex-display]
  1. [latex]{p}^{2}-5p+5\text{ when }p=-1[/latex]

    Answer: 8

  2. [latex]{q}^{2}-2q+9\text{ when }q=-2[/latex]
  3. [latex]2{w}^{2}-3w+7\text{ when }w=-2[/latex]

    Answer: 21

  4. [latex]3{u}^{2}-4u+5\text{ when }u=-3[/latex]
  5. [latex]6x - 5y+15\text{ when }x=3\text{ and }y=-1[/latex]

    Answer: 38

  6. [latex]3p - 2q+9\text{ when }p=8\text{ and }q=-2[/latex]
  7. [latex]9a - 2b - 8\text{ when }a=-6\text{ and }b=-3[/latex]

    Answer: −56

  8. [latex]7m - 4n - 2\text{ when }m=-4\text{ and }n=-9[/latex]

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.
  1. The product of [latex]-3[/latex] and 15

    Answer: −3·15 = −45

  2. The product of [latex]-4[/latex] and [latex]16[/latex]
  3. The quotient of [latex]-60[/latex] and [latex]-20[/latex]

    Answer: −60 ÷ (−20) = 3

  4. The quotient of [latex]-40[/latex] and [latex]-20[/latex]
  5. The quotient of [latex]-6[/latex] and the sum of [latex]a[/latex] and [latex]b[/latex]

    Answer: [latex]\Large\frac{-6}{a+b}[/latex]

  6. The quotient of [latex]-7[/latex] and the sum of [latex]m[/latex] and [latex]n[/latex]
  7. The product of [latex]-10[/latex] and the difference of [latex]p\text{ and }q[/latex]

    Answer: −10 (p − q)

  8. The product of [latex]-13[/latex] and the difference of [latex]c\text{ and }d[/latex]

Everyday Math

Stock market

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

Answer: –$3,600

Weight loss

In the first week of a diet program, eight women lost an average of [latex]3\text{ pounds}[/latex] each. What was the total weight change for the eight women?

Writing Exercises

In your own words, state the rules for multiplying two integers. [practice-area rows="4"][/practice-area]

Answer: Sample answer: Multiplying two integers with the same sign results in a positive product. Multiplying two integers with different signs results in a negative product.

In your own words, state the rules for dividing two integers. [practice-area rows="4"][/practice-area] Why is [latex]{-2}^{4}\ne {\left(-2\right)}^{4}[/latex]? [practice-area rows="4"][/practice-area]

Answer: Sample answer: In one expression the base is positive and then we take the opposite, but in the other the base is negative.

Why is [latex]{-4}^{2}\ne {\left(-4\right)}^{2}[/latex]? [practice-area rows="4"][/practice-area]

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