Problem Set 3: Integers
Practice Makes Perfect
Locate Positive and Negative Numbers on the Number Line
In the following exercises, locate and label the given points on a number line.
- ⓐ [latex]5[/latex]
- ⓑ [latex]-5[/latex]
- ⓒ [latex]-2[/latex]
- ⓐ [latex]-8[/latex]
- ⓑ [latex]8[/latex]
- ⓒ [latex]-6[/latex]
- ⓐ [latex]-7[/latex]
- ⓑ [latex]7[/latex]
- ⓒ [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 [latex]<[/latex]; or [latex]\text{>.}[/latex]
- ⓐ >
- ⓑ <
- ⓒ <
- ⓓ >
- ⓐ [latex]6\text{__}2[/latex];
- ⓑ [latex]-7\text{__}4[/latex];
- ⓒ [latex]-9\text{__}-1[/latex];
- ⓓ [latex]9\text{__}-3[/latex]
- ⓐ [latex]-5\text{__}1[/latex];
- ⓑ [latex]-4\text{__}-9[/latex];
- ⓒ [latex]6\text{__}10[/latex];
- ⓓ [latex]3\text{__}-8[/latex]
- ⓐ <
- ⓑ >
- ⓒ <
- ⓓ >
- ⓐ [latex]-7\text{__}3[/latex];
- ⓑ [latex]-10\text{__}-5[/latex];
- ⓒ [latex]2\text{__}-6[/latex];
- ⓓ [latex]8\text{__}9[/latex]
Find Opposites
In the following exercises, find the opposite of each number.
- ⓐ [latex]2[/latex]
- ⓑ [latex]-6[/latex]
- ⓐ −2
- ⓑ 6
- ⓐ [latex]9[/latex]
- ⓑ [latex]-4[/latex]
- ⓐ [latex]-8[/latex]
- ⓑ [latex]1[/latex]
- ⓐ 8
- ⓑ −1
- ⓐ [latex]-2[/latex]
- ⓑ [latex]6[/latex]
In the following exercises, simplify.
[latex]-\left(-4\right)[/latex]
4
[latex]-\left(-8\right)[/latex]
[latex]-\left(-15\right)[/latex]
15
[latex]-\left(-11\right)[/latex]
In the following exercises, evaluate.
[latex]-m\text{when}[/latex]
- ⓐ −3
- ⓑ 3
[latex]-p\text{when}[/latex]
[latex]-c\text{when}[/latex]
- ⓐ −12;
- ⓑ 12
[latex]-d\text{when}[/latex]
Simplify Expressions with Absolute Value
In the following exercises, simplify each absolute value expression.
- ⓐ [latex]|7|[/latex]
- ⓑ [latex]|-25|[/latex]
- ⓒ [latex]|0|[/latex]
- ⓐ 7
- ⓑ 25
- ⓒ 0
- ⓐ [latex]|5|[/latex]
- ⓑ [latex]|20|[/latex]
- ⓒ [latex]|-19|[/latex]
- ⓐ [latex]|-32|[/latex]
- ⓑ [latex]|-18|[/latex]
- ⓒ [latex]|16|[/latex]
- ⓐ 32
- ⓑ 18
- ⓒ 16
- ⓐ [latex]|-41|[/latex]
- ⓑ [latex]|-40|[/latex]
- ⓒ [latex]|22|[/latex]
In the following exercises, evaluate each absolute value expression.
- ⓐ [latex]|x|\text{when}x=-28[/latex]
- ⓑ [latex]|-u|\text{when}u=-15[/latex]
- ⓐ 28
- ⓑ 15
- ⓐ [latex]|y|\text{when}y=-37[/latex]
- ⓑ [latex]|-z|\text{when}z=-24[/latex]
- ⓐ [latex]-|p|\text{when}p=19[/latex]
- ⓑ [latex]-|q|\text{when}q=-33[/latex]
- ⓐ −19
- ⓑ −33
- ⓐ [latex]-|a|\text{when}a=60[/latex]
- ⓑ [latex]-|b|\text{when}b=-12[/latex]
In the following exercises, fill in [latex]\text{<},\text{>},\text{or}=[/latex] to compare each expression.
- ⓐ [latex]-6\text{__}|-6|[/latex]
- ⓑ [latex]-|-3|\text{__}-3[/latex]
- ⓐ <
- ⓑ =
- ⓐ [latex]-8\text{__}|-8|[/latex]
- ⓑ [latex]-|-2|\text{__}-2[/latex]
- ⓐ [latex]|-3|\text{__}-|-3|[/latex]
- ⓑ [latex]4\text{__}-|-4|[/latex]
- ⓐ >
- ⓑ >
- ⓐ [latex]|-5|\text{__}-|-5|[/latex]
- ⓑ [latex]9\text{__}-|-9|[/latex]
In the following exercises, simplify each expression.
