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*