This document provides an introduction to solving inequalities. It begins with essential questions and key vocabulary related to inequalities, including set-builder and interval notation. It then shows four examples of solving different types of one-step and multi-step inequalities. Each example solves the inequality, expresses the solution set using set-builder and interval notation, and graphs it on a number line. The examples demonstrate how to isolate the variable, determine if the inequality sign flips, and write the solution set in the two different notations.

1. SOLVING
INEQUALITIES
SECTION 1-2

2. ESSENTIAL QUESTIONS
• How do you solve one step inequalities?
• How do you solve multi-step inequalities?

3. VOCABULARY
1. Set-Builder Notation:
2. Interval Notation:

4. VOCABULARY
1. Set-Builder Notation: The solution set of an
inequality
2. Interval Notation:

5. VOCABULARY
1. Set-Builder Notation: The solution set of an
inequality
2. Interval Notation:
x | x < 3{ }; x | x ≥ −2{ }

6. VOCABULARY
1. Set-Builder Notation: The solution set of an
inequality
2. Interval Notation: An alternate way to write
out the solution set of an inequality
x | x < 3{ }; x | x ≥ −2{ }

7. VOCABULARY
1. Set-Builder Notation: The solution set of an
inequality
2. Interval Notation: An alternate way to write
out the solution set of an inequality
x | x < 3{ }; x | x ≥ −2{ }
(−∞,3); [−2,+∞)

8. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2

9. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y

10. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2

11. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y

12. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5

13. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5
y | y > −5{ }

14. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5
y | y > −5{ }
(−5,+∞)

15. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5
y | y > −5{ }
(−5,+∞)

16. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5
y | y > −5{ }
(−5,+∞)
-5 -4 -3-6-7

17. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5
y | y > −5{ }
(−5,+∞)
-5 -4 -3-6-7

18. EXAMPLE 1
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
4y − 3 < 5y + 2
−4y −4y −2−2
−5 < y
y > −5
y | y > −5{ }
(−5,+∞)
-5 -4 -3-6-7

19. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p

20. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−0.3−0.3

21. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
−0.3−0.3

22. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
−0.3−0.3
p ≥ −40

23. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
p | p ≥ −40{ }
−0.3−0.3
p ≥ −40

24. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
p | p ≥ −40{ }
[−40,+∞)
−0.3−0.3
p ≥ −40

25. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
p | p ≥ −40{ }
[−40,+∞)
−0.3−0.3
p ≥ −40

26. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
p | p ≥ −40{ }
[−40,+∞)
-40-39-38-41-42
−0.3−0.3
p ≥ −40

27. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
p | p ≥ −40{ }
[−40,+∞)
-40-39-38-41-42
−0.3−0.3
p ≥ −40

28. EXAMPLE 2
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
12 ≥ −0.3p
−40 ≤ p
p | p ≥ −40{ }
[−40,+∞)
-40-39-38-41-42
−0.3−0.3
p ≥ −40

29. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2

30. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2

31. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2

32. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x

33. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7

34. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3

35. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3

36. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3
x | x <
7
3
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪

37. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3
x | x <
7
3
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
−∞, 7
3( )

38. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3
x | x <
7
3
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
−∞, 7
3( )

39. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3
7
3
8
3
325
3
x | x <
7
3
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
−∞, 7
3( )

40. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3
7
3
8
3
325
3
x | x <
7
3
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
−∞, 7
3( )

41. EXAMPLE 3
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
−x >
x − 7
2
−2x > x − 7
2(−x) >
x − 7
2
⎛
⎝
⎜
⎞
⎠
⎟ 2
−x−x
−3x > −7
−3x
−3
>
−7
−3
x <
7
3
7
3
8
3
325
3
x | x <
7
3
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
−∞, 7
3( )

42. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?

43. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
g = gallons

44. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
2.59 + 2.59g ≤ 15
g = gallons

45. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
2.59 + 2.59g ≤ 15
g = gallons
−2.59 −2.59

46. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
2.59 + 2.59g ≤ 15
g = gallons
−2.59 −2.59
2.59g ≤ 12.41

47. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
2.59 + 2.59g ≤ 15
g = gallons
−2.59 −2.59
2.59g ≤ 12.41
2.59g
2.59
≤
12.41
2.59

48. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
2.59 + 2.59g ≤ 15
g = gallons
−2.59 −2.59
2.59g ≤ 12.41
2.59g
2.59
≤
12.41
2.59
g ≤ 4.79

49. EXAMPLE 4
Matt Mitarnowski has at most $15 to spend
today. He buys a bag of pretzels and a bottle of
water for $2.59. Matt also needs some gas in
his car. If the cost of gas is $2.59 per gallon,
how many gallons (to the nearest tenth of a
gallon) can Matt buy?
2.59 + 2.59g ≤ 15
g = gallons
−2.59 −2.59
2.59g ≤ 12.41
2.59g
2.59
≤
12.41
2.59
g ≤ 4.79
Matt can buy at most
4.7 gallons

50. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11

51. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11

52. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2

53. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5

54. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and

55. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9

56. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9

57. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }

58. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }
(5,9]

59. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }
(5,9]

60. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }
(5,9]
5 6 7 8 9

61. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }
(5,9]
5 6 7 8 9

62. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }
(5,9]
5 6 7 8 9

63. EXAMPLE 5
Solve the inequality, then graph the solution set
on a number line. Write your solution in both
set-builder notation and interval notation.
7 < x + 2 ≤ 11
x + 2 > 7 and x + 2 ≤ 11
−2 −2 −2 −2
x > 5 and x ≤ 9
5 < x ≤ 9
x | 5 < x ≤ 9{ }
(5,9]
5 6 7 8 9

64. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)

65. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)

66. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)
5 6 7 84

67. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)
5 6 7 84

68. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)
5 6 7 84

69. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)
5 6 7 84

70. EXAMPLE 6
Graph the solution set on a number line.
(−∞,5]∪ (7,+∞)
5 6 7 84

71. PROBLEM SET

72. PROBLEM SET
p. 16 #3-27 multiples of 3, 22, p. 20 #3-18
multiples of 3