1. 1
Venn diagram
A U
C
B
• The universal set U is usually represented by a rectangle.
• Inside this rectangle, subsets of the universal set are represented by
geometrical figures.

2. 2
Venn diagrams help us identify some useful formulas in set operations.

3. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
• To represent (A ∩ C) ∪ (B ∩ C):

4. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):

5. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):

6. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):

7. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):
A
C
B

8. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):
A
C
B

9. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):
A
C
B

10. 2
Venn diagrams help us identify some useful formulas in set operations.
• To represent (A ∪ B) ∩ C :
A
C
B
• To represent (A ∩ C) ∪ (B ∩ C):
A
C
B
To prove (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) in a rigorous manner, should use
formal mathematical logic.

11. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces

12. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces
Example {1, 3, 5, 7, 9}

13. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces
Example {1, 3, 5, 7, 9}
• Because intervals have infinitely many elements , we can’t use listing method
to express intervals. We introduce new notations:

14. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces
Example {1, 3, 5, 7, 9}
• Because intervals have infinitely many elements , we can’t use listing method
to express intervals. We introduce new notations:
square bracket means endpoint included
round bracket means endpoint excluded

15. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces
Example {1, 3, 5, 7, 9}
• Because intervals have infinitely many elements , we can’t use listing method
to express intervals. We introduce new notations:
square bracket means endpoint included
round bracket means endpoint excluded
Example [7, 11], (−2, 5]

16. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces
Example {1, 3, 5, 7, 9}
• Because intervals have infinitely many elements , we can’t use listing method
to express intervals. We introduce new notations:
square bracket means endpoint included
round bracket means endpoint excluded
Example [7, 11], (−2, 5]
Note The interval [7, 11] contains ALL numbers between 7 and 11 (including
integers, rational numbers, irrational numbers)

17. 3
A few remarks about set notations
• If a set has finitely many elements , use the listing method to express the set
write down all the elements
enclosed the elements by braces
Example {1, 3, 5, 7, 9}
• Because intervals have infinitely many elements , we can’t use listing method
to express intervals. We introduce new notations:
square bracket means endpoint included
round bracket means endpoint excluded
Example [7, 11], (−2, 5]
Note The interval [7, 11] contains ALL numbers between 7 and 11 (including
integers, rational numbers, irrational numbers)
For example, 10 ∈ [7, 11]
9.123 ∈ [7, 11]
√
50 ∈ [7, 11]

18. 4
Inequalities
To solve an inequality (or inequalities) in an unknown x means to find all real
numbers x such that the inequality is satisfied .
The set of all such x is called the solution set to the inequality.

19. 4
Inequalities
To solve an inequality (or inequalities) in an unknown x means to find all real
numbers x such that the inequality is satisfied .
The set of all such x is called the solution set to the inequality.
Polynomial inequalities
an xn + an−1 xn−1 + · · · + a1 x + a0 < 0 ( or > 0, or ≤ 0, or ≥ 0)
where n ≥ 1 and an 0.

20. 4
Inequalities
To solve an inequality (or inequalities) in an unknown x means to find all real
numbers x such that the inequality is satisfied .
The set of all such x is called the solution set to the inequality.
Polynomial inequalities
an xn + an−1 xn−1 + · · · + a1 x + a0 < 0 ( or > 0, or ≤ 0, or ≥ 0)
where n ≥ 1 and an 0.
(1) n = 1 Linear inequalities
(2) n = 2 Quadratic inequalities
(3) n ≥ 3 Higher degree inequalities

21. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9

22. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.

23. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x

24. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x
2x ≤ 3 − 1

25. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x
2x ≤ 3 − 1
x ≤ 1

26. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1
x ≤ 1

27. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1

28. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1 −3 ≤ x

29. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1 −3 ≤ x
Solution set = {x ∈ R : x ≤ 1 and − 3 ≤ x}

30. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1 −3 ≤ x
Solution set = {x ∈ R : x ≤ 1 and − 3 ≤ x}
= {x ∈ R : −3 ≤ x ≤ 1}

31. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1 −3 ≤ x
Solution set = {x ∈ R : x ≤ 1 and − 3 ≤ x}
= {x ∈ R : −3 ≤ x ≤ 1}
= [−3, 1]

32. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1 −3 ≤ x
Solution set = {x ∈ R : x ≤ 1 and − 3 ≤ x}
= {x ∈ R : −3 ≤ x ≤ 1}
= [−3, 1]
Be careful 1 ≤ 3 − 2x
1 − 3 ≤ −2x

33. 5
Example Find the solution set to the following compound inequality:
1 ≤ 3 − 2x ≤ 9
Solution The inequality means 1 ≤ 3 − 2x and 3 − 2x ≤ 9.
Solving separately: 1 ≤ 3 − 2x 3 − 2x ≤ 9
2x ≤ 3 − 1 3 − 9 ≤ 2x
x ≤ 1 −3 ≤ x
Solution set = {x ∈ R : x ≤ 1 and − 3 ≤ x}
= {x ∈ R : −3 ≤ x ≤ 1}
= [−3, 1]
Be careful 1 ≤ 3 − 2x
1 − 3 ≤ −2x
−2
≥ x
−2

34. 6
Example Find the solution set to the following:
2x + 1 < 3 and 3x + 10 < 4

35. 6
Example Find the solution set to the following:
2x + 1 < 3 and 3x + 10 < 4
Solution Solve separately: 2x + 1 < 3
2x < 2
x < 1

36. 6
Example Find the solution set to the following:
2x + 1 < 3 and 3x + 10 < 4
Solution Solve separately: 2x + 1 < 3 3x + 10 < 4
2x < 2 3x < −6
x < 1 x < −2

37. 6
Example Find the solution set to the following:
2x + 1 < 3 and 3x + 10 < 4
Solution Solve separately: 2x + 1 < 3 3x + 10 < 4
2x < 2 3x < −6
x < 1 x < −2
Solution set = {x ∈ R : x < 1 and x < −2}

38. 6
Example Find the solution set to the following:
2x + 1 < 3 and 3x + 10 < 4
Solution Solve separately: 2x + 1 < 3 3x + 10 < 4
2x < 2 3x < −6
x < 1 x < −2
Solution set = {x ∈ R : x < 1 and x < −2}
= {x ∈ R : x < −2}

39. 6
Example Find the solution set to the following:
2x + 1 < 3 and 3x + 10 < 4
Solution Solve separately: 2x + 1 < 3 3x + 10 < 4
2x < 2 3x < −6
x < 1 x < −2
Solution set = {x ∈ R : x < 1 and x < −2}
= {x ∈ R : x < −2}
= (−∞, −2)

40. 7
Example Find the solution set to the following:
2x + 1 > 9 and 3x + 4 < 10

41. 7
Example Find the solution set to the following:
2x + 1 > 9 and 3x + 4 < 10
Solution Solve separately: 2x + 1 > 9
2x > 8
x > 4

42. 7
Example Find the solution set to the following:
2x + 1 > 9 and 3x + 4 < 10
Solution Solve separately: 2x + 1 > 9 3x + 4 < 10
2x > 8 3x < 6
x > 4 x < 2

43. 7
Example Find the solution set to the following:
2x + 1 > 9 and 3x + 4 < 10
Solution Solve separately: 2x + 1 > 9 3x + 4 < 10
2x > 8 3x < 6
x > 4 x < 2
Solution set = {x ∈ R : x > 4 and x < 2}

44. 7
Example Find the solution set to the following:
2x + 1 > 9 and 3x + 4 < 10
Solution Solve separately: 2x + 1 > 9 3x + 4 < 10
2x > 8 3x < 6
x > 4 x < 2
Solution set = {x ∈ R : x > 4 and x < 2}
= ∅

45. 8
Wording
(1) Find the solution(s) to the inequality 2x − 1 > 0.
(2) Find the solution set to the inequality 2x − 1 > 0.

