The document outlines a series of mathematical inequalities and equations that need to be solved, using methods such as the wavy curve method. Each inequality is paired with specific solutions that define the ranges of x in set notation. The document serves as a guide for solving inequalities through various mathematical techniques.

1. Wavy curve
method S1- DPP

2. Q1. Solve the inequality 5x + 12 ≥ 32

3. Q2. Solve the inequality 7(3x - 2) - 5 < 8(x - 1)

4. Q3. Solve the inequality

5. Q4. Solve for x : (x - 6) (x - 1) > 0

6. Q5. Solve for x : (11x - 6) (2x - 3) ≤ 0

7. Q6. Solve the inequality x2
- 4x +3 ≥ 0

8. Q7. Solve the inequality x2
+ 3x - 28 ≤ 0

9. x ∈(-∞, 0) ∪ (5, ∞)
x ∈(-∞, 0) ∪ (5,
6)
x ∈ (-1, 6)
x ∈(-∞, -1] ∪ [6, ∞)
A
B
D
C
A
B
D
C
D
Q8. Solve for x : x ( x - 5) ≥ 6

10. (-∞, -2] ∪ [2, ∞)
(-∞, -2] ∪ [0, 2]
None of these
[-2, 0] ∪[2, ∞)
A
B
D
C
A
B
D
C
D
Q9. Solve for x: x(x2
- 4) ≤ 0
B

11. x ∈ (2, 5)
x ∈ [2, 5]
x ∈ (-∞, 2) ∪ (5, ∞)
x ∈(-∞, 2] ∪ [5, ∞)
A
B
D
C
A
B
D
C
D
Q10. Solve for x : (2 - x) (x - 5) ≥ 0

12. Q11. Solve to x: (x - 2) (5x - 3) (x + 3) ≤ 0

13. Q12. Solve the inequality (x + 3) (x - 2) ≤ (x - 2)

14. Q13. Solve for x:

15. Q14. Solve

16. Q15. Solve

17. x ∈ (-∞, -5] ∪ [-2, 0] ∪ [1, ∞)
x ∈ (-∞, -5] ∪ (-2, 0) ∪ [1, ∞)
x ∈ (-∞, -5] ∪ (-2, 0) ∪ (1, ∞)
x ∈ (-∞, -5] ∪ [1, ∞)
A
B
D
C
A
B
D
C
D
Q16. Solve for x:

18. x ∈ (-∞, -2] ∪ (5, ∞)
x ∈ (-∞, -2] ∪ (5, ∞) ∪ {2}
None of these
x ∈ [-2, -5]
A
B
D
C
A
B
D
C
D
Q17. Solve for x : (x + 2)(x - 1)2
(x - 5) ≥ 0

19. Q18. Solve for x:

20. Q19. find the values of x.

21. x ∈ [-3, -1] ∪ [2, ∞)
x ∈ (-∞, -3] ∪ (2,
∞)
x ∈ [-3, -1] ∪ [-1, 2]
x ∈ (-∞, -3] ∪(2, ∞) ∪ {-1}
A
B
D
C
A
B
D
C
D
Q20. Solve for x :

22. Q21. find the values of x.

23. Q22. Solver for x:

24. Q23. Solve for x:

25. Q24. Solve for x:

26. x ∈ (-7, -4) ∪ (5, ∞) ∪ {-1, 0,
2}
x ∈ (-∞, -4) ∪ (5, -∞) ∪ {-1, 0,
-2}
None of these
x ∈ (-∞, -4) ∪ (5, ∞)
A
B
D
C
A
B
D
C
D
Q25. Solve for x:

27. Q26. Find the set of all x for which

28. Wavy curve
method S1- DPP
solutions

29. Q1. Solve the inequality 5x + 12 ≥ 32

30. Solution:

31. Q2. Solve the inequality 7(3x - 2) - 5 < 8(x - 1)

32. Solution:

33. Q3. Solve the inequality

34. Solution:

35. Q4. Solve for x : (x - 6) (x - 1) > 0

36. Solution:
x ∈ (-∞, 1) ∪ (6, ∞)
+ - +
-∞ 1 6 +∞
(x - 6) (x - 1) > 0
x = 1, 6

37. Q5. Solve for x : (11x - 6) (2x - 3) ≤ 0

38. + - +
-∞ +∞
Solution:
(11x - 6) (2x - 3) ≤ 0

39. Q6. Solve the inequality x2
- 4x +3 ≥ 0

40. (x - 3) (x - 1) ≥ 0
x = 1, 3
x ∈ (-∞, 1] ∪ [3, ∞)
Solution:
+ - +
1 3
x2
- 4x +3 ≥ 0

41. Q7. Solve the inequality x2
+ 3x - 28 ≤ 0

42. Solution:
+ - +
-7 4
(x+ 7) (x- 4) ≤ 0
x = -7, 4
x ∈ [-7, 4]
x2
+ 3x - 28 ≤ 0

43. x ∈(-∞, 0) ∪ (5, ∞)
x ∈(-∞, 0) ∪ (5,
6)
x ∈ (-1, 6)
x ∈(-∞, -1] ∪ [6, ∞)
A
B
D
C
A
B
D
C
D
Q8. Solve for x : x ( x - 5) ≥ 6

