# 3.E: Integers (Exercises)

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### 3.1 - Introduction to Integers

#### Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the integer on the number line.

- 5
- −5
- −3
- 3
- −8
- −7

#### Order Positive and Negative Numbers

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

- 4__8
- −6__3
- −5__−10
- −9__−4
- 2__−7
- −3__1

#### Find Opposites

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

- 6
- −2
- −4
- 3

In the following exercises, simplify.

- (a) −(8) (b) −(−8)
- (a) −(9) (b) −(−9)

In the following exercises, evaluate.

- −x, when (a) x = 32 (b) x = −32
- −n, when (a) n = 20 (b) n = −20

#### Simplify Absolute Values

In the following exercises, simplify.

- |−21|
- |−42|
- |36|
- −|15|
- |0|
- −|−75|

In the following exercises, evaluate.

- |x| when x = −14
- −|r| when r = 27
- −|−y| when y = 33
- |−n| when n = −4

In the following exercises, fill in <, >, or = for each of the following pairs of numbers.

- −|−4|__4
- −2__|−2|
- −|−6|__−6
- −|−9|__|−9|

In the following exercises, simplify.

- −(−55) and − |−55|
- −(−48) and − |−48|
- |12 − 5|
- |9 + 7|
- 6|−9|
- |14−8| − |−2|
- |9 − 3| − |5 − 12|
- 5 + 4|15 − 3|

#### Translate Phrases to Expressions with Integers

In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

- the opposite of 16
- the opposite of −8
- negative 3
- 19 minus negative 12
- a temperature of 10 below zero
- an elevation of 85 feet below sea level

### 3.2 - Add Integers

#### Model Addition of Integers

In the following exercises, model the following to find the sum.

- 3 + 7
- −2 + 6
- 5 + (−4)
- −3 + (−6)

#### Simplify Expressions with Integers

In the following exercises, simplify each expression.

- 14 + 82
- −33 + (−67)
- −75 + 25
- 54 + (−28)
- 11 + (−15) + 3
- −19 + (−42) + 12
- −3 + 6(−1 + 5)
- 10 + 4(−3 + 7)

#### Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

- n + 4 when (a) n = −1 (b) n = −20
- x + (−9) when (a) x = 3 (b) x = −3
- (x + y)
^{3}when x = −4, y = 1 - (u + v)
^{2}when u = −4, v = 11

#### Translate Word Phrases to Algebraic Expressions

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

- the sum of −8 and 2
- 4 more than −12
- 10 more than the sum of −5 and −6
- the sum of 3 and −5, increased by 18

#### Add Integers in Applications

In the following exercises, solve.

**Temperature**On Monday, the high temperature in Denver was −4 degrees. Tuesday’s high temperature was 20 degrees more. What was the high temperature on Tuesday?**Credit**Frida owed $75 on her credit card. Then she charged $21 more. What was her new balance?

### 3.3 - Subtract Integers

#### Model Subtraction of Integers

In the following exercises, model the following.

- 6 − 1
- −4 − (−3)
- 2 − (−5)
- −1 − 4

#### Simplify Expressions with Integers

In the following exercises, simplify each expression.

- 24 − 16
- 19 − (−9)
- −31 − 7
- −40 − (−11)
- −52 − (−17) − 23
- 25 − (−3 − 9)
- (1 − 7) − (3 − 8)
- 3
^{2}− 7^{2}

#### Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

- x − 7 when (a) x = 5 (b) x = −4
- 10 − y when (a) y = 15 (b) y = −16
- 2n
^{2}− n + 5 when n = −4 - −15 − 3u
^{2 }when u = −5

#### Translate Phrases to Algebraic Expressions

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

- the difference of −12 and 5
- subtract 23 from −50

#### Subtract Integers in Applications

In the following exercises, solve the given applications.

**Temperature**One morning the temperature in Bangor, Maine was 18 degrees. By afternoon, it had dropped 20 degrees. What was the afternoon temperature?**Temperature**On January 4, the high temperature in Laredo, Texas was 78 degrees, and the high in Houlton, Maine was −28 degrees. What was the difference in temperature of Laredo and Houlton?

### 3.4 - Multiply and Divide Integers

#### Multiply Integers

In the following exercises, multiply.

- −9 • 4
- 5(−7)
- (−11)(−11)
- −1 • 6

#### Divide Integers

In the following exercises, divide.

- 56 ÷ (−8)
- −120 ÷ (−6)
- −96 ÷ 12
- 96 ÷ (−16)
- 45 ÷ (−1)
- −162 ÷ (−1)

#### Simplify Expressions with Integers

In the following exercises, simplify each expression.

- 5(−9) − 3(−12)
- (−2)
^{5} - −3
^{4} - (−3)(4)(−5)(−6)
- 42 − 4(6 − 9)
- (8 − 15)(9 − 3)
- −2(−18) ÷ 9
- 45 ÷ (−3) − 12

#### Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

- 7x − 3 when x = −9
- 16 − 2n when n = −8
- 5a + 8b when a = −2, b = −6
- x
^{2}+ 5x + 4 when x = −3

#### Translate Word Phrases to Algebraic Expressions

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

- the product of −12 and 6
- the quotient of 3 and the sum of −7 and s

### 3.5 - Solve Equations using Integers; The Division Property of Equality

#### Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

- 5x − 10 = −35
- x = −9
- x = −5
- x = 5

- 8u + 24 = −32
- u = −7
- u = −1
- u = 7

#### Using the Addition and Subtraction Properties of Equality

In the following exercises, solve.

- a + 14 = 2
- b − 9 = −15
- c + (−10) = −17
- d − (−6) = −26

#### Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.

#### Solve Equations Using the Division Property of Equality

In the following exercises, solve each equation using the division property of equality and check the solution.

- 8p = 72
- −12q = 48
- −16r = −64
- −5s = −100

#### Translate to an Equation and Solve.

In the following exercises, translate and solve.

- The product of −6 and y is −42
- The difference of z and −13 is −18.
- Four more than m is −48.
- The product of −21 and n is 63.

#### Everyday Math

- Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

### PRACTICE TEST

- Locate and label 0, 2, −4, and −1 on a number line.

In the following exercises, compare the numbers, using < or > or =.

- (a) −6__3 (b) −1__−4
- (a) −5__|−5| (b) −|−2|__−2

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

- (a) −7 (b) 8

In the following exercises, simplify.

- −(−22)
- |4 − 9|
- −8 + 6
- −15 + (−12)
- −7 − (−3)
- 10 − (5 − 6)
- −3 • 8
- −6(−9)
- 70 ÷ (−7)
- (−2)
^{3} - −4
^{2} - 16−3(5−7)
- |21 − 6| − |−8|

In the following exercises, evaluate.

- 35 − a when a = −4
- (−2r)
^{2}when r = 3 - 3m − 2n when m = 6, n = −8
- −|−y| when y = 17

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

- the difference of −7 and −4
- the quotient of 25 and the sum of m and n.

In the following exercises, solve.

- Early one morning, the temperature in Syracuse was −8°F. By noon, it had risen 12°. What was the temperature at noon?
- Collette owed $128 on her credit card. Then she charged $65. What was her new balance?

In the following exercises, solve.

- n + 6 = 5
- p − 11 = −4
- −9r = −54

In the following exercises, translate and solve.

- The product of 15 and x is 75.
- Eight less than y is −32.