46. 8
Wording
(1) Find the solution(s) to the inequality 2x − 1 > 0.
(2) Find the solution set to the inequality 2x − 1 > 0.
Answer
1
(1) Solutions x> 2

47. 8
Wording
(1) Find the solution(s) to the inequality 2x − 1 > 0.
(2) Find the solution set to the inequality 2x − 1 > 0.
Answer
1
(1) Solutions x> 2
1 1
(2) Solution set x∈R:x> 2
= 2
,∞

48. 8
Wording
(1) Find the solution(s) to the inequality 2x − 1 > 0.
(2) Find the solution set to the inequality 2x − 1 > 0.
Answer
1
(1) Solutions x> 2
1 1
(2) Solution set x∈R:x> 2
= 2
,∞
• Solve the inequality 2x − 1 > 0.
Can give solution or solution set.

49. 9
Quadratic Inequalities ax2 + bx + c > 0

50. 9
Quadratic Inequalities ax2 + bx + c > 0
ax2 + bx + c ≥ 0
ax2 + bx + c < 0
ax2 + bx + c ≤ 0

51. 9
Quadratic Inequalities ax2 + bx + c > 0
ax2 + bx + c ≥ 0
ax2 + bx + c < 0
ax2 + bx + c ≤ 0
Properties of real numbers
(1) m > 0 and n > 0 =⇒ m · n > 0

52. 9
Quadratic Inequalities ax2 + bx + c > 0
ax2 + bx + c ≥ 0
ax2 + bx + c < 0
ax2 + bx + c ≤ 0
Properties of real numbers
(1) m > 0 and n > 0 =⇒ m · n > 0
(2) m < 0 and n < 0 =⇒ m · n > 0

53. 9
Quadratic Inequalities ax2 + bx + c > 0
ax2 + bx + c ≥ 0
ax2 + bx + c < 0
ax2 + bx + c ≤ 0
Properties of real numbers
(1) m > 0 and n > 0 =⇒ m · n > 0
(2) m < 0 and n < 0 =⇒ m · n > 0
(3) m > 0 and n < 0 =⇒ m · n < 0

54. 9
Quadratic Inequalities ax2 + bx + c > 0
ax2 + bx + c ≥ 0
ax2 + bx + c < 0
ax2 + bx + c ≤ 0
Properties of real numbers
(1) m > 0 and n > 0 =⇒ m · n > 0
(2) m < 0 and n < 0 =⇒ m · n > 0
(3) m > 0 and n < 0 =⇒ m · n < 0
From these we get
(4) m · n > 0 ⇐⇒ (m > 0 and n > 0) or (m < 0 and n < 0)

55. 9
Quadratic Inequalities ax2 + bx + c > 0
ax2 + bx + c ≥ 0
ax2 + bx + c < 0
ax2 + bx + c ≤ 0
Properties of real numbers
(1) m > 0 and n > 0 =⇒ m · n > 0
(2) m < 0 and n < 0 =⇒ m · n > 0
(3) m > 0 and n < 0 =⇒ m · n < 0
From these we get
(4) m · n > 0 ⇐⇒ (m > 0 and n > 0) or (m < 0 and n < 0)
(5) m · n < 0 ⇐⇒ (m > 0 and n < 0) or (m < 0 and n > 0)

56. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0

57. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial

58. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial
x2 + 2x − 15 > 0
(x + 5)(x − 3) > 0

59. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial
x2 + 2x − 15 > 0
(x + 5)(x − 3) > 0
Apply Rule (4)
(x + 5 > 0 and x − 3 > 0) or (x + 5 < 0 and x − 3 < 0)

60. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial
x2 + 2x − 15 > 0
(x + 5)(x − 3) > 0
Apply Rule (4)
(x + 5 > 0 and x − 3 > 0) or (x + 5 < 0 and x − 3 < 0)
(x > −5 and x > 3) or (x < −5 and x < 3)

61. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial
x2 + 2x − 15 > 0
(x + 5)(x − 3) > 0
Apply Rule (4)
(x + 5 > 0 and x − 3 > 0) or (x + 5 < 0 and x − 3 < 0)
(x > −5 and x > 3) or (x < −5 and x < 3)
x>3 or x < −5

62. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial
x2 + 2x − 15 > 0
(x + 5)(x − 3) > 0
Apply Rule (4)
(x + 5 > 0 and x − 3 > 0) or (x + 5 < 0 and x − 3 < 0)
(x > −5 and x > 3) or (x < −5 and x < 3)
x>3 or x < −5
Solution set = {x ∈ R : x < −5 or x > 3}

63. 10
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Solution
Method 1 Factorize the quadratic polynomial
x2 + 2x − 15 > 0
(x + 5)(x − 3) > 0
Apply Rule (4)
(x + 5 > 0 and x − 3 > 0) or (x + 5 < 0 and x − 3 < 0)
(x > −5 and x > 3) or (x < −5 and x < 3)
x>3 or x < −5
Solution set = {x ∈ R : x < −5 or x > 3}
= (−∞, −5) ∪ (3, ∞)

64. 11
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 2 Graphical method

65. 11
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 2 Graphical method
• Graph of y = x2 + 2x − 15

66. 11
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 2 Graphical method
• Graph of y = x2 + 2x − 15
-5 3

67. 11
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 2 Graphical method
• Graph of y = x2 + 2x − 15
-5 3
• To solve the inequality x2 + 2x − 15 > 0 means
to find all (real numbers) x such that y > 0

68. 11
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 2 Graphical method
• Graph of y = x2 + 2x − 15
-5 3
• To solve the inequality x2 + 2x − 15 > 0 means
to find all (real numbers) x such that y > 0
• Solution set: (−∞, −5) ∪ (3, ∞)

69. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3

70. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +

71. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5
x−3
(x + 5)(x − 3)

72. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 0
x−3
(x + 5)(x − 3)

73. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 0 + + +
x−3
(x + 5)(x − 3)

74. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3
(x + 5)(x − 3)

75. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 0
(x + 5)(x − 3)

76. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 0 +
(x + 5)(x − 3)

77. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3)

78. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) +

79. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0

80. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 −

81. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0

82. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +

83. 12
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
(x + 5)(x − 3) + 0 − 0 +
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Solution set: (−∞, −5) ∪ (3, ∞)

84. 13
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Steps

85. 13
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Steps
• Factorize left-side x2 + 2x − 15 = (x + 5)(x − 3)

86. 13
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Steps
• Factorize left-side x2 + 2x − 15 = (x + 5)(x − 3)
• Zeros of left-side −5 and 3

87. 13
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Steps
• Factorize left-side x2 + 2x − 15 = (x + 5)(x − 3)
• Zeros of left-side −5 and 3
• Divide real number line into three parts: (−∞, −5), (−5, 3), (3, ∞)

88. 13
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Steps
• Factorize left-side x2 + 2x − 15 = (x + 5)(x − 3)
• Zeros of left-side −5 and 3
• Divide real number line into three parts: (−∞, −5), (−5, 3), (3, ∞)
• On each of these intervals, determine the sign of (x + 5) and (x − 3),

89. 13
Example Find the solution set to the inequality
x2 + 2x − 15 > 0
Method 3
x < −5 x = −5 −5 < x < 3 x=3 x>3
x+5 − 0 + + +
x−3 − − − 0 +
(x + 5)(x − 3) + 0 − 0 +
Steps
• Factorize left-side x2 + 2x − 15 = (x + 5)(x − 3)
• Zeros of left-side −5 and 3
• Divide real number line into three parts: (−∞, −5), (−5, 3), (3, ∞)
• On each of these intervals, determine the sign of (x + 5) and (x − 3),
hence the sign of (x + 5)(x − 3)

90. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0

91. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0
≥ 0
< 0
≤ 0

92. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0
≥ 0
< 0
Method 1 2n−1 cases ≤ 0
For example, n = 3: a·b·c>0
4 cases: +++ +−− −+− −−+

93. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0
≥ 0
< 0
Method 1 2n−1 cases ≤ 0
For example, n = 3: a·b·c>0
4 cases: +++ +−− −+− −−+
n = 4: a·b·c·d >0
8 cases: + + ++ + + −− − + +− − − ++
+ − +− − + −+ + − −+ − − −−

94. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0
≥ 0
< 0
Method 1 2n−1 cases ≤ 0
For example, n = 3: a·b·c>0
4 cases: +++ +−− −+− −−+
n = 4: a·b·c·d >0
8 cases: + + ++ + + −− − + +− − − ++
+ − +− − + −+ + − −+ − − −−
Method 2 Need graphs polynomials of degrees ≥ 3

95. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0
≥ 0
< 0
Method 1 2n−1 cases ≤ 0
For example, n = 3: a·b·c>0
4 cases: +++ +−− −+− −−+
n = 4: a·b·c·d >0
8 cases: + + ++ + + −− − + +− − − ++
+ − +− − + −+ + − −+ − − −−
Method 2 Need graphs polynomials of degrees ≥ 3
Method 3 By table

96. 14
Polynomial inequalities (degree ≥ 3)
an xn + an−1 xn−1 + · · · a1 x + a0 > 0
≥ 0
< 0
Method 1 2n−1 cases ≤ 0
For example, n = 3: a·b·c>0
4 cases: +++ +−− −+− −−+
n = 4: a·b·c·d >0
8 cases: + + ++ + + −− − + +− − − ++
+ − +− − + −+ + − −+ − − −−
Method 2 Need graphs polynomials of degrees ≥ 3
Method 3 By table
ALL three methods need to factorize L.S.

97. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0

98. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)

99. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).

100. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)

101. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)

102. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
−3 −2 2
x−2
x+2
x+3
p(x)

103. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2
x+2
x+3
p(x)

104. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 0
x+2 0
x+3 0
p(x) 0 0 0

105. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 0 +
x+2 0
x+3 0
p(x) 0 0 0

106. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 0
x+3 0
p(x) 0 0 0

107. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 0 + + +
x+3 0
p(x) 0 0 0

108. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 0
p(x) 0 0 0

109. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 0 + + + + +
p(x) 0 0 0

110. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) 0 0 0

111. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) − 0 0 0

112. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) − 0 + 0 0

113. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) − 0 + 0 − 0

114. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) − 0 + 0 − 0 +

115. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) − 0 + 0 − 0 +
• Solution: x < −3 or x = −3 or x = −2 or −2 < x < 2 or x = 2, that is,

116. 15
Example Solve the following inequality x3 + 3x2 − 4x − 12 ≤ 0
Solution Denote p(x) = x3 + 3x2 − 4x − 12.
• Factorize p(x)
p(2) = 23 + 3(22) − 4(2) − 12 = 0
By Factor Theorem, (x − 2) is a factor of p(x).
Long division or compare coeffficients p(x) = (x − 2)(x2 + 5x + 6)
= (x − 2)(x + 2)(x + 3)
•
x < −3 −3 −3 < x < −2 −2 −2 < x < 2 2 2<x
x−2 − − − − − 0 +
x+2 − − − 0 + + +
x+3 − 0 + + + + +
p(x) − 0 + 0 − 0 +
• Solution: x < −3 or x = −3 or x = −2 or −2 < x < 2 or x = 2, that is,
x ≤ −3 or − 2 ≤ x ≤ 2.