44. Solution:
x2
- 5x - 6 ≥ 0
(x -6) (x + 1) ≥ 0
+ - +
-1 6
x ∈(-∞, -1] ∪ [6, ∞)

45. (-∞, -2] ∪ [2, ∞)
(-∞, -2] ∪ [0, 2]
None of these
[-2, 0] ∪[2, ∞)
A
B
D
C
A
B
D
C
D
Q9. Solve for x: x(x2
- 4) ≤ 0
B

46. Solution:
(x) (x - 2) (x + 2) ≤ 0
x ∈ (-∞, -2] ∪ [0, 2]
x(x2
- 4) ≤ 0
- + - +
-∞ -2 0 2 +∞

47. x ∈ (2, 5)
x ∈ [2, 5]
x ∈ (-∞, 2) ∪ (5, ∞)
x ∈(-∞, 2] ∪ [5, ∞)
A
B
D
C
A
B
D
C
D
Q10. Solve for x : (2 - x) (x - 5) ≥ 0

48. + +
-
-∞ 2 5 +∞
Solution:

49. Q11. Solve to x: (x - 2) (5x - 3) (x + 3) ≤ 0

50. Solution:
+ - - +
-∞ -3 2 +∞
(x-2) (5x - 3) (x + 3) ≤ 0

51. Q12. Solve the inequality (x + 3) (x - 2) ≤ (x - 2)

52. Solution:
(x + 3) (x - 2) ≤ (x - 2)
(x + 3) (x - 2) - (x - 2) ≤ 0
(x - 2) (x + 2) ≤ 0
x = -2, 2
∴ x ∈ [-2, 2]
+ - +
-2 2

53. Q13. Solve for x:

54. Solution:
+
–
+
–
x ∈ (–∞ ,–1) ∪ (1,
2)
1 2
–1

55. Q14. Solve

56. Solution:
- + - +
-∞ 1 2 3 +∞

57. Q15. Solve

58. Solution:
- + - +
-∞ -4 2 +∞

59. x ∈ (-∞, -5] ∪ [-2, 0] ∪ [1, ∞)
x ∈ (-∞, -5] ∪ (-2, 0) ∪ [1, ∞)
x ∈ (-∞, -5] ∪ (-2, 0) ∪ (1, ∞)
x ∈ (-∞, -5] ∪ [1, ∞)
A
B
D
C
A
B
D
C
D
Q16. Solve for x:

60. Solution:
+ - + - +
-5 -2 0 1
x ∈ (-∞, -5] ∪ (-2, 0) ∪ [1, ∞)

61. x ∈ (-∞, -2] ∪ (5, ∞)
x ∈ (-∞, -2] ∪ (5, ∞) ∪ {2}
None of these
x ∈ [-2, -5]
A
B
D
C
A
B
D
C
D
Q17. Solve for x : (x + 2)(x - 1)2
(x - 5) ≥ 0

62. Solution:
x ∈(-∞, -2] ∪ [5, ∞) ∪ {1}
+ - - +
-∞ -2 1 5 +∞
(x + 2)(x - 1)2
(x - 5) ≥ 0

63. Q18. Solve for x:

64. Solution:
We can write ( 2 − x )3
= (− 1 (x − 2) )3
= − ( x − 2 )3
− ( x −2 )3
( x – 21)6
( 2x – 3 ) 2
≥ 0
( x −2 )3
( x – 21)6
( 2x – 3 ) 2
≤ 0

65. Solution:
3/2 2 +
– 21
( x −2 )3
( x – 21)6
( 2x – 3 ) 2
≤ 0
+
–
x ∈ ( − ∞ , 3/2 ) ∪ ( 3/2 , 2
]
∪
{21}

66. Q19. find the values of x.

67. Solution:
+ - - +
0 1 3

68. x ∈ [-3, -1] ∪ [2, ∞)
x ∈ (-∞, -3] ∪ (2,
∞)
x ∈ [-3, -1] ∪ [-1, 2]
x ∈ (-∞, -3] ∪(2, ∞) ∪ {-1}
A
B
D
C
A
B
D
C
D
Q20. Solve for x :

69. Solution:
+ - - +
-3 -1 2
x ∈ (-∞, -3] ∪(2, ∞) ∪ {-1}

70. Q21. find the values of x.

71. Solution:
- + - -
0 2 4
+ +

72. Q22. Solver for x:

73. Solution:
2
1 3 +
– + 4
(x-4)1
–
(x-3)23
(x-2)100
(x-1)2001
+
x ∈ (1,2) ∪ (2, 3) ∪ (4, ∞
)

74. Q23. Solve for x:

75. Solution:
x ∈ (-∞, 1] ∪ [2, 4) ∪ (4, 5)
-∞ - + - - - + +∞
1 2 3 4 5

76. Q24. Solve for x:

77. Solution:
x ∈ (1, 2) ∪ (5, +∞)
-∞ - + - - - + +∞
1 2 3 4 5

78. x ∈ (-7, -4) ∪ (5, ∞) ∪ {-1, 0,
2}
x ∈ (-∞, -4) ∪ (5, -∞) ∪ {-1, 0,
-2}
None of these
x ∈ (-∞, -4) ∪ (5, ∞)
A
B
D
C
A
B
D
C
D
Q25. Solve for x:

79. Solution:
- + - - - - +
-7 -4 -1 0 2 5
x ∈ (-7, -4) ∪ (5, ∞) ∪ {-1, 0,
2}

80. Q26. Find the set of all x for which

81. Solution:
+ - + - +
-∞ -2 -1 +∞

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