The document defines and provides examples of absolute value. Absolute value is the distance of a number from 0, regardless of its direction. It is solved as the number if it is greater than or equal to 0, and as the negative of the number if it is less than 0. Examples of evaluating absolute value expressions are provided. The document also discusses solving absolute value equations and inequalities, providing examples of setting up and solving different absolute value equations for the variable.

1. Absolute Value

2. Absolute Value
 a, a  0
a 
  a, a  0

3. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.

4. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.

5. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5

6. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5

7. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5
(ii ) 6  8  2

8. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5
(ii ) 6  8  2  4
4

9. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5
(ii ) 6  8  2  4
4
(iii )7  6  3  20

10. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5
(ii ) 6  8  2  4
4
(iii )7  6  3  20  7  2
72
5

11. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5 (iv)3 6
(ii ) 6  8  2  4
4
(iii )7  6  3  20  7  2
72
5

12. Absolute Value
 a, a  0
a 
  a, a  0
Absolute value is distance of a number from 0.
It is magnitude only, direction is NOT considered.
e.g.  i  5  5 (iv)3 6  3  6
 18
(ii ) 6  8  2  4
4
(iii )7  6  3  20  7  2
72
5

13. Absolute Value Equations/Inequations

14. Absolute Value Equations/Inequations
e.g.  i  x  7

15. Absolute Value Equations/Inequations
e.g.  i  x  7
x  7 or x  7

16. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7

17. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7

18. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10
x5

19. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2

20. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5

21. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1

22. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1
2 x  6  3 x  1 or   2 x  6   3 x  1

23. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1
2 x  6  3 x  1 or   2 x  6   3 x  1
 x  7
x7

24. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1
2 x  6  3 x  1 or   2 x  6   3 x  1
 x  7 2 x  6  3 x  1
x7 5 x  5
x  1

25. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1
2 x  6  3 x  1 or   2 x  6   3 x  1
 x  7 2 x  6  3 x  1
x7 5 x  5
x  1
(NOT a solution)

26. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1
2 x  6  3 x  1 or   2 x  6   3 x  1
 x  7 2 x  6  3 x  1
x7 5 x  5
x  1
(NOT a solution)
x  7

27. Absolute Value Equations/Inequations
e.g.  i  x  7 (ii ) 2 x  3  7
x  7 or x  7 2 x  3  7 or   2 x  3  7
2 x  10 2 x  3  7
x5 2 x  4
x  2
x  2 or x  5
(iii ) 2 x  6  3 x  1
2 x  6  3 x  1 or   2 x  6   3 x  1
 x  7 2 x  6  3 x  1 Note : the equation
x7 5 x  5  something with pronumerals
x  1 may produce an answer that
(NOT a solution) is not a solution
x  7

28. (iv) x  5  2

29. (iv) x  5  2
x  5  2 or   x  5   2

30. (iv) x  5  2
x  5  2 or   x  5   2
x  3

31. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7

32. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
–7 –3

33. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
–7 –3

34. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3

35. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1

36. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1
3 x  2  1 or   3 x  2   1

37. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1
3 x  2  1 or   3 x  2   1
3 x  1
1
x
3

38. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1
3 x  2  1 or   3 x  2   1
3 x  1 3 x  2  1
1 3 x  3
x
3 x  1

39. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1
3 x  2  1 or   3 x  2   1
3 x  1 3 x  2  1
1 3 x  3
x
3 x  1
–1 1

3

40. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1
3 x  2  1 or   3 x  2   1
3 x  1 3 x  2  1
1 3 x  3
x
3 x  1
–1 1

3

41. (iv) x  5  2
x  5  2 or   x  5   2
x  3 x  5  2
x  7
x  7
7  x  3
–7 –3
(v ) 3 x  2  1
3 x  2  1 or   3 x  2   1
3 x  1 3 x  2  1
1 3 x  3
x
3 x  1 1
 x  1 or x  
3
–1 1

3

42. Absolute Value Graphs

43. Absolute Value Graphs
y  f  x
the part of f  x  below the x axis is reflected above the x axis

44. Absolute Value Graphs
y  f  x
the part of f  x  below the x axis is reflected above the x axis
y f x
the left side of f ( x) disappears and the right side is reflected in the y axis

45. Absolute Value Graphs
y  f  x
the part of f  x  below the x axis is reflected above the x axis
y f x
the left side of f ( x) disappears and the right side is reflected in the y axis
e.g.  i  y  x

46. Absolute Value Graphs
y  f  x
the part of f  x  below the x axis is reflected above the x axis
y f x
the left side of f ( x) disappears and the right side is reflected in the y axis
e.g.  i  y  x
y
x

47. Absolute Value Graphs
y  f  x
the part of f  x  below the x axis is reflected above the x axis
y f x
the left side of f ( x) disappears and the right side is reflected in the y axis
e.g.  i  y  x
y
(–1,1) (1,1)
x

48. (ii ) y  x  2

49. (ii ) y  x  2 y
1. basic curve : y  x
x

50. (ii ) y  x  2 y
1. basic curve : y  x
2. shift left 2 units
2 x

51. (ii ) y  x  2 y
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x

52. (ii ) y  x  2 y
2
1. basic curve : y  x
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2

53. (ii ) y  x  2 y
2
1. basic curve : y  x
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis

54. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2

55. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2
1. basic curve : y  x
y
x

56. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2
1. basic curve : y  x
y
2. shift up 2 units
2
x

57. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2
1. basic curve : y  x
y
2. shift up 2 units (–1,3) (1,3)
2
x

58. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2
1. basic curve : y  x
y
2. shift up 2 units
OR 2
1. basic curve : y  x  2 x

59. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2
1. basic curve : y  x
y
2. shift up 2 units
OR 2
1. basic curve : y  x  2 x
2. reflect left in the y axis

60. (ii ) y  x  2 y
2
1. basic curve : y  x
(–3,1) (–1,1)
2. shift left 2 units
–2 x
OR
1. basic curve : y  x  2
2. reflect up in the x axis
(iii ) y  x  2
1. basic curve : y  x
y
2. shift up 2 units (–1,3) (1,3)
OR 2
1. basic curve : y  x  2 x
2. reflect left in the y axis

61. (iv) y  x 2  2

62. (iv) y  x 2  2 y
1. basic curve : y  x 2  2
 2 2 x
–2

63. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x

64. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2

65. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2
y5 y  x  5
–5 x

66. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2
y5 y  x  5
y2
–5 x

67. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2
y5 y  x  5
(–7,2) (–3,2) y2
–5 x

68. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2
Q: for what values of x is the
absolute value curve below the
y5 y  x  5
line y = 2?
(–7,2) (–3,2) y2
–5 x

69. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2
Q: for what values of x is the
absolute value curve below the
y5 y  x  5
line y = 2?
(–7,2) (–3,2) y2
–5 x
7  x  3

70. (iv) y  x 2  2 y
1. basic curve : y  x 2  2 2
2. reflect up in the x axis
 2 2 x
(v ) x  5  2
Q: for what values of x is the
absolute value curve below the
y5 y  x  5
line y = 2?
(–7,2) (–3,2) y2
–5 x Exercise 3D; 2acfh, 3bdfh, 5adfh,
7bd, 8bdf, 9b i, iii, 11, 14, 18, 21*
7  x  3 Exercise 3E; 1, 2ab, 3ac, 4ac, 5ac,
14, 17c, 20*