The document discusses solving inequalities involving quadratic and rational expressions in three steps: (1) identify values where the denominator is zero, (2) solve the equality, (3) plot solutions on a number line and test regions to determine the solution interval or values. For quadratic inequalities, the solutions are always in the form -6 < x < 1 or x < -4 or x > 1.

1. Inequations & Inequalities

2. Inequations & Inequalities
1. Quadratic Inequations
e.g. (i ) x 2  5 x  6  0

3. Inequations & Inequalities
1. Quadratic Inequations
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0

4. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
–6 1 x

5. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
–6 1 x
Q: for what values of x is the
parabola below the x axis?

6. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
–6 1 x
Q: for what values of x is the
parabola below the x axis?

7. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
parabola below the x axis?

8. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?

9. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0

10. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0
y
 x  4  x  1  0
–4 1 x

11. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0
y
 x  4  x  1  0
–4 1 x
Q: for what values of x is the
parabola above the x axis?

12. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0
y
 x  4  x  1  0
–4 1 x
Q: for what values of x is the
parabola above the x axis?

13. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0
y
 x  4  x  1  0
x  4 or x  1
Note: quadratic inequalities –4 1 x
always have solutions in the form Q: for what values of x is the
? < x < ? OR x < ? or x > ? parabola above the x axis?

14. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0
y
 x  4  x  1  0
x  4 or x  1
–4 1 x
Q: for what values of x is the
parabola above the x axis?

15. Inequations & Inequalities
1. Quadratic Inequations y
e.g. (i ) x 2  5 x  6  0
 x  6  x  1  0
6  x  1 –6 1 x
Q: for what values of x is the
(ii )  x 2  3 x  4  0 parabola below the x axis?
x 2  3x  4  0
y
 x  4  x  1  0
x  4 or x  1
Note: quadratic inequalities –4 1 x
always have solutions in the form Q: for what values of x is the
? < x < ? OR x < ? or x > ? parabola above the x axis?

16. 2. Inequalities with Pronumerals in the Denominator

17. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3
x

18. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero
x

19. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x

20. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x
2) Solve the “equality”

21. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3

22. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line

23. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
3

24. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3

25. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
Test x  1 3
1

26. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3
Test x  1 3 Test x  1
1 4
4

27. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3
Test x  1 3 Test x  1
1 4
4

28. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4

29. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
3

30. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5
x3

31. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5
x3
x3 0
x  3

32. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5 2
x3 5
x3
x3 0
x  3

33. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5 2
x3 5
x3
x3 0 2  5 x  15
x  3 5 x  13
13
x
5

34. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5 2
x3 5
x3
x3 0 2  5 x  15 –3 13

x  3 5 x  13 5
13
x
5

35. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5 2
x3 5
x3
x3 0 2  5 x  15 –3 13

x  3 5 x  13 5
13
x
5

36. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5 2
x3 5
x3
x3 0 2  5 x  15 –3 13

x  3 5 x  13 5
13
x
5

37. 2. Inequalities with Pronumerals in the Denominator
1
e.g. (i )  3 1) Find the value where the denominator is zero x  0
x 1
2) Solve the “equality” 3
x 1
x
3
3) Plot these values on a number line
0 1
4) Test regions 3
1
1 1 3 1
Test x  1 3 Test x  1 Test x  1 3
1 4 1
4
1
0  x 
2 3
(ii ) 5 2
x3 5
x3
x3 0 2  5 x  15 –3 13

x  3 5 x  13 5
13
x
13  x  3 or x  
5 5

38. 3. Proving Inequalities

39. 3. Proving Inequalities
(I) