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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. ⓐ = $2$
2. ⓑ = $-2$
3. ⓒ = $-5$

Exercise 2

1. ⓐ = $5$
2. ⓑ = $-5$
3. ⓒ = $-2$

Exercise 3

1. ⓐ = $-8$
2. ⓑ = $8$
3. ⓒ = $-6$

Exercise 4

1. ⓐ = $-7$
2. ⓑ = $7$
3. ⓒ = $-1$

Order Positive and Negative Numbers on the Number Line

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

2. $-3\text{__}6$

3. $-8\text{__}-2$

4. $1\text{__}-10$

5. $6\text{__}2$
6. $-7\text{__}4$
7. $-9\text{__}-1$
8. $9\text{__}-3$
9. $-5\text{__}1$

10. $-4\text{__}-9$

11. $6\text{__}10$

12. $3\text{__}-8$

13. $-7\text{__}3$
14. $-10\text{__}-5$
15. $2\text{__}-6$
16. $8\text{__}9$

Find Opposites

In the following exercises, find the opposite of each number.
1. $2$

2. $-6$

3. $9$
4. $-4$
5. $-8$

6. $1$

7. $-2$
8. $6$

Simplify Negatives

In the following exercises, simplify.
1. $-\left(-4\right)$

2. $-\left(-8\right)$
3. $-\left(-15\right)$

4. $-\left(-11\right)$

Simplify Negatives

In the following exercises, evaluate.

Exercise 1

$-m$ when
1.  $m=3$

2. $m=-3$

Exercise 2

$-p$ when
1.  $p=6$
2. $p=-6$

Exercise 3

$-c$ when
1.  $c=12$

2. $c=-12$

Exercise 3

$-d$ when
1. $d=21$
2. $d=-21$

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

Exercise 1

1. $|7|$

2. $|-25|$

3.  $|0|$

4.  $|5|$
5.  $|20|$
6.  $|-19|$
7.  $|-32|$

8. $|-18|$

9.  $|16|$

10. $|-41|$
11.  $|-40|$
12. $|22|$

Simplify Expressions with Absolute Value

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

2.  $|-u|\text{ when }u=-15$

3. $|y|\text{ when }y=-37$
4. $|-z|\text{ when }z=-24$
5. $-|p|\text{ when }p=19$

6. $-|q|\text{ when }q=-33$

7. $-|a|\text{ when }a=60$
8. $-|b|\text{ when }b=-12$

Simplify Expressions with Absolute Value

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

2. $-|-3|\text{__}-3$

3. $-8\text{__}|-8|$
4. $-|-2|\text{__}-2$
5. $|-3|\text{__}-|-3|$

6. $4\text{__}-|-4|$

7. $|-5|\text{__}-|-5|$
8. $9\text{__}-|-9|$

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. $18-|2\left(8 - 3\right)|$

8. $15-|3\left(8 - 5\right)|$
9. $8\left(14 - 2|-2|\right)$

10. $6\left(13 - 4|-2|\right)$

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 $6\text{degrees}$ below zero

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

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

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

2. deficit

Credit Cards

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

Weight Loss

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

Weight Loss

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

Football

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

Football

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

Calories

Ozzie rode his bike for $\text{30 minutes,}$ burning $\text{168 calories.}$ Then he had a $\text{140-calorie}$ 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: [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?

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 $-8$ and $\text{2}$ is negative, but the sum of $\text{8}$ and $-2$ 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. $8 - 2$

2. $9 - 3$
3. $-5-\left(-1\right)$

4. $-6-\left(-4\right)$
5. $-5 - 4$

6. $-7 - 2$
7. $8-\left(-4\right)$

8. $7-\left(-3\right)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.
1.  $15 - 6$

2. $15+\left(-6\right)$

3. $12 - 9$
4. $12+\left(-9\right)$
5. $44 - 28$

6. $44+\left(-28\right)$

7.  $35 - 16$
8. $35+\left(-16\right)$
9. $8-\left(-9\right)$

10. $8+9$

11. $4-\left(-4\right)$
12. $4+4$
13. $27-\left(-18\right)$

14. $27+18$

15. $46-\left(-37\right)$
16. $46+37$

Simplify Expressions with Integers

In the following exercises, simplify each expression.
1. $15-\left(-12\right)$

2. $14-\left(-11\right)$
3. $10-\left(-19\right)$

4. $11-\left(-18\right)$
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. $-103-\left(-52\right)$

14. $-105-\left(-68\right)$
15. $-45-\left(-54\right)$

16. $-58-\left(-67\right)$
17. $8 - 3 - 7$

18. $9 - 6 - 5$
19. $-5 - 4+7$

20. $-3 - 8+4$
21. $-14-\left(-27\right)+9$

22. $-15-\left(-28\right)+5$
23. $71+\left(-10\right)-8$

24. $64+\left(-17\right)-9$
25. $-16-\left(-4+1\right)-7$

26. $-15-\left(-6+4\right)-3$
27. $\left(2 - 7\right)-\left(3 - 8\right)$

28. $\left(1 - 8\right)-\left(2 - 9\right)$
29. $-\left(6 - 8\right)-\left(2 - 4\right)$

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

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

34. $5\cdot 7 - 8\cdot 2 - 4\cdot 9$
35. ${5}^{2}-{6}^{2}$

36. ${6}^{2}-{7}^{2}$

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.  $x=3$

2. $x=-3$

Exercise 2

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

Exercise 3

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

2. $y=-2$

Exercise 4

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

Exercise 5

1. $4{x}^{2}-15x+1\text{ when }x=3$

2. $5{x}^{2}-14x+7\text{ when }x=2$
3. $-12 - 5{x}^{2}\text{ when }x=6$

4. $-19 - 4{x}^{2}\text{ when }x=5$

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 $-10$

Answer: −3 − (−10) = 13

2. Subtract $-20$ from $45$

Answer: 45 − (−20) = 65

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

Answer: −6 − 9 = −15

6.  Subtract $-12$ from $-16$

Answer: −16 − (−12) = −4

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

Answer: −17 − 8 = −25

10. $-24$ minus $37$

Answer: −24 − 37 = −61

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

Answer: 6 − 21 = −15

14. $31$ subtracted from $-19$

Answer: −19 − 31 = −50

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 $28^{\circ}\text{ Fahrenheit}$. By evening, the temperature had dropped $38^{\circ}\text{ Fahrenheit}$. What was the temperature that evening?

Temperature

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

Temperature

On January 15, the high temperature in Anaheim, California, was $84^{\circ}\text{ Fahrenheit}$. That same day, the high temperature in Embarrass, Minnesota was $-12^{\circ}\text{ Fahrenheit}$. 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 $89^{\circ}$, and the high temperature in Whitefield, New Hampshire was $-31^{\circ}$. 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 $30\text{-yard line}$. On the next three downs, they gained $2\text{ yards}$, lost $7\text{ yards}$, and lost $4\text{ yards}$. 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 $20\text{-yard line}$. On the next three downs, they lost $\text{8 yards,}$ gained $5\text{ yards}$, and lost $6\text{ yards}$. 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

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?

Weight loss

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

Writing Exercises

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

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