[latex]|8 - 4|[/latex]
4
[latex]|9 - 6|[/latex]
[latex]8|-7|[/latex]
56
[latex]5|-5|[/latex]
[latex]|15 - 7|-|14 - 6|[/latex]
0
[latex]|17 - 8|-|13 - 4|[/latex]
[latex]18-|2\left(8 - 3\right)|[/latex]
8
[latex]15-|3\left(8 - 5\right)|[/latex]
[latex]8\left(14 - 2|-2|\right)[/latex]
80
[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.
- ⓐ the opposite of [latex]8[/latex]
- ⓑ the opposite of [latex]-6[/latex]
- ⓒ negative three
- ⓓ [latex]4[/latex] minus negative [latex]3[/latex]
- ⓐ −8
- ⓑ −(−6), or 6
- ⓒ −3
- ⓓ 4−(−3)
- ⓐ the opposite of [latex]11[/latex]
- ⓑ the opposite of [latex]-4[/latex]
- ⓒ negative nine
- ⓓ [latex]8[/latex] minus negative [latex]2[/latex]
- ⓐ the opposite of [latex]20[/latex]
- ⓑ the opposite of [latex]-5[/latex]
- ⓒ negative twelve
- ⓓ [latex]18[/latex] minus negative [latex]7[/latex]
- ⓐ −20
- ⓑ −(−5), or 5
- ⓒ −12
- ⓓ 18−(−7)
- ⓐ the opposite of [latex]15[/latex]
- ⓑ the opposite of [latex]-9[/latex]
- ⓒ negative sixty
- ⓓ [latex]12[/latex] minus [latex]5[/latex]
a temperature of [latex]6\text{degrees}[/latex] below zero
−6 degrees
a temperature of [latex]14\text{degrees}[/latex] below zero
an elevation of [latex]40\text{feet}[/latex] below sea level
−40 feet
an elevation of [latex]65\text{feet}[/latex] below sea level
a football play loss of [latex]12\text{yards}[/latex]
−12 yards
a football play gain of [latex]4\text{yards}[/latex]
a stock gain of [latex]\text{$3}[/latex]
$3
a stock loss of [latex]\text{$5}[/latex]
a golf score one above par
+1
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:
- ⓐ 20,320 feet
- ⓑ −282 feet
Extreme temperatures The highest recorded temperature on Earth is [latex]\text{58^\circ Celsius,}[/latex] recorded in the Sahara Desert in 1922. The lowest recorded temperature is [latex]\text{90^\circ }[/latex] below [latex]\text{0^\circ Celsius,}[/latex] recorded in Antarctica in 1983. Use integers to write the:
State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of [latex]\text{$540 million.}[/latex] That same month, Texas estimated it would have a budget deficit of [latex]\text{$27 billion.}[/latex] Use integers to write the budget:
- ⓐ $540 million
- ⓑ −$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:
Writing Exercises
Give an example of a negative number from your life experience.
Sample answer: I have experienced negative temperatures.
What are the three uses of the "−" sign in algebra? Explain how they differ.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.
Practice Makes Perfect
Model Addition of Integers
In the following exercises, model the expression to simplify.
[latex]7+4[/latex]
11
[latex]8+5[/latex]
[latex]-6+\left(-3\right)[/latex]
−9
[latex]-5+\left(-5\right)[/latex]
[latex]-7+5[/latex]
−2
[latex]-9+6[/latex]
[latex]8+\left(-7\right)[/latex]
1
[latex]9+\left(-4\right)[/latex]
Simplify Expressions with Integers
In the following exercises, simplify each expression.
[latex]-21+\left(-59\right)[/latex]
−80
[latex]-35+\left(-47\right)[/latex]
[latex]48+\left(-16\right)[/latex]
32
[latex]34+\left(-19\right)[/latex]
[latex]-200+65[/latex]
−135
[latex]-150+45[/latex]
[latex]2+\left(-8\right)+6[/latex]
0
[latex]4+\left(-9\right)+7[/latex]
[latex]-14+\left(-12\right)+4[/latex]
−22
[latex]-17+\left(-18\right)+6[/latex]
[latex]135+\left(-110\right)+83[/latex]
108
[latex]140+\left(-75\right)+67[/latex]
[latex]-32+24+\left(-6\right)+10[/latex]
−4
[latex]-38+27+\left(-8\right)+12[/latex]
[latex]19+2\left(-3+8\right)[/latex]
29
[latex]24+3\left(-5+9\right)[/latex]
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
[latex]x+8[/latex] when
[latex]y+9[/latex] when
[latex]y+\left(-14\right)[/latex] when
[latex]x+\left(-21\right)[/latex] when
When [latex]a=-7[/latex], evaluate:
When [latex]b=-11[/latex], evaluate:
When [latex]c=-9[/latex], evaluate:
When [latex]d=-8[/latex], evaluate:
[latex]m+n[/latex] when, [latex]m=-15[/latex] , [latex]n=7[/latex]
−8
[latex]p+q[/latex] when, [latex]p=-9[/latex] , [latex]q=17[/latex]
[latex]r - 3s[/latex] when, [latex]r=16[/latex] , [latex]s=2[/latex]
10
[latex]2t+u[/latex] when, [latex]t=-6[/latex] , [latex]u=-5[/latex]
[latex]{\left(a+b\right)}^{2}[/latex] when, [latex]a=-7[/latex] , [latex]b=15[/latex]
64
[latex]{\left(c+d\right)}^{2}[/latex] when, [latex]c=-5[/latex] , [latex]d=14[/latex]
[latex]{\left(x+y\right)}^{2}[/latex] when, [latex]x=-3[/latex] , [latex]y=14[/latex]
121
[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.
The sum of [latex]-14[/latex] and [latex]5[/latex]
−14 + 5 = −9
The sum of [latex]-22[/latex] and [latex]9[/latex]
[latex]8[/latex] more than [latex]-2[/latex]
−2 + 8 = 6
[latex]5[/latex] more than [latex]-1[/latex]
[latex]-10[/latex] added to [latex]-15[/latex]
−15 + (−10) = −25
[latex]-6[/latex] added to [latex]-20[/latex]
[latex]6[/latex] more than the sum of [latex]-1[/latex] and [latex]-12[/latex]
[−1 + (−12)] + 6 = −7
[latex]3[/latex] more than the sum of [latex]-2[/latex] and [latex]-8[/latex]
the sum of [latex]10[/latex] and [latex]-19[/latex], increased by [latex]4[/latex]
[10 + (−19)] + 4 = −5
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\text{^\circ F}[/latex] at sunrise. By noon the temperature had risen [latex]\text{26^\circ F.}[/latex] What was the temperature at noon?
7°F
Temperature The temperature in Chicago was [latex]-15\text{^\circ F}[/latex] at 6 am. By afternoon the temperature had risen [latex]\text{28^\circ F.}[/latex] What was the afternoon temperature?
Credit Cards Lupe owes [latex]\text{$73}[/latex] on her credit card. Then she charges [latex]\text{$45}[/latex] more. What is the new balance?
−$118
Credit Cards Frank owes [latex]\text{$212}[/latex] on his credit card. Then he charges [latex]\text{$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?
−8 pounds
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?
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?
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]\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]
What was the overall change for the week?
−32
Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:
[latex]\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]
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.
Sample 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.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
Practice Makes Perfect
Model Subtraction of Integers
In the following exercises, model each expression and simplify.
[latex]8 - 2[/latex]
6
[latex]9 - 3[/latex]
[latex]-5-\left(-1\right)[/latex]
−4
[latex]-6-\left(-4\right)[/latex]
[latex]-5 - 4[/latex]
−9
[latex]-7 - 2[/latex]
[latex]8-\left(-4\right)[/latex]
12
[latex]7-\left(-3\right)[/latex]
Simplify Expressions with Integers
In the following exercises, simplify each expression.
- ⓐ [latex]4-\left(-4\right)[/latex]
- ⓑ [latex]4+4[/latex]
- ⓐ [latex]27-\left(-18\right)[/latex]
- ⓑ [latex]27+18[/latex]
- ⓐ 45
- ⓑ 45
- ⓐ [latex]46-\left(-37\right)[/latex]
- ⓑ [latex]46+37[/latex]
In the following exercises, simplify each expression.
[latex]15-\left(-12\right)[/latex]
27
[latex]14-\left(-11\right)[/latex]
[latex]10-\left(-19\right)[/latex]
29
[latex]11-\left(-18\right)[/latex]
[latex]48 - 87[/latex]
−39
[latex]45 - 69[/latex]
[latex]31 - 79[/latex]
−48
[latex]39 - 81[/latex]
[latex]-31 - 11[/latex]
−42
[latex]-32 - 18[/latex]
[latex]-17 - 42[/latex]
−59
[latex]-19 - 46[/latex]
[latex]-103-\left(-52\right)[/latex]
−51
[latex]-105-\left(-68\right)[/latex]
[latex]-45-\left(-54\right)[/latex]
9
[latex]-58-\left(-67\right)[/latex]
[latex]8 - 3 - 7[/latex]
−2
[latex]9 - 6 - 5[/latex]
[latex]-5 - 4+7[/latex]
−2
[latex]-3 - 8+4[/latex]
[latex]-14-\left(-27\right)+9[/latex]
22
[latex]-15-\left(-28\right)+5[/latex]
[latex]71+\left(-10\right)-8[/latex]
53
[latex]64+\left(-17\right)-9[/latex]
[latex]-16-\left(-4+1\right)-7[/latex]
−20
[latex]-15-\left(-6+4\right)-3[/latex]
[latex]\left(2 - 7\right)-\left(3 - 8\right)[/latex]
0
[latex]\left(1 - 8\right)-\left(2 - 9\right)[/latex]
[latex]-\left(6 - 8\right)-\left(2 - 4\right)[/latex]
4
[latex]-\left(4 - 5\right)-\left(7 - 8\right)[/latex]
[latex]25-\left[10-\left(3 - 12\right)\right][/latex]
6
[latex]32-\left[5-\left(15 - 20\right)\right][/latex]
[latex]6\cdot 3 - 4\cdot 3 - 7\cdot 2[/latex]
−8
[latex]5\cdot 7 - 8\cdot 2 - 4\cdot 9[/latex]
[latex]{5}^{2}-{6}^{2}[/latex]
−11
[latex]{6}^{2}-{7}^{2}[/latex]
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression for the given values.
[latex]x - 6\text{when}[/latex]
- ⓐ −3
- ⓑ −9
[latex]x - 4\text{when}[/latex]
[latex]5-y\text{when}[/latex]
- ⓐ 3
- ⓑ 7
[latex]8-y\text{when}[/latex]
[latex]4{x}^{2}-15x+1\text{when}x=3[/latex]
−8
[latex]5{x}^{2}-14x+7\text{when}x=2[/latex]
[latex]-12 - 5{x}^{2}\text{when}x=6[/latex]
−192
[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.
- ⓐ The difference of [latex]3[/latex] and [latex]-10[/latex]
- ⓑ Subtract [latex]-20[/latex] from [latex]45[/latex]
- ⓐ −3 − (−10) = 13
- ⓑ 45 − (−20) = 65
- ⓐ The difference of [latex]8[/latex] and [latex]-12[/latex]
- ⓑ Subtract [latex]-13[/latex] from [latex]50[/latex]
- ⓐ The difference of [latex]-6[/latex] and [latex]9[/latex]
- ⓑ Subtract [latex]-12[/latex] from [latex]-16[/latex]
- ⓐ −6 − 9 = −15
- ⓑ −16 − (−12) = −4
- ⓐ The difference of [latex]-8[/latex] and [latex]9[/latex]
- ⓑ Subtract [latex]-15[/latex] from [latex]-19[/latex]
- ⓐ [latex]8[/latex] less than [latex]-17[/latex]
- ⓑ [latex]-24[/latex] minus [latex]37[/latex]
- ⓐ −17 − 8 = −25
- ⓑ −24 − 37 = −61
- ⓐ [latex]5[/latex] less than [latex]-14[/latex]
- ⓑ [latex]-13[/latex] minus [latex]42[/latex]
- ⓐ [latex]21[/latex] less than [latex]6[/latex]
- ⓑ [latex]31[/latex] subtracted from [latex]-19[/latex]
- ⓐ 6 − 21 = −15
- ⓑ −19 − 31 = −50
- ⓐ [latex]34[/latex] less than [latex]7[/latex]
- ⓑ [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]\text{28^\circ Fahrenheit.}[/latex] By evening, the temperature had dropped [latex]\text{38^\circ Fahrenheit.}[/latex] What was the temperature that evening?
−10°
Temperature On Thursday, the temperature in Spincich Lake, Michigan, was [latex]\text{22^\circ Fahrenheit.}[/latex] By Friday, the temperature had dropped [latex]\text{35^\circ Fahrenheit.}[/latex] What was the temperature on Friday?
Temperature On January 15, the high temperature in Anaheim, California, was [latex]\text{84^\circ Fahrenheit.}[/latex] That same day, the high temperature in Embarrass, Minnesota was [latex]\text{-12^\circ Fahrenheit.}[/latex] What was the difference between the temperature in Anaheim and the temperature in Embarrass?
96°
Temperature On January 21, the high temperature in Palm Springs, California, was [latex]\text{89^\circ ,}[/latex] and the high temperature in Whitefield, New Hampshire was [latex]\text{-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]\text{30-yard line.}[/latex] On the next three downs, they gained [latex]\text{2 yards,}[/latex] lost [latex]\text{7 yards,}[/latex] and lost [latex]\text{4 yards.}[/latex] What was the yard line at the end of the third down?
21-yard line
Football At the first down, the Barons football team had the ball on their [latex]\text{20-yard line.}[/latex] On the next three downs, they lost [latex]\text{8 yards,}[/latex] gained [latex]\text{5 yards,}[/latex] and lost [latex]\text{6 yards.}[/latex] What was the yard line at the end of the third down?
Checking Account John has [latex]\text{$148}[/latex] in his checking account. He writes a check for [latex]\text{$83.}[/latex] What is the new balance in his checking account?
$65
Checking Account Ellie has [latex]\text{$426}[/latex] in her checking account. She writes a check for [latex]\text{$152.}[/latex] What is the new balance in her checking account?
Checking Account Gina has [latex]\text{$210}[/latex] in her checking account. She writes a check for [latex]\text{$250.}[/latex] What is the new balance in her checking account?
−$40
Checking Account Frank has [latex]\text{$94}[/latex] in his checking account. He writes a check for [latex]\text{$110.}[/latex] What is the new balance in his checking account?
Checking Account Bill has a balance of [latex]\text{-$14}[/latex] in his checking account. He deposits [latex]\text{$40}[/latex] to the account. What is the new balance?
$26
Checking Account Patty has a balance of [latex]\text{-$23}[/latex] in her checking account. She deposits [latex]\text{$80}[/latex] to the account. What is the new balance?
Everyday Math
Camping Rene is on an Alpine hike. The temperature is [latex]-\mathbf{\text{7}}\mathbf{\text{^\circ }}[/latex]. Rene’s sleeping bag is rated "comfortable to [latex]-\mathbf{\text{20}}\text{^\circ ".}[/latex] How much can the temperature change before it is too cold for Rene’s sleeping bag?
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].
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]
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
Practice Makes Perfect
Multiply Integers
In the following exercises, multiply each pair of integers.
[latex]-4\cdot 8[/latex]
−32
[latex]-3\cdot 9[/latex]
[latex]-5\left(7\right)[/latex]
−35
[latex]-8\left(6\right)[/latex]
[latex]-18\left(-2\right)[/latex]
36
[latex]-10\left(-6\right)[/latex]
[latex]9\left(-7\right)[/latex]
−63
[latex]13\left(-5\right)[/latex]
[latex]-1\cdot 6[/latex]
−6
[latex]-1\cdot 3[/latex]
[latex]-1\left(-14\right)[/latex]
14
[latex]-1\left(-19\right)[/latex]
Divide Integers
In the following exercises, divide.
[latex]-24\div 6[/latex]
−4
[latex]-28\div 7[/latex]
[latex]56\div \left(-7\right)[/latex]
−8
[latex]35\div \left(-7\right)[/latex]
[latex]-52\div \left(-4\right)[/latex]
13
[latex]-84\div \left(-6\right)[/latex]
[latex]-180\div 15[/latex]
−12
[latex]-192\div 12[/latex]
[latex]49\div \left(-1\right)[/latex]
−49
[latex]62\div \left(-1\right)[/latex]
Simplify Expressions with Integers
In the following exercises, simplify each expression.
[latex]5\left(-6\right)+7\left(-2\right)-3[/latex]
−47
[latex]8\left(-4\right)+5\left(-4\right)-6[/latex]
[latex]-8\left(-2\right)-3\left(-9\right)[/latex]
43
[latex]-7\left(-4\right)-5\left(-3\right)[/latex]
[latex]{\left(-5\right)}^{3}[/latex]
−125
[latex]{\left(-4\right)}^{3}[/latex]
[latex]{\left(-2\right)}^{6}[/latex]
64
[latex]{\left(-3\right)}^{5}[/latex]
[latex]-{4}^{2}[/latex]
−16
[latex]-{6}^{2}[/latex]
[latex]-3\left(-5\right)\left(6\right)[/latex]
90
[latex]-4\left(-6\right)\left(3\right)[/latex]
[latex]-4\cdot 2\cdot 11[/latex]
−88
[latex]-5\cdot 3\cdot 10[/latex]
[latex]\left(8 - 11\right)\left(9 - 12\right)[/latex]
9
[latex]\left(6 - 11\right)\left(8 - 13\right)[/latex]
[latex]26 - 3\left(2 - 7\right)[/latex]
41
[latex]23 - 2\left(4 - 6\right)[/latex]
[latex]-10\left(-4\right)\div \left(-8\right)[/latex]
−5
[latex]-8\left(-6\right)\div \left(-4\right)[/latex]
[latex]65\div \left(-5\right)+\left(-28\right)\div \left(-7\right)[/latex]
−9
[latex]52\div \left(-4\right)+\left(-32\right)\div \left(-8\right)[/latex]
[latex]9 - 2\left[3 - 8\left(-2\right)\right][/latex]
−29
[latex]11 - 3\left[7 - 4\left(-2\right)\right][/latex]
[latex]{\left(-3\right)}^{2}-24\div \left(8 - 2\right)[/latex]
5
[latex]{\left(-4\right)}^{2}-32\div \left(12 - 4\right)[/latex]
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
[latex]-2x+17\text{when}[/latex]
[latex]-5y+14\text{when}[/latex]
[latex]10 - 3m\text{when}[/latex]
[latex]18 - 4n\text{when}[/latex]
[latex]{p}^{2}-5p+5\text{when}p=-1[/latex]
8
[latex]{q}^{2}-2q+9\text{when}q=-2[/latex]
[latex]2{w}^{2}-3w+7\text{when}w=-2[/latex]
21
[latex]3{u}^{2}-4u+5\text{when}u=-3[/latex]
[latex]6x - 5y+15\text{when}x=3\text{and}y=-1[/latex]
38
[latex]3p - 2q+9\text{when}p=8\text{and}q=-2[/latex]
[latex]9a - 2b - 8\text{when}a=-6\text{and}b=-3[/latex]
−56
[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.
The product of [latex]-3[/latex] and 15
−3·15 = −45
The product of [latex]-4[/latex] and [latex]16[/latex]
The quotient of [latex]-60[/latex] and [latex]-20[/latex]
−60 ÷ (−20) = 3
The quotient of [latex]-40[/latex] and [latex]-20[/latex]
The quotient of [latex]-6[/latex] and the sum of [latex]a[/latex] and [latex]b[/latex]
[latex]\frac{-6}{a+b}[/latex]
The quotient of [latex]-7[/latex] and the sum of [latex]m[/latex] and [latex]n[/latex]
The product of [latex]-10[/latex] and the difference of [latex]p\text{and}q[/latex]
−10 (p − q)
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]\text{$12}[/latex] per share. What was the total effect on Javier’s portfolio?
−$3,600
Weight loss In the first week of a diet program, eight women lost an average of [latex]\text{3 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.
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.
Why is [latex]{-2}^{4}\ne {\left(-2\right)}^{4}?[/latex]
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]
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ 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?
Chapter Review Exercises
Use the Language of Algebra
Use Variables and Algebraic Symbols
In the following exercises, translate from algebra to English.
[latex]3\cdot 8[/latex]
the product of 3 and 8
[latex]12-x[/latex]
[latex]24\div 6[/latex]
the quotient of 24 and 6
[latex]9+2a[/latex]
[latex]50\ge 47[/latex]
50 is greater than or equal to 47
[latex]3y<15[/latex]
[latex]n+4=13[/latex]
The sum of n and 4 is equal to 13
[latex]32-k=7[/latex]
Identify Expressions and Equations
In the following exercises, determine if each is an expression or equation.
[latex]5+u=84[/latex]
equation
[latex]36 - 6s[/latex]
[latex]4y - 11[/latex]
expression
[latex]10x=120[/latex]
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
[latex]2\cdot 2\cdot 2[/latex]
23
[latex]a\cdot a\cdot a\cdot a\cdot a[/latex]
[latex]x\cdot x\cdot x\cdot x\cdot x\cdot x[/latex]
x6
[latex]10\cdot 10\cdot 10[/latex]
In the following exercises, write in expanded form.
[latex]{8}^{4}[/latex]
8 ⋅ 8 ⋅ 8 ⋅ 8
[latex]{3}^{6}[/latex]
[latex]{y}^{5}[/latex]
y ⋅ y ⋅ y ⋅ y ⋅ y
[latex]{n}^{4}[/latex]
In the following exercises, simplify each expression.
[latex]{3}^{4}[/latex]
81
[latex]{10}^{6}[/latex]
[latex]{2}^{7}[/latex]
128
[latex]{4}^{3}[/latex]
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.
[latex]10+2\cdot 5[/latex]
20
[latex]\left(10+2\right)\cdot 5[/latex]
[latex]\left(30+6\right)\div 2[/latex]
18
[latex]30+6\div 2[/latex]
[latex]{7}^{2}+{5}^{2}[/latex]
74
[latex]{\left(7+5\right)}^{2}[/latex]
[latex]4+3\left(10 - 1\right)[/latex]
31
[latex]\left(4+3\right)\left(10 - 1\right)[/latex]
Evaluate, Simplify, and Translate Expressions
Evaluate an Expression
In the following exercises, evaluate the following expressions.
[latex]9x - 5\text{when}x=7[/latex]
58
[latex]{y}^{3}\text{when}y=5[/latex]
[latex]3a - 4b[/latex] when [latex]a=10,b=1[/latex]
26
[latex]bh\text{when}b=7,h=8[/latex]
Identify Terms, Coefficients and Like Terms
In the following exercises, identify the terms in each expression.
[latex]12{n}^{2}+3n+1[/latex]
12n2,3n, 1
[latex]4{x}^{3}+11x+3[/latex]
In the following exercises, identify the coefficient of each term.
[latex]6y[/latex]
6
[latex]13{x}^{2}[/latex]
In the following exercises, identify the like terms.
[latex]5{x}^{2},3,5{y}^{2},3x,x,4[/latex]
3, 4, and 3x, x
[latex]8,8{r}^{2},\text{8}r,3r,{r}^{2},3s[/latex]
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
[latex]15a+9a[/latex]
24a
[latex]12y+3y+y[/latex]
[latex]4x+7x+3x[/latex]
14x
[latex]6+5c+3[/latex]
[latex]8n+2+4n+9[/latex]
12n + 11
[latex]19p+5+4p - 1+3p[/latex]
[latex]7{y}^{2}+2y+11+3{y}^{2}-8[/latex]
10y2 + 2y + 3
[latex]13{x}^{2}-x+6+5{x}^{2}+9x[/latex]
Translate English Phrases to Algebraic Expressions
In the following exercises, translate the following phrases into algebraic expressions.
the difference of [latex]x[/latex] and [latex]6[/latex]
x − 6
the sum of [latex]10[/latex] and twice [latex]a[/latex]
the product of [latex]3n[/latex] and [latex]9[/latex]
3n ⋅ 9
the quotient of [latex]s[/latex] and [latex]4[/latex]
[latex]5[/latex] times the sum of [latex]y[/latex] and [latex]1[/latex]
5(y + 1)
[latex]10[/latex] less than the product of [latex]5[/latex] and [latex]z[/latex]
Jack bought a sandwich and a coffee. The cost of the sandwich was [latex]\text{$3}[/latex] more than the cost of the coffee. Call the cost of the coffee [latex]c[/latex]. Write an expression for the cost of the sandwich.
c + 3
The number of poetry books on Brianna’s bookshelf is [latex]5[/latex] less than twice the number of novels. Call the number of novels [latex]n[/latex]. Write an expression for the number of poetry books.
Solve Equations Using the Subtraction and Addition Properties of Equality
Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each number is a solution to the equation.
[latex]y+16=40[/latex]
[latex]d - 6=21[/latex]
[latex]4n+12=36[/latex]
[latex]20q - 10=70[/latex]
[latex]15x - 5=10x+45[/latex]
[latex]22p - 6=18p+86[/latex]
Model the Subtraction Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.
x + 3 = 5; x = 2
Solve Equations using the Subtraction Property of Equality
In the following exercises, solve each equation using the subtraction property of equality.
[latex]c+8=14[/latex]
6
[latex]v+8=150[/latex]
[latex]23=x+12[/latex]
11
[latex]376=n+265[/latex]
Solve Equations using the Addition Property of Equality
In the following exercises, solve each equation using the addition property of equality.
[latex]y - 7=16[/latex]
23
[latex]k - 42=113[/latex]
[latex]19=p - 15[/latex]
34
[latex]501=u - 399[/latex]
Translate English Sentences to Algebraic Equations
In the following exercises, translate each English sentence into an algebraic equation.
The sum of [latex]7[/latex] and [latex]33[/latex] is equal to [latex]40[/latex].
7 + 33 = 44
The difference of [latex]15[/latex] and [latex]3[/latex] is equal to [latex]12[/latex].
The product of [latex]4[/latex] and [latex]8[/latex] is equal to [latex]32[/latex].
4 ⋅ 8 = 32
The quotient of [latex]63[/latex] and [latex]9[/latex] is equal to [latex]7[/latex].
Twice the difference of [latex]n[/latex] and [latex]3[/latex] gives [latex]76[/latex].
2(n − 3) = 76
The sum of five times [latex]y[/latex] and [latex]4[/latex] is [latex]89[/latex].
Translate to an Equation and Solve
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Eight more than [latex]x[/latex] is equal to [latex]35[/latex].
x + 8 = 35; x = 27
[latex]21[/latex] less than [latex]a[/latex] is [latex]11[/latex].
The difference of [latex]q[/latex] and [latex]18[/latex] is [latex]57[/latex].
q − 18 = 57; q = 75
The sum of [latex]m[/latex] and [latex]125[/latex] is [latex]240[/latex].
Mixed Practice
In the following exercises, solve each equation.
[latex]h - 15=27[/latex]
h = 42
[latex]k - 11=34[/latex]
[latex]z+52=85[/latex]
z = 33
[latex]x+93=114[/latex]
[latex]27=q+19[/latex]
q = 8
[latex]38=p+19[/latex]
[latex]31=v - 25[/latex]
v = 56
[latex]38=u - 16[/latex]
Find Multiples and Factors
Identify Multiples of Numbers
In the following exercises, list all the multiples less than [latex]50[/latex] for each of the following.
[latex]3[/latex]
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48
[latex]2[/latex]
[latex]8[/latex]
8, 16, 24, 32, 40, 48
[latex]10[/latex]
Use Common Divisibility Tests
In the following exercises, using the divisibility tests, determine whether each number is divisible by [latex]2,\text{by}3,\text{by}5,\text{by}6,\text{and by}10[/latex].
[latex]96[/latex]
2, 3, 6
[latex]250[/latex]
[latex]420[/latex]
2, 3, 5, 6, 10
[latex]625[/latex]
Find All the Factors of a Number
In the following exercises, find all the factors of each number.
[latex]30[/latex]
1, 2, 3, 5, 6, 10, 15, 30
[latex]70[/latex]
[latex]180[/latex]
1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
[latex]378[/latex]
Identify Prime and Composite Numbers
In the following exercises, identify each number as prime or composite.
[latex]19[/latex]
prime
[latex]51[/latex]
[latex]121[/latex]
composite
[latex]219[/latex]
Prime Factorization and the Least Common Multiple
Find the Prime Factorization of a Composite Number
In the following exercises, find the prime factorization of each number.
[latex]84[/latex]
2 ⋅ 2 ⋅ 3 ⋅ 7
[latex]165[/latex]
[latex]350[/latex]
2 ⋅ 5 ⋅ 5 ⋅ 7
[latex]572[/latex]
Find the Least Common Multiple of Two Numbers
In the following exercises, find the least common multiple of each pair of numbers.
[latex]9,15[/latex]
45
[latex]12,20[/latex]
[latex]25,35[/latex]
350
[latex]18,40[/latex]
Everyday Math
Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month.
Answers will vary
Chapter Practice Test
In the following exercises, translate from an algebraic equation to English phrases.
[latex]6\cdot 4[/latex]
[latex]15-x[/latex]
fifteen minus x
In the following exercises, identify each as an expression or equation.
[latex]5\cdot 8+10[/latex]
[latex]x+6=9[/latex]
equation
[latex]3\cdot 11=33[/latex]
In the following exercises, simplify, using the order of operations.
[latex]4+3\cdot 5[/latex]
[latex]\left(8+1\right)\cdot 4[/latex]
36
[latex]1+6\left(3 - 1\right)[/latex]
[latex]\left(8+4\right)\div 3+1[/latex]
5
[latex]{\left(1+4\right)}^{2}[/latex]
[latex]5\left[2+7\left(9 - 8\right)\right][/latex]
45
In the following exercises, evaluate each expression.
[latex]8x - 3\text{when}x=4[/latex]
[latex]{y}^{3}\text{when}y=5[/latex]
125
[latex]6a - 2b\text{when}a=5,b=7[/latex]
[latex]hw\text{when}h=12,w=3[/latex]
36
Simplify by combining like terms.
In the following exercises, translate each phrase into an algebraic expression.
[latex]5[/latex] more than [latex]x[/latex]
x + 5
the quotient of [latex]12[/latex] and [latex]y[/latex]
three times the difference of [latex]a\text{and}b[/latex]
3(a − b)
Caroline has [latex]3[/latex] fewer earrings on her left ear than on her right ear. Call the number of earrings on her right ear, [latex]r[/latex]. Write an expression for the number of earrings on her left ear.
In the following exercises, solve each equation.
[latex]n - 6=25[/latex]
n = 31
[latex]x+58=71[/latex]
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
[latex]15[/latex] less than [latex]y[/latex] is [latex]32[/latex].
y − 15 = 32; y = 47
the sum of [latex]a[/latex] and [latex]129[/latex] is [latex]164[/latex].
List all the multiples of [latex]4[/latex], that are less than [latex]50[/latex].
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48
Find all the factors of [latex]90[/latex].
Find the prime factorization of [latex]1080[/latex].
23 ⋅ 33 ⋅ 5
Find the LCM (Least Common Multiple) of [latex]24[/latex] and [latex]40[/latex].
Glossary
- least common multiple
- The smallest number that is a multiple of two numbers is called the least common multiple (LCM).)
- prime factorization
- The prime factorization of a number is the product of prime numbers that equals the number.
Licenses & Attributions
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
