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  • Solving Equations

Prealgebra

Charles P. McKeague

Chapter 4

Solving Equations - all with Video Answers

Educators

Section 1

The Distributive Property and Algebraic Expressions

Problem 1

Use the distributive property to combine each of the following pairs of similar terms.
$$2 x+8 x$$

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Problem 2

Use the distributive property to combine each of the following pairs of similar terms.
$$3 x+7 x$$

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Problem 3

Use the distributive property to combine each of the following pairs of similar terms.
$$-4 y+5 y$$

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Problem 4

Use the distributive property to combine each of the following pairs of similar terms.
$$-3 y+10 y$$

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Problem 5

Use the distributive property to combine each of the following pairs of similar terms.
$$4 a-a$$

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Problem 6

Use the distributive property to combine each of the following pairs of similar terms.
$$9 a-a$$

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Problem 7

Use the distributive property to combine each of the following pairs of similar terms.
$$8(x+2)$$

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Problem 8

Use the distributive property to combine each of the following pairs of similar terms.
$$8(x-2)$$

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Problem 9

Use the distributive property to combine each of the following pairs of similar terms.
$$2(3 a+7)$$

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Problem 10

Use the distributive property to combine each of the following pairs of similar terms.
$$5(3 a+2)$$

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Problem 11

Use the distributive property to combine each of the following pairs of similar terms.
$$\frac{1}{3}(3 x+6)$$

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Problem 12

Use the distributive property to combine each of the following pairs of similar terms.
$$\frac{1}{2}(2 x+4)$$

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Problem 13

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$4 x+2 x+3+8$$

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Problem 14

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$7 x+5 x+2+9$$

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Problem 15

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$7 x-5 x+6-4$$

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Problem 16

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$10 x-7 x+9-6$$

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Problem 17

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$-2 a+a+7+5$$

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Problem 18

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$-8 a+3 a+12+1$$

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Problem 19

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$6 y-2 y-5+1$$

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Problem 20

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$4 y-3 y-7+2$$

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Problem 21

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$4 x+2 x-8 x+4$$

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Problem 22

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$6 x+5 x-12 x+6$$

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Problem 23

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$9 x-x-5-1$$

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Problem 24

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$2 x-x-3-8$$

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Problem 25

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$2 a+4+3 a+5$$

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Problem 26

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$9 a+1+2 a+6$$

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Problem 27

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$3 x+2-4 x+1$$

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Problem 28

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$7 x+5-2 x+6$$

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Problem 29

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$12 y+3+5 y$$

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Problem 30

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$8 y+1+6 y$$

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Problem 31

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$4 a-3-5 a+2 a$$

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Problem 32

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property.
$$6 a-4-2 a+6 a$$

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Problem 33

Apply the distributive property to each expression and then simplify.
$$2(3 x+4)+8$$

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Problem 34

Apply the distributive property to each expression and then simplify.
$$2(5 x+1)+10$$

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Problem 35

Apply the distributive property to each expression and then simplify.
$$5(2 x-3)+4$$

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Problem 36

Apply the distributive property to each expression and then simplify.
$$6(4 x-2)+7$$

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Problem 37

Apply the distributive property to each expression and then simplify.
$$8(2 y+4)+3 y$$

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Problem 38

Apply the distributive property to each expression and then simplify.
$$2(5 y+1)+2 y$$

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Problem 39

Apply the distributive property to each expression and then simplify.
$$6(4 y-3)+6 y$$

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Problem 40

Apply the distributive property to each expression and then simplify.
$$5(2 y-6)+4 y$$

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Problem 41

Apply the distributive property to each expression and then simplify.
$$2(x+3)+4(x+2)$$

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Problem 42

Apply the distributive property to each expression and then simplify.
$$3(x+1)+2(x+5)$$

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Problem 43

Apply the distributive property to each expression and then simplify.
$$3(2 a+4)+7(3 a-1)$$

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Problem 44

Apply the distributive property to each expression and then simplify.
$$7(2 a+2)+4(5 a-1)$$

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Problem 45

Find the value of each of the following expressions when $x = 5$.
$$2 x+4$$

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Problem 46

Find the value of each of the following expressions when $x = 5$.
$$3 x+2$$

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Problem 47

Find the value of each of the following expressions when $x = 5$.
$$7 x-8$$

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Problem 48

Find the value of each of the following expressions when $x = 5$.
$$8 x-9$$

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Problem 49

Find the value of each of the following expressions when $x = 5$.
$$-4 x+1$$

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Problem 50

Find the value of each of the following expressions when $x = 5$.
$$-3 x+7$$

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Problem 51

Find the value of each of the following expressions when $x = 5$.
$$-8+3 x$$

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Problem 52

Find the value of each of the following expressions when $x = 5$.
$$-7+2 x$$

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Problem 53

Find the value of each of the following expressions when $a = -2$.
$$2 a+5$$

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Problem 54

Find the value of each of the following expressions when $a = -2$.
$$3 a+4$$

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Problem 55

Find the value of each of the following expressions when $a = -2$.
$$-7 a+4$$

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Problem 56

Find the value of each of the following expressions when $a = -2$.
$$-9 a+3$$

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Problem 57

Find the value of each of the following expressions when $a = -2$.
$$-a+10$$

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Problem 58

Find the value of each of the following expressions when $a = -2$.
$$-a+8$$

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Problem 59

Find the value of each of the following expressions when $a = -2$.
$$-4+3 a$$

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Problem 60

Find the value of each of the following expressions when $a = -2$.
$$-6+5 a$$

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Problem 61

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$3 x+5 x+4$$

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Problem 62

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$6 x+8 x+7$$

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Problem 63

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$9 x+x+3+7$$

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Problem 64

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$5 x+3 x+2+4$$

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Problem 65

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$4 x+3+2 x+5$$

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Problem 66

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$7 x+6+2 x+9$$

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Problem 67

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$3 x-8+2 x-3$$

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Problem 68

Find the value of each of the following expressions when $x=3 .$ You may substitute 3 for $x$ in each expression the way it is written, or you may simplify each expression first and then substitute 3 for $x .$
$$7 x-2+4 x-1$$

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Problem 69

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{1}{2}$$

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Problem 70

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{1}{3}$$

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Problem 71

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{1}{4}$$

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Problem 72

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{1}{6}$$

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Problem 73

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{3}{2}$$

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Problem 74

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{2}{3}$$

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Problem 75

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{3}{4}$$

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Problem 76

Find the value of each of $12 x-3$ for each of the following values of $x .$
$$\frac{5}{6}$$

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Problem 77

Use the distributive property to write two equivalent expressions for the area of each figure.
CAN'T COPY THE FIGURE

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Problem 78

Use the distributive property to write two equivalent expressions for the area of each figure.
CAN'T COPY THE FIGURE

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Problem 79

Write an expression for the perimeter of each figure.
CAN'T COPY THE FIGURE

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Problem 80

Write an expression for the perimeter of each figure.
CAN'T COPY THE FIGURE

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Problem 81

Write an expression for the perimeter of each figure.
CAN'T COPY THE FIGURE

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Problem 82

Write an expression for the perimeter of each figure.
CAN'T COPY THE FIGURE

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Problem 83

This Google Earth image shows the Leaning Tower of Pisa. Most buildings stand at a right angle, but the tower is sinking on one side. The angle of inclination is the angle between the vertical and the tower. If the angle between the tower and the ground is $85^{\circ}$ what is the angle of inclination?
CAN'T COPY THE IMAGE

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Problem 84

This Google Earth image shows the Pentagon. The interior angles of a regular pentagon are all the same and sum to $540^{\circ} .$ Find the size of each angle.
CAN'T COPY THE IMAGE

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Problem 85

Find x in each figure and decide if the two angles are complementary or supplementary.
CAN'T COPY THE FIGURE

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Problem 86

Find x in each figure and decide if the two angles are complementary or supplementary.
CAN'T COPY THE FIGURE

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Problem 87

Find x in each figure and decide if the two angles are complementary or supplementary.
CAN'T COPY THE FIGURE

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Problem 88

Geometry This Google Earth image shows the Pentagon. The interior angles of a regular pentagon are all the same and sum
to $540^{\circ} .$ Find the size of
each angle.

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Problem 89

Luke earns $\$ 12$ per hour working as a math tutor. We can express the amount he earns each week for working $x$ hours with the expression $12 x .$ Indicate with a yes or no, which of the following could be one of Luke's paychecks. If you answer no, explain your answer.
a. $\$ 60$ for working five hours
b. $\$ 100$ for working nine hours
c. $\$ 80$ for working seven hours
d. $\$ 168$ for working 14 hours

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Problem 90

Kelly earns $\$ 15$ per hour working as a graphic designer. We can express the amount she earns each week for working $x$ hours with the expression $15 x .$ Indicate with a yes or no which of the following could be one of Kelly's paychecks. If you answer no, explain your answer.
a. $\$ 75$ for working five hours
b. $\$ 125$ for working nine hours
c. $\$ 90$ for working six hours
d. $\$ 500$ for working 35 hours

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Problem 91

On a certain day, the temperature on the ground is 72 degrees Fahrenheit, and the temperature at an altitude of $A$ feet above the ground is found from the expression $72-\frac{A}{300} .$ Find the temperature at the following altitudes.
a. $12,000$ feet
b. $15,000$ feet
c. $27,000$ feet
CAN'T COPY THE FIGURE

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Problem 92

As you know, the expression $2 l+2 w$ gives the perimeter of a rectangle with length $I$ and width $w .$ The garden below has a width of $3 \frac{1}{2}$ feet and a length of 8 feet. What is the length of the fence that surrounds the garden?
CAN'T COPY THE FIGURE

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Problem 93

cost of Bottled Water A water bottling company charges
$\$ 7$ per month for their water dispenser and $\$ 2$ for each gallon of water delivered. If you have $g$ gallons of water delivered in a month, then the expression $7+2 g$ gives the amount of your bill for that month. Find the monthly bill for each of the following deliveries.
a. 10 gallons
b. 20 gallons
CAN'T COPY THE FIGURE

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Problem 94

Cellular Phone Rates A cellular phone company charges $\$ 35$ per month plus 25 cents for each minute, or fraction of a minute, that you use one of their cellular phones. The expression $\frac{3500+25 t}{100}$ gives the amount of money, in dollars, you will pay for using one of their phones for $t$ minutes a month. Find the monthly bill for using one of their phones:
a. 20 minutes in a month
b. 40 minutes in a month
CAN'T COPY THE FIGURE

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Problem 95

Add.
$$4+(-4)$$

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00:10

Problem 96

Add.
$$2+(-2)$$

Tiffany Tran
Tiffany Tran
Numerade Educator

Problem 97

Add.
$$-2+(-4)$$

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Problem 98

Add.
$$-2+(-5)$$

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Problem 99

Add.
$$-5+2$$

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Problem 100

Add.
$$-3+12$$

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Problem 101

Add.
$$\frac{5}{8}+\frac{3}{4}$$

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Problem 102

Add.
$$\frac{5}{6}+\frac{2}{3}$$

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Problem 103

Add.
$$-\frac{3}{4}+\frac{3}{4}$$

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Problem 104

Add.
$$-\frac{2}{3}+\frac{2}{3}$$

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Problem 105

Simplify.
$$x+0$$

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Problem 106

Simplify.
$$y+0$$

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Problem 107

Simplify.
$$y+4-6$$

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Problem 108

Simplify.
$$y+6-2$$

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Problem 109

Give the opposite of each number.
$$9$$

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Problem 110

Give the opposite of each number.
$$12$$

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Problem 111

Give the opposite of each number.
$$-6$$

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Problem 112

Give the opposite of each number.
$$-5$$

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Problem 113

Match each statement on the left with the property that justifies it on the right.
a. Distributive property
b. Associative property
c. Commutative property
d. Commutative and associative properties
$$2(6+5)=2(6)+2(5)$$

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Problem 114

Match each statement on the left with the property that justifies it on the right.
a. Distributive property
b. Associative property
c. Commutative property
d. Commutative and associative properties
$$3+(4+1)=(3+4)+1$$

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Problem 115

Match each statement on the left with the property that justifies it on the right.
a. Distributive property
b. Associative property
c. Commutative property
d. Commutative and associative properties
$$x+5=5+x$$

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Problem 116

Match each statement on the left with the property that justifies it on the right.
a. Distributive property
b. Associative property
c. Commutative property
d. Commutative and associative properties
$$(a+3)+2=a+(3+2)$$

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Problem 117

Match each statement on the left with the property that justifies it on the right.
a. Distributive property
b. Associative property
c. Commutative property
d. Commutative and associative properties
$$(x+5)+1=1+(x+5)$$

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Problem 118

Match each statement on the left with the property that justifies it on the right.
a. Distributive property
b. Associative property
c. Commutative property
d. Commutative and associative properties
$$(a+4)+2=(4+2)+a$$

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Problem 119

Perform the indicated operation.
$$-\frac{5}{4}\left(\frac{8}{15}\right)$$

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Problem 120

Perform the indicated operation.
$$-\frac{4}{3}\left(\frac{6}{5}\right)$$

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Problem 121

Perform the indicated operation.
$$12 \div \frac{2}{3}$$

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Problem 122

Perform the indicated operation.
$$6 \div \frac{3}{5}$$

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Problem 123

Perform the indicated operation.
$$\frac{2}{3}-\frac{3}{4}$$

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Problem 124

Perform the indicated operation.
$$\frac{3}{5}-\frac{5}{8}$$

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