1. Expressions & Equations

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3. Click on a topic to go to that section.
Table of Contents
Commutative and Associative Properties
Combining Like Terms
Click on a topic to go to that section.
The Distributive Property and Factoring
Simplifying Algebraic Expressions
Inverse Operations
One Step Equations
Two Step Equations
Multi-Step Equations
Distributing Fractions in Equations
Translating Between Words and Equations
Using Numerical and Algebraic Expressions and Equations
Graphing & Writing Inequalities with One Variable
Simple Inequalities involving Addition & Subtraction
Simple Inequalities involving Multiplication & Division
Common Core Standards: 7.EE.1, 7.EE.3, 7.EE.4

4. Commutative and Associative Properties
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5. Commutative Property of Addition: The order in which the terms of a sum are added does not change the sum.
a + b = b + a
5 + 7 = 7 + 5
12= 12
Commutative Property of Multiplication: The order in which the terms of a product are multiplied does not change the product.
ab = ba
4(5) = 5(4)
Teacher’s instructions: When you commute to work you are going back and forth but it is the same route. So it is the same backwards and forwards.

6. Associative Property of Addition: The order in which the terms of a sum are grouped does not change the sum.
(a + b) + c = a + (b + c)
(2 + 3) + 4 = 2 + (3 + 4)
5 + 4 = 2 + 7
9 = 9
Teacher’s instructions: You are at a party. You associate with Alan and Barbara and then with Chris. This is the same as associating with Barbara and Chris and then Alan, just in a different order.

7. The Commutative Property is particularly useful when you are combining integers.
Example:
(-4)=
-15 + (-4) + 9= Changing it this way allows for the
= negatives to be added together first.
-10

8. Associative Property of Multiplication: The order in which the terms of a product are grouped does not change the product.

9. Identify the property of -5 + 3 = 3 + (-5)1
Identify the property of = 3 + (-5) A
Commutative Property of Addition B
Commutative Property of Multiplication C
Associative Property of Addition D
Associative Property of Multiplication
Answer: A

10. Identify the property of a + (b + c) = (a + c) + b2
Identify the property of a + (b + c) = (a + c) + b A
Commutative Property of Addition B
Commutative Property of Multiplication C
Associative Property of Addition D
Associative Property of Multiplication
Answer: C

11. Identify the property of (3 * (-4)) * 8 = 3 * ((-4) * 8)A
Commutative Property of Addition B
Commutative Property of Multiplication C
Associative Property of Addition D
Asociative Property of Multiplication
Answer: D

12. Discuss why using the Commutative Property would be useful with the following problems:
(-4)
x 3 x 0
x 7 x -2
(-6)

13. Combining Like Terms
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14. An Algebraic Expression - contains numbers, variables and at least one operation.

15. Like terms: terms in an expression that have the
same variable raised to the same power
Examples:
LIKE TERMS NOT LIKE TERMS
6x and 2x x2 and 2x
5y and 8y x and 8y
4x2 and 7x x2y and 7xy2

16. Identify all of the terms like 2x4
Identify all of the terms like 2x A 5x B
3x2 C 5y D
12y E 2
Answer: A

17. Identify all of the terms like 8y5
Identify all of the terms like 8y A 9y B
4y2 C 7y D 8 E
-18x
Answer: A, C

18. Identify all of the terms like 8xy6
Identify all of the terms like 8xy A 8x B
3x2y C
39xy D 4y E
-8xy
Answer: C, E

19. Identify all of the terms like 2y7
Identify all of the terms like 2y A
51w B 2x C 3y D 2w E
-10y
Answer: C, E

20. Identify all of the terms like 14x28
Identify all of the terms like 14x2 A
-5x B
8x2 C
13y2 D x E
-x2
Answer: B, E

21. If two or more like terms are being added or subtracted, they can be combined.
To combine like terms add/subtract the coefficient but leave the variable alone.
7x +8x =15x
9v-2v = 7v

22. Sometimes there are constant terms that can be combined. 9 + 2f + 6 =
Sometimes there will be both coeffients and constants to be combined.
3g g - 2
11g + 5
Notice that the sign before a given term goes with the number.

23. Try These: 1.) 2b +6g(3) + 4f + 9f 2.) 9j + 3 + 24h + 6 + 7h + 3
3.) 7a a c c
4.) 8x + 56xy + 5y
Answers:
1. 2b+18g +13f
2. 9j+31h+12
3. 9a-27+13c
4. 8x +56xy+5y

24. 9
8x + 3x = 11x
A True
B False
Answer: A True

25. 10
7x + 7y = 14xy
A True
B False
Answer: B False

26. 11
2x + 3x = 5x
A True
B False
Answer: A True

27. 12
9x + 5y = 14xy
A True
B False
Answer: B False

28. 13
6x + 2x = 8x2
A True
B False
Answer: B False

29. 14
-15y + 7y = -8y
A True
B False
Answer: A True

30. 15
-6 + y + 8 = 2y
A True
B False
Answer: B False

31. 16
-7y + 9y = 2y
A True
B False
Answer: A True

32. 17
9x x = A
15x B
11x + 4 C
13x + 2x D
9x + 6x
Answer: B

33. 18
12x + 3x A
15x B
13x C
17x D
15x + 2
Answer: D

34. 19
-4x x - 14 A
-22x B
-2x - 20 C
-6x +20 D
22x
Answer: B

35. The Distributive Property
and Factoring
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36. An Area Model
Imagine that you have two rooms next to each other. Both are 4 feet long. One is 7 feet wide and the other is 3 feet wide .
How could you express the area of those two rooms together? 4 7 3

37. Either way, the area is 40 feet2:
7 + 3 4 4 7 3
You could multiply 4 by 7, then 4 by 3 and add them
4(7) + 4(3) = = 40
You could add and then multiply by 4
4(7+3)=
4(10)= 40 OR
Either way, the area is 40 feet2:

38. An Area Model
Imagine that you have two rooms next to each other. Both are 4 yards long. One is 3 yards wide and you don't know how wide the other is.
How could you express the area of those two rooms together? 4 x 3

39. The area of the two rooms is
You cannot add x and 3 because they aren't like terms, so you can only do it by multiplying 4 by x and 4 by 3 and adding
4(x) + 4(3)=
4x + 12
The area of the two rooms is
(Note: 4x cannot be combined with 12) 4
x + 3

40. The Distributive Property
Finding the area of the rectangles demonstrates the distributive property. Use the distributive property when expressions are
written like so: a(b + c)
4(x + 2)
4(x) + 4(2)
4x + 8
The 4 is distributed to each term of the sum (x + 2)

41. Write an expression equivalent to: 5(y + 4) 5(y) + 5(4) 5y + 20
6(x + 2) 3(x + 4)
4(x - 5) 7(x - 1)
Remember to distribute the 5 to the y and the 4

42. -2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6
The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions.
EXAMPLE:
-2(x + 3) = -2(x) + -2(3) = -2x + -6 or -2x - 6
3(4x - 6) = 3(4x) - 3(6) = 12x - 18
-2 (x - 3) = -2(x) - (-2)(3) = -2x + 6
TRY THESE:
3(4x + 2) =
-1(6m + 4) =
-3(2x - 5) =
Be careful with your signs!

43. -(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7
Keep in mind that when there is a negative sign on the outside of the parenthesis it really is a -1.
For example:
-(2x + 7) = -1(2x + 7) = -1(2x) + -1(7) = -2x - 7
What do you notice about the original problem and its answer?
Remove to see answer.
The numbers are turned to their opposites.
Try these:
-(9x + 3) =
-(-5x + 1) =
-(2x - 4) =
-(-x - 6) =

44. 20
4(2 + 5) = 4(2) + 5
A True
B False
Answer: B False

45. 21
8(x + 9) = 8(x) + 8(9)
A True
B False
Answer: A True

46. 22
-4(x + 6) = (6)
A True
B False
Answer: B False

47. 23
3(x - 4) = 3(x) - 3(4)
A True
B False
Answer: A True

48. 24
Use the distributive property to rewrite the expression without parentheses
3(x + 4) A
3x + 4 B
3x + 12 C
x + 12 D 7x
Answer: B

49. 25
Use the distributive property to rewrite the expression without parentheses
5(x + 7) A
x + 35 B
5x + 7 C
5x + 35 D
40x
Answer: C

50. 26
Use the distributive property to rewrite the expression without parentheses
(x + 5)2 A
2x + 5 B
2x + 10 C
x + 10 D
12x
Answer: B

51. 27
Use the distributive property to rewrite the expression without parentheses
3(x - 4) A
3x - 4 B
x - 12 C
3x - 12 D 9x
Answer: C

52. 28
Use the distributive property to rewrite the expression without parentheses
2(w - 6) A
2w - 6 B
w - 12 C
2w - 12 D
10w
Answer: C

53. 29
Use the distributive property to rewrite the expression without parentheses
-4(x - 9) A
-4x - 36 B
x - 36 C
4x - 36 D
-4x + 36
Answer: D

54. 30
Use the distributive property to rewrite the expression without parentheses
5.2(x - 9.3) A
-5.2x B
5.2x C
-5.2x D
-48.36x
Answer: B

55. 31
Use the distributive property to rewrite the expression without parentheses A B C D
Answer: D

56. We can also use the Distributive Property in reverse
We can also use the Distributive Property in reverse. This is called Factoring.
When we factor an expression, we find all numbers or variables that divide into all of the parts of an expression.
Example:
7x + 35 Both the 7x and 35 are divisible by 7
7(x + 5) By removing the 7 we have factored the
problem
We can check our work by using the distributive property to see that the two expressions are equal.

57. We can factor with numbers, variables, or both.
2x + 4y = 2(x + 2y)
9b + 3 = 3(3b + 1)
-5j - 10k + 25m = -5(j + 2k - 5m) *Careful of your signs
4a + 6a + 8ab = 2a( b)

58. Factor the following expressions: 1.) 6b + 9c = 2.) -2h - 10j =
Try these:
Factor the following expressions:
1.) 6b + 9c =
2.) -2h - 10j =
3.) 4a + 20ab + 12abc =
Answers:
1. 3(2b+3c)
2. -2(h+5j)
3. 4a(1+5b+3bc)

59. Factor the following: 4p + 24q32
Factor the following: 4p + 24q A
4 (p + 24q) B
2 (2p + 12q) C
4(p + 6q) D
2 (2p + 24q)
Answer: C

60. Factor the following: 5g + 15h33
Factor the following: 5g + 15h A
3(g + 5h) B
5(g + 3h) C
5(g + 15h) D
5g (1 + 3h)
Answer: B

61. Factor the following: 3r + 9rt + 15rx34
Factor the following: 3r + 9rt + 15rx A
3(r+ 3rt + 5rx) B
3r(1 + 3t + 5x) C
3r (3t + 5x) D
3 (r + 9rt + 15rx)
Answer: B

62. Factor the following: 2v+7v+14v35
Factor the following: 2v+7v+14v A
7(2v + v + 2v) B
7v( ) C
7v (1 + 2) D
v( )
Answer: D

63. Factor the following: -6a - 15ab - 18abc36
Factor the following: -6a - 15ab - 18abc A
-3a(2 + 5b + 6bc) B
3a(2+ 5b + 6bc) C
-3(2a - 5b - 6bc) D
-3a (2 -5b - 6bc)
Answer: A

64. What divides into the expression: -5n - 20mn - 10np
Answer: -5n

65. -
If a regular pentagon has a perimeter of 10x + 25, what does each side equal?
Answer: 2x + 5

66. Simplifying Algebraic Expressions
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67. Now we will use what we know about combining like terms and the distributive property to simplify algebraic expressions.
Remember, like terms have the same variable and same exponent.

68. To simplify:
4 + 5(x + 3) First Distribute
4 + 5(x) + 5(3)
4 + 5x Then combine Like Terms
5x + 19
Notice that when combining like terms, you add/subtract the coefficients but the variable remains the same. Remember that you can combine coefficient or constant terms.

69. 37
7x +3(x - 4) = 10x - 4
A True
B False
Answer: B False

70. 38
8 +(x + 3)5 = 5x + 11
A True
B False
Answer: B False

71. 39
4 +(x - 3)6 = 6x -14
A True
B False
Answer: A True

72. 40
2x + 3y + 5x + 12 = 10xy + 12
A True
B False
Answer: B False

73. 41 5x2 + 2x + 7(x + 1) + x2 = 6x2 + 9x + 7 A True B False
Answer: A True

74. 42 2x3 + 4x2 + 6(x2 + 3x) + x = 2x3 + 10x2 + 4x A True B False
Answer: B False

75. Which expression represents the perimeter of the figure?43
The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure. yz y y
Which expression represents the perimeter of the figure? x x A
5xyz B
x2 + y3z
Answer: D C
2x + 3yz D
2x + 2y + yz
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from accessed 17, June, 2011.

76. 44
A rectangle has a width of x and a length that is double that. What is the perimeter of the rectangle? A 4x B 6x C 8x D
10x
Answer: B

77. Inverse Operations
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78. What is an equation?
An equation is a mathematical statement containing an equal sign to show that two expressions are equal.
2 + 3 = 5
9 – 2 = 7
5 + 3 = 1 + 7
An algebraic equation is just an equation that has algebraic symbols in one or both of the expressions.
4x = 24
9 + h = 15

79. Equations can also be used to state the equality of two
expressions containing one or more variables.
In real numbers we can say, for example, that for any given value of x it is true that
4x + 1 = 13
x = 3 because
4(3) + 1 = 13
= 13
13 = 13

80. An equation can be compared to a balanced scale.
Both sides need to contain the same quantity in order for it to be "balanced".

81. For example, 9+ 11 = 6 + 14 represents an equation because both sides simplify to 20.=
20 = 20
Any of the numerical values in the equation can be represented by a variable.
Examples:
15 + c = 25
x + 10 = 25
= y

82. When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).

83. In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation.
Let's review the inverses of each operation:
Addition Subtraction
Multiplication Division
Square Square Root

84. There are two questions to ask when solving an equation:
*What operation is in the equation?
*What is the inverse of that operation (This will be the operation you use to solve the equation.)?

85. A good phrase to remember when doing equations is:
Whatever you do to one side of the equation, you do to the other.
For example, if you add three on one side of the equal sign you must add three to the other side as well…
to keep the equation in balance.

86. To solve for "x" in the following equation...
Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides (in this case, it is subtraction).
x = 25
In the original equation, replace x with 25 and see if it makes the equation true.
x + 7 = 32
= 32
32 = 32

87. a.) y +7 = 14 subtract 7 b.) a - 21 = 10 add 21
For each equation, write the inverse operation needed to solve for the variable.
a.) y +7 = 14 subtract b.) a - 21 = add 21
c.) 5s = divide by 5 d.) x = multiply by 12 12
move
move
move
move

88. Which method is better? Why?
Think about this...
To solve c - 3 = 12
Which method is better? Why?
Kendra
Added 3 to each side of the equation
c - 3 = 12
c = 15
Ted
Subtracted 12 from each side, then added 15.
c - 3 = 12
c - 15 = 0
c = 15

89. What is the inverse operation needed to solve this equation?45
What is the inverse operation needed to solve this equation?
2x = 14 A
Addition B
Subtraction C
Multiplication D
Division
Answer: D

90. What is the inverse operation needed to solve this equation?46
What is the inverse operation needed to solve this equation?
x - 3 = -12 A
Addition B
Subtraction C
Multiplication D
Division
Answer: A

91. What is the inverse operation needed to solve this problem? -2 + x = 947
What is the inverse operation needed to solve this problem?
-2 + x = 9 A
Addition B
Subtraction C
Multiplication D
Division
Answer: A, B
Add 2
Subtract -2

92. One Step Equations
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93. To solve equations, you must work backwards through the order of operations to find the value of the variable.
Remember to use inverse operations in order to isolate the variable on one side of the equation.
Whatever you do to one side of an equation, you MUST do to the other side!

94. Examples:
y + 3 = 13
The inverse of adding 3 is subtracting 3
y = 10
4m = 32
The inverse of multiplying by 4 is dividing by 4
m = 8
Remember - whatever you do to one side of an equation, you MUST do to the other!!!

95. Solve each equation then click the box to see work & solution.
One Step Equations
Solve each equation then click the box to see work & solution.
x - 5 = 2
x = 7
2 = x - 4
6 = x
click to show
inverse operation
click to show
inverse operation
x + 5 = -14
x = -19
6 = x + 1
5 = x
click to show
inverse operation
click to show
inverse operation
x + 9 = 5
x = -4
12 = x + 17
-5 = x
click to show
inverse operation
click to show
inverse operation

96. One Step Equations 4x = 16 4 4 x = 4 x (2) (2) = 9 2 x = 18 -2x = -12
x = 4
-2x = -12
x = 6
-20 = 5x
-4 = x x
click to show
inverse operation
(2)
= 9
(2) 2
x = 18
click to show
inverse operation x
click to show
inverse operation
= 36
(-6)
click to show
inverse operation
(-6) -6
x = -216
click to show
inverse operation

97. 48
Solve.
x - 7 = 19
Answer: 26

98. 49
Solve.
j + 15 = 17
Answer: 2

99. 50
Solve.
42 = 6y
Answer: 7

100. 51
Solve.
-115 = -5x
Answer: 23

101. 52
Solve.
= 12 x 9
Answer: 108

102. 53
Solve.
w - 17 = 37
Answer: 54

103. 54
Solve.
-3 = x 7
Answer: -21

104. 55
Solve.
23 + t = 11
Answer: -12

105. 56
Solve.
108 = 12r
Answer: 9

106. Sometimes the operation can be confusing.
For example: x = 7
This looks like you should use subtraction to undo the problem. However, -2 + x = 7 is the same as x - 2 = 7
so while it appears to be addition, it is really subtraction. In order to undo this we can add.
-2 + x = 7
x - 2 = 7
+2 +2
x = 9
OR OR
-2 + x = x = 7
- (-2) -(-2)
x = x = 9

107. -2 + x = 7
= -2
-4 + x = 5
This did not
cancel out
anything.
+2 +2
x = 9
This did
cancel out to
find the
answer.
x - 2 = 7
This is the same
as the middle problem

108. Try these: 1.) -4 + b = 7 2.) -2 + r = 4 3.) -3 + w = 6 4.) -5 + c = 9
Answers:
1. b=11
2. r=6
3. w=9
4. c=14

109. To which does the "-" belong?
Think about this...
In the expression
To which does the "-" belong?
Does it belong to the x? The 3? Both?
The answer is that there is one negative so it is used once with either the variable or the 3. Generally, we assign it to the 3 to avoid creating a negative variable.
So:

110. 57
Solve.
Answer: 15

111. 58
Solve.
-5 + q = 15
Answer: 20

112. 59
Solve.
Answer: -96

113. 60
Solve
Answer: -63

114. 61
Solve.
Answer: -1.4

115. 62
Solve.
Answer: -19.1

116. 63
Solve.
Answer: 7.5

117. Sometimes you will have an equation where you are multiplying a variable by a fraction.
𝟑 𝟒𝒙 =𝟗

118. To undo the fraction you: Multiply by the Reciprocal of the Coefficent
This means that you will flip the fraction and then multiply
**Dividing by a fraction is the same
as multiplying by its reciprocal

119. 1 times any number is itself so this is why it can cancel out.

120. 64
Solve.
Answer: 15

121. 65
Solve
Answer: 12

122. 66
Solve.
Answer: .4

123. Two-Step Equations
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124. Sometimes it takes more than one step to solve an equation
Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.
This means that you undo in the opposite order (PEMDAS):
1st: Addition & Subtraction
2nd: Multiplication & Division
3rd: Exponents
4th: Parentheses
Whatever you do to one side of an equation, you MUST do to the other side!

125. 4 4 Undo multiplication second x = 2 -2y - 9 = -13
Examples:
4x + 2 = 10
Undo addition first
4x = 8
Undo multiplication second
x = 2
-2y - 9 = -13
Undo subtraction first
-2y = -4
Undo multiplication second
y = 2
Remember - whatever you do to one side
of an equation, you MUST do to the other!!!

126. Solve each equation then click the box to see
Two Step Equations
Solve each equation then click the box to see
work & solution.
5b + 3 = 18
5b = 15
b = 3
w + 6 = 10 2
w 2 =
w = 8
3j - 4 = 14
3j = 18
j = 6
-2x + 3 = -1
-2x = -4
x = 2
-2m - 4 = 22
-2m = 26
m = -13
+5 = +5
t = 15

127. 67
Solve the equation.
5x - 6 = -56
Answer: -10

128. 68
Solve the equation.
14 = 3c + 2
Answer: 4

129. 69
Solve the equation. x 5
- 4 = 24
Answer: 140

130. 70
Solve the equation.
5r - 2 = -12
Answer: -2

131. 71
Solve the equation.
14 = -2n - 6
Answer: -10

132. 72
Solve the equation.
+ 7 = 13 x 5
Answer: 30

133. 73
Solve the equation.
+ 2 = -10 x 3 -
Answer: 36

134. 74
Solve the equation.
Answer: 30

135. 75
Solve the equation.
Answer: 1.5

136. 76
Solve the equation.
Answer: -3

137. 77
Solve the equation.
Answer: -3

138. 78
Solve -3 5 1 2 1 10
x =
Answer: 2/3

139. 79
Solve the equation.
Answer: -16.8

140. 80
Solve the equation.
Answer: 2

141. Multi-Step Equations
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142. Steps for Solving Multiple Step Equations
As equations become more complex, you should:
1. Simplify each side of the equation.
(Combining like terms and the distributive property)
2. Use inverse operations to solve the equation.
Remember, whatever you do to one side of an equation,
you MUST do to the other side!

143. Examples:
5x + 7x + 4 = 28
12x + 4 = 28 Combine Like Terms
Undo Addition
12x = 24
Undo Multiplication
x = 2
-1 = 2r - 7r +19
-1 = -5r Combine Like Terms
-19 = Undo Subtraction
-20 = -5r
Undo Multiplication
4 = r

144. Try these.
12h - 10h + 7 = 25
-17q + 7q -13 = 27
17 - 9f + 6 = 140
h = 9
q = - 4
f = -13

145. Always check to see that both sides of the equation are simplified before you begin solving the equation.
Sometimes, you need to use the distributive property in order to simplify part of the equation.
Remember: The distributive property is a(b + c) = ab + ac
Examples
5(20 + 6) = 5(20) + 5(6)
9(30 - 2) = 9(30) - 9(2)
3(5 + 2x) = 3(5) + 3(2x)
-2(4x - 7) = -2(4x) - (-2)(7)

146. Examples:
2(b - 8) = 28
2b - 16 = Distribute the 2 through (b - 8)
Undo subtraction
2b = 44
Undo multiplication
b = 22
3r + 4(r - 2) = 13
3r + 4r - 8 = 13 Distribute the 4 through (r - 2)
7r - 8 = Combine Like Terms
Undo subtraction
7r = 21
Undo multiplication
r = 3

147. Try these.
3(w - 2) = 9
4(2d + 5) = 92
6m + 2(2m + 7) = 54
w = 5
d = 9
m = 4

148. 81
Solve.
9 + 3x + x = 25
Answer: 4

149. 82
Solve
-8e e = -13
Answer: 4

150. 83
Solve.
-27 = 8x x - 11
Answer: -2

151. 84
Solve.
n n - 5 = 13
Answer: 4

152. 85
Solve.
32 = f - 3f + 6f
Answer: 8

153. 86
Solve.
6g - 15g = -38
Answer: 3

154. 87
Solve.
3(a - 5) = -21
Answer: -2

155. 88
Solve.
4(x + 3) = 20
Answer: 2

156. 89
Solve.
3 = 7(k - 2) + 17
Answer: 0

157. 90
Solve.
2(p + 7) -7 = 5
Answer: -1

158. 91
Solve.
3m -1m + 3(m-2) = 19.75
Answer: 5.15

159. 92
Solve.
Answer: 2.1

160. 93
Solve.
Answer: 4

161. 94
Solve.
Answer: 7

162. Distributing Fractions in Equations
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163. Remember...
1. Simplify each side of the equation.
2. Solve the equation.
(Undo addition and subtraction first, multiplication and division second)
Remember, whatever you do to one side of an equation, you MUST do to the other side!

164. There is more than one way to solve an equation with a fraction coefficient. While you can, you don't need to distribute.
Multiply by the reciprocal Multiply by the LCD 3 5 72 5
(-3 + 3x) =
(-3 + 3x) = 3 5 72
 (-3 + 3x) = 
3(-3 + 3x) = 72
-9 + 9x = 72
9x = 81
x = 9
(-3 + 3x) = 3 5 72
 (-3 + 3x) = 
-3 + 3x = 24
3x = 27
x = 9

165. Some problems work better when you multiply by the reciprocal and some work better multiplying by the LCM.
Which strategy would you use for the following? Why?

166. 95
Solve.
Answer: 13

167. 96
Solve.
Answer: -7 5/6

168. 97
Solve.
(8 - 3c) = 2 3 16
Answer: 0

169. 98
Solve.
Answer: 3

170. 99
Solve.
Answer: 3/4

171. Translating Between Words and Expressions
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172. List words that indicate
addition
Answers:
Sum More than
Total All together
Add
Plus
increased by

173. List words that indicate
subtraction
Answers:
Minus Decreased by
Difference Less
Take away Fewer Than
Less than Subtracted From
Subtract

174. List words that indicate
multiplication
Answers:
Product Multiplied by
Times Of
Twice...
Double...

175. List words that indicate
division
Answers:
Divided by Divisible by
Quotient of Divisibility
Half...
Fractions

176. List words that indicate
equals
Answers:
Total The same value as Is
The same as

177. Be aware of the difference between "less" and "less than".
For example:
"Eight less three" and "Three less than Eight" are equivalent expressions. So what is the difference in wording?
Eight less three:
Three less than eight:
When you see "less than", you need to switch the order of the numbers.

178. As a rule of thumb, if you see the words "than" or
"from" it means you have to reverse the order
of the two items on either side of the word.
Examples:
·8 less than b means b - 8
·3 more than x means x + 3
·x less than 2 means x
_______________
click to reveal

179. The many ways to represent multiplication...
How do you represent "three times a"?
(3)(a) 3(a) 3  a 3a
The preferred representation is 3a
When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable.
The following are not allowed:
3xa ... The multiplication sign looks like another
variable
a3 ... The number is always written in front of the

180. Representation of division...
How do you represent "b divided by 12"?
b ÷ 12
b ∕ 12 b 12

181. When choosing a variable, there are some that are often avoided:
l, i, t, o, O, s, S
Why might these be avoided?
It is best to avoid using letters that might be confused for numbers or operations. In the case above (1, +, 0, 5)

182. TRANSLATE THE WORDS INTO AN ALGEBRAIC EXPRESSION
Three times j
Eight divided by j
j less than 7
5 more than j
4 less than j 1 + + + 2 3 j - - - - 4 5 6 7     8 9 ÷ ÷ ÷ ÷

183. Write the expression for each statement.
Then check your answer.
The sum of twenty-three and m
23 + m

184. Write the expression for each statement.
Then check your answer.
Twenty-four less than d
d - 24

185. Write the expression for each statement.
Remember, sometimes you need to use parentheses for a quantity.
Four times the difference of eight and j
4(8-j)

186. Write the expression for each statement. Then check your answer.
The product of seven and w, divided by 12 7w 12

187. Write the expression for each statement.
Then check your answer.
The square of the sum of six and p
(6+p)2

188. The quotient of 200 and the quantity of p times 7
100
The quotient of 200 and the quantity of p times 7 A
200 7p B
200 - (7p) C
200 ÷ 7p D 7p
200
Answer: A

189. 35 multiplied by the quantity r less 45
101
35 multiplied by the quantity r less 45 A
35r - 45 B
35(45) - r C
35(45 - r) D
35(r - 45)
Answer: D

190. Mary had 5 jellybeans for each of 4 friends.
102
Mary had 5 jellybeans for each of 4 friends. A
5 + 4 B
5 - 4 C
5 x 4 D
5 ÷ 4
Answer: C

191. 103
If n + 4 represents an odd integer, the next larger odd integer is represented by A
n + 2 B
n + 3 C
n + 5 D
n + 6
Answer: D
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

192. 104
a less than 27 A
27 - a B a 27 C
a - 27 D
27 + a
Answer: A

193. If h represents a number, which equation is a correct translation of:
105
If h represents a number, which equation is a correct translation of:
“Sixty more than 9 times a number is 375”? A
9h = 375 B
9h + 60 = 375 C
9h - 60 = 375 D
60h + 9 = 375
Answer: B
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

194. Using Numerical and Algebraic Expressions and Equations
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195. We can use our algebraic translating skills to solve other problems.
We can use a variable to show an unknown.
A constant will be any fixed amount.
If there are two separate unknowns, relate one to the other.

196. The cafeteria sold 150 grilled chicken sandwiches and 75 tenders.
The school cafeteria sold 225 chicken meals today. They sold twice the number of grilled chicken sandwiches than chicken tenders. How many of each were sold?
2c + c = 225
c + 2c = 225
3c = 225
c = 75
The cafeteria sold 150 grilled chicken sandwiches and 75 tenders.
chicken
sandwiches
chicken
tenders
total meals

197. Julie needs 1 inches on each side.
Julie is matting a picture in a frame. Her frame is 9 inches wide and her picture is 7 inches wide. How much matting should she put on either side?
2m + 7 = 1 2 9 1 2 9
2m + 7 =
2m = 2
m = 1
Julie needs 1 inches on each side. 1 4 2
both sides
of the mat
size of
picture
size of
frame

198. Many times with equations there will be one number that will be the same no matter what (constant) and one that can be changed based on the problem (variable and coefficient).
Example: George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all?

199. George is buying video games online. The cost of the video is $30
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all?
Notice that the video games are "per game" so that means there could be many different amounts of games and therefore many different prices. This is shown by writing the amount for one game next to a variable to indicate any number of games.
30g
cost of
one video
game
number
of games

200. George is buying video games online. The cost of the video is $30
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all?
Notice also that there is a specific amount that is charged no matter what, the flat fee. This will not change so it is the constant and it will be added (or subtracted) from the other part of the problem.
30g
cost of
one video
game
number
of games
the cost
of the
shipping

201. George is buying video games online. The cost of the video is $30
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all?
"Total" means equal so here is how to write the rest of the equation.
30g = 127
cost of
one video
game
number
of games
the total amount
the cost
of the
shipping

202. George is buying video games online. The cost of the video is $30
George is buying video games online. The cost of the video is $30.00 per game and shipping is a flat fee of $7.00. He spent a total of $ How many games did he buy in all?
Now we can solve it.
30g + 7 = 127
30g = 120
g = 4 George bought 4 video games.

203. 106
Lorena has a garden and wants to put a gate to her fence directly in the middle of one side. The whole length of the fence is 24 feet. If the gate is 4 feet, how many feet should be on either side of the fence? 1 2
Answer: 9 ¾
2x+41/2= 24
x= 93/4 feet

204. 107
Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. Which equation shows this problem? A
12p + 27 = 147 B
12p + 27p = 147 C
27p + 12 = 147 D
39p = 147
Answer: C

205. 108
Lewis wants to go to the amusement park with his family. The cost is $12.00 for parking plus $27.00 per person to enter the park. Lewis and his family spent $147. How many people went to the amusement park WITH Lewis?
Answer: 4
Careful with this type of problem. It asks how many people are going with Lewis. This means that you have to find the answer and then subtract one because Lewis is one of the people going and the problem asks how many are going with him.
4 people went with Lewis

206. 109
Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle? Which equation represents the situation? A
= 239 B
9d + 68 = 239
Answer: B C
68d + 9 = 239 D
77d = 239

207. 110
Mary is saving up for a new bicycle that is $239. She has $68.00 already saved. If she wants to put away $9.00 per week, how many weeks will it take to save enough for her bicycle?
Answer: 19 weeks

208. Be prepared to show your equation!
111
You are selling t-shirts for $15 each as a fundraiser. You sold 17 less today then you did yesterday. Altogether you have raised $675.
Write and solve an equation to determine the number of t-shirts you sold today.
Be prepared to show your equation!
Answer: 14
15[x+(x-17)]=675
14 shirts today
31 shirts yesterday

209. Be prepared to show your equation!
112
Rachel bought $12.53 worth of school supplies. She still needs to buy pens which are $2.49 per pack. She has a total of $20.00 to spend on school supplies. How many packs of pens can she buy?
Write and solve an equation to determine the number of packs of pens Rachel can buy.
Be prepared to show your equation!
Answer: 3
2.49p = 20
p=3 packs

210. Write and solve an equation to determine the width of the rectangle.
113
The length of a rectangle is 9 cm greater than its width and its perimeter is 82 cm.
Write and solve an equation to determine the width of the rectangle.
Be prepared to show your equation!
Answer: 16
2w+2(w+9)=82
w= 16cm

211. The product of -4 and the sum of 7 more than a number is -96.
114
The product of -4 and the sum of 7 more than a number is -96.
Write and solve an equation to determine the number. Be prepared to show your equation!
Answer: 17
-4(7+x)= -96
x=17

212. Be prepared to show your equation! How many subscribers last year?
115
A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200.
Write and solve an equation to determine the number of subscribers they had each year.
Be prepared to show your equation!
How many subscribers last year?
Answer:
182x+ 182(x+2100)=2,566,200
x=6000
6000 subscribers last year
8100 subscribers this year

213. Be prepared to show your equation! How many subscribers this year?
116
A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200.
Write and solve an equation to determine the number of subscribers they had each year.
Be prepared to show your equation!
How many subscribers this year?
Answer:
182x+ 182(x+2100)=2,566,200
x=6000
6000 subscribers last year
8100 subscribers this year

214. The perimeter of a hexagon is 13.2 cm.
117
The perimeter of a hexagon is 13.2 cm.
Write and solve an equation to determine the length of a side of the hexagon.
Be prepared to show your equation!
Answer: 2.2
6x=13.2
x=2.2 cm

215. Graphing and Writing Inequalities with One Variable
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216. When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.
Important Words
Sample Sentence
Equivalent
Translation
is more than
Trenton is more than 10 miles away.
t > 10
is greater than
A is greater than B.
A > B
must exceed
The speed must exceed 25 mph.
The speed is greater than 25 mph.
s > 25

217. Important Words Sample Sentence Equivalent Translation cannot exceed
When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.
Important Words
Sample Sentence
Equivalent
Translation
cannot exceed
Time cannot exceed 60 minutes.
Time must be less than or equal to 60 minutes.
t < 60
is at most
At most, 7 students were late for class.
Seven or fewer students were late for class.
n < 7
is at least
Bob is at least 14 years old.
Bob's age is greater than or equal to 14.
B > 14

218. How are these inequalities read?
2 + 2 > Two plus two is greater than 3
2 + 2 > Two plus two is greater than or equal to 3
2 + 2 ≥ Two plus two is greater than or equal to 4
2 + 2 < Two plus two is less than 5
2 + 2 ≤ Two plus two is less than or equal to 5
2 + 2 ≤ Two plus two is less than or equal to 4

219. Writing inequalities
Let's translate each statement into an inequality.
words
x is less than 10
translate to
inequality statement x 10
<
20 is greater than or equal to y 20
> y

220. You try a few: 1. 14 is greater than a 2. b is less than or equal to 8
3. 6 is less than the product of f and 20
4. The sum of t and 9 is greater than or equal to 36
5. 7 more than w is less than or equal to 10
decreased by p is greater than or equal to 2
7. Fewer than 12 items
8. No more than 50 students
9. At least 275 people attended the play
Answers:
> a
2. b ≤ 8
3. 6 < 20f
4. t + 9 ≥ 36
w ≤ 10
p ≥ 2
7. n < 12
8. s < 50
9. p > 275

221. Do you speak math?
Change the following expressions from English into math.
Answer
Double a number is at most four.
2x ≤ 4
Answer
Three plus a number is at least six.
3 + x ≥ 6

222. Five less than a number is less than twice that number.
Answer
x - 5 < 2x
The sum of two consecutive numbers is at least thirteen.
Answer
x + (x + 1) ≥ 13
Three times a number plus seven is at least nine.
Answer
3x + 7 > 9

223. Let e represent an employee's wages.
A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour.
Let e represent an employee's wages.
An employee earns e
at least
>
$7.50
7.5
7.5

224. Try this:
The speed limit on a road is 55 miles per hour. Define a variable and write an inequality.
Answer:
Let s = speed
0 < s < 55

225. 118
You have $200 to spend on clothes. You already spent $140 and shirts cost $12. Which equation shows this scenario? A
200 < 12x + 140 B
x + 140 ≤ C
200 > 12x + 140 D
x + 140 ≥
Answer: D

226. 119
A sea turtle can live up to 125 years. If one is already 37 years old, which scenario shows how many more years could it live? A
125 < 37 + x x ≤ B C
125 > 37 + x D x ≥
Answer: D

227. 120
The width of a rectangle is 3 in longer than the length. The perimeter is no less than 25 inches. A
4a + 6 < 25 B
4a + 6  25 ≤ C
4a + 6 > 25 D
4a + 6 ≥ 25
Answer: D

228. 121
The absolute value of the sum of two numbers is less than or equal to the sum of the absolute values of the same two numbers. A B C D
Answer: B

229. Solution Sets
A solution to an inequality is NOT a single number. It will have more than one value. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
This would be read as the solution set is all numbers greater than or equal to negative 5.

230. Let's name the numbers that are solutions of the given inequality.
Which of the following are solutions? {5, 10, 15, 20}
5 > 10 is not true
So, not a solution
10 > 10 is not true
So, not a solution
15 > 10 is true
So, 15 is a solution
20 > 10 is true
So, 20 is a solution
Answer:
{15, 20} are solutions of the inequality r > 10

231. Let's try another one. 30 ≥ 4d; {3, 4, 5, 6, 7, 8} 30 ≥ 4d 30 ≥ (4)3
30 ≥ 12
30 ≥ 4d
30 ≥ (4)4
30 ≥ 16
30 ≥ 4d
30 ≥ (4)5
30 ≥ 20
click to reveal
click to reveal
click to reveal
30 ≥ 4d
30 ≥ (4) 6
30 ≥ 24
30 ≥ 4d
30 ≥ (4)7
30 ≥ 28
30 ≥ 4d
30 ≥ (4)8
30 ≥ 32
click to reveal
click to reveal
click to reveal

232. Graphing Inequalities - The Circle
An open circle on a number shows that the number is not part of the solution. It is used with "greater than" and "less than". The word equal is not included.< >
A closed circle on a number shows that the number is part of the solution. It is used with "greater than or equal to" and "less than or equal to". < >

233. Graphing Inequalities - The Arrow
The arrow should always point in the direction of those numbers who satisfy the inequality.
*If the variable is on the left side of the inequality, then < and ≤ will show an arrow pointing left.
*If the variable is on the left side of the inequality, then > and ≥ will show an arrow pointing right.

234. Notice that < and ≤ look like an arrow pointing left
and that > and ≥ look like an arrow pointing right.
But what if the variable isn't on the left?
Do the opposite of where the inequality symbol points. -1 -2 -3 -4 -5 1 2 3 4 5

235. Graphing Inequalities
What is the number in the inequality?
What kind of circle should be used?
In what direction does the line go?

236. Graphing Inequalities
x is less than 5
Step 1: Rewrite this as x < 5.
Step 2: What kind of circle? Because it is less than, it does not include the number 5 and so it is an open circle. -1 -2 -3 -4 -5 1 2 3 4 5

237. x < 5
Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left. -1 -2 -3 -4 -5 1 2 3 4 5
Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality. -1 -2 -3 -4 -5 1 2 3 4 5

238. Graphing Inequalities x is less than or equal to 5
Step 1: Rewrite this as x ≤ 5.
Step 2: What kind of circle? Because it is less than or equal to, it does include the number 5 and so it is a closed circle. -1 -2 -3 -4 -5 1 2 3 4 5

239. x ≤ 5
Step 3: Draw an arrow on the number line showing all possible solutions. Numbers greater than the variable, go to the right. Numbers less than the variable, go to the left. -1 -2 -3 -4 -5 1 2 3 4 5
Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality. -1 -2 -3 -4 -5 1 2 3 4 5

240. You try Graph the inequality x > 2 Graph the inequality -3 > x1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
click 2 on the number line for answer
click -3 on the number line for answer
.05.05
Graph the inequality
-3 > x 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Teacher’s instructions: Notice that the inequality symbol is pointing right but the arrow is pointing left. This is because the variable is on the right side.

241. Graph the inequalities.
Try these.
Graph the inequalities.
1. x > -3 -1 -2 -3 -4 -5 1 2 3 4 5
.05.
2. x < 4 -1 -2 -3 -4 -5 1 2 3 4 5

242. State the inequality shown.
Try these.
State the inequality shown. 1. -1 -2 -3 -4 -5 1 2 3 4 5 2. -1 -2 -3 -4 -5 1 2 3 4 5
Answers:
1. x  5
2. x  -1

243. This solution set would be x > -4.
122
This solution set would be x > -4.
A True
B False 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: B False

244. State the inequality shown.
123
State the inequality shown. -1 -2 -3 -4 -5 1 2 3 4 5 A
x > 3 B
x < 3 C
x < 3
Answer: D D
x > 3

245. State the inequality shown.
124
State the inequality shown. A
11 < x B
11 > x C
11 > x D
11 < x
Answer: A

246. State the inequality shown.
125
State the inequality shown. -1 -2 -3 -4 -5 1 2 3 4 5 A
x > -1 B
x < -1 C
x < -1 D
x > -1
Answer: C

247. State the inequality shown.
126
State the inequality shown. -1 -5 1 5 -2 -3 -4 2 3 4 A
-4 < x B
-4 > x C
-4 < x D
-4 > x
Answer: C

248. State the inequality shown.
127
State the inequality shown. -1 -2 -3 -4 -5 1 2 3 4 5 A
x > 0 B
x < 0 C
x < 0 D
x > 0
Answer: B

249. Simple Inequalities Involving Addition
and Subtraction
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250. Remembers how to solve an algebraic equation?
x + 3 = 13
x = 10
Use the inverse of addition
Be sure to check your answer!
Does = 13
13 = 13

251. · Solving one-step inequalities is much like solving
one-step equations.
·To solve an inequality, you need to isolate the
variable using the properties of inequalities and
inverse operations.
· Remember, whatever you do to one side, you do to
the other.

252. To find the solution, isolate the variable x.
Remember, it is isolated when it appears by itself on one side of the equation.
12 > x + 5
Subtract to undo addition
7 > x

253. 7 > x
The symbol is > so it is an open circle and it is numbers less than 7 so it goes to the left. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

254. Solve and graph. A. j + 7 > -2 A. j + 7 > -2 -7 -7 j > -9
j > -9 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
-9 is not included in solution set; therefore we graph with an open circle.

255. Solve and graph. B. r - 2 > 4 r - 2 > 4 +2 +2 r > 6 11 10 12
r > 6 11 10 12 13 14 9 8 7 6 5 4 3 2 1

256. Solve and graph. 9 > w + 4 C. 9 > w + 4 - 4 5 > w w < 5 12 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

257. 128 Solve the inequality. 3 < s + 4 ____ < s Answer: -1

258. Solve the inequality and graph the solution. -4 + b < -2
129
Solve the inequality and graph the solution.
-4 + b < -2 A 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: B
b < 2 D 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

259. Solve the inequality and graph the solution. -8 > b - 5
130
Solve the inequality and graph the solution.
-8 > b - 5 A 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: C
-3 > b D 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

260. 131 Solve the inequality. m + 6.4 < 9.6 m < ______ Answer: 3.2

261. Simple Inequalities Involving Multiplication
and Division
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262. Multiplying or Dividing by a Positive Number
3x > -36
x > -12
Since x is multiplied by 3, divide both sides by 3 for the inverse operation.

263. ( ) ( ) Solve the inequality. 2 3 r < 4
Since r is multiplied by 2/3,
multiply both sides by the reciprocal of 2/3. ( ) ( ) 3 2 2 3 3 2
r < 4
r < 6

264. 132
3k > 18 A 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 D 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: B
k > 6

265. 133 -30 > 3q A 10 > q B -10 < q C -10 > q D 10 < q
Answer: C

266. 134 X 2 < -3 A B C D Answer: D x < -6 3 4 5 6 7 8 9 -3 -4 -5 -62 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 B 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 C 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: D
x < -6 D

267. 135 3 4 g > 27 A g > 36 B g > 108 C g > 36 D g > 108
Answer: A

268. 136 -21 > 3d A d > -7 B d > -7 C d < -7 D d < -7
Answer: D

269. Now let's see what happens when we multiply or divide by negative numbers.
·Sometimes you must multiply or divide to isolate
the variable.
·Multiplying or dividing both sides of an inequality
by a negative number gives a surprising result.

270. Write down two numbers and put the appropriate
inequality (< or >) between them.
Apply each rule to your original two numbers from
step 1 and simplify. Write the correct inequality
(< or >) between the answers.
A. Add 4
B. Subtract 4
C. Multiply by 4
D. Multiply by -5
E. Divide by 4
F. Divide by -4

271. What happened with the inequality symbol in your
results?
4. Compare your results with the rest of the class.
5. What pattern(s) do you notice in the inequalities?
How do different operations affect inequalities?
Write a rule for inequalities.

272. Let's see what happens when we multiply this inequality
by -1.
5 > -1
-1 • 5 ? -1 • -1
-5 ? 1
-5 < 1
We know 5 is greater than -1
Multiply both sides by -1
Is -5 less than or greater than 1?
You know -5 is less than 1, so you should use <
What happened to the inequality symbol to keep the inequality statement true?

273. Helpful Hint
The direction of the inequality changes only if the number you are using to multiply or divide by is negative.

274. Dividing each side by -3 changes the > to <.
Solve and graph. A.
-3y > 18
-3y < 18
y < -6
Dividing each side by -3 changes the > to <. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

275. Flip the sign because you divided by a negative.
Solve and graph. B.
-7m > -28
-7m < -28
m < 4
Divide each side by -7
Flip the sign because you divided by a negative. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

276. The sign does NOT change because you did not divide by a negative.
Solve and graph. C.
5m > -25
m > -5
Divide each side by 5.
The sign does NOT change because you did not divide by a negative. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

277. Solve and graph. D. -8y > 32 E. -9f > -54 1 2 3 4 5 6 7 8 9 102 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
E. -9f > -54 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answers:
y  -4
f < 6

278. -r 2
< 5 ( ) ( ) -r 2
Multiply both sides by the reciprocal of -1/2. -2
> 5 -2
Why did the inequality change?
r > -10
You multiplied by a negative. 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10

279. Solve and graph each inequality.
Try these.
Solve and graph each inequality.
h < 42 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
2. 4x > -20 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answers:
h > -7
x > -5

280. Solve and graph each inequality.
Try these.
Solve and graph each inequality.
3. 5m < 30 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10 a -2
> -3 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answers:
m < 6
4. a  6

281. 137 Solve and graph. 3y < -6 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -62 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: y < -2

282. Solve and graph. 138 x < -2 -4 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -52 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: x > 8

283. 139 Solve and graph. -5y ≤ -25 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -62 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: y  5

284. 140 Solve and graph. n > 2 -2 1 2 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -52 3 4 5 6 7 8 9 10 -1 -2 -3 -4 -5 -6 -7 -8 -9
-10
Answer: n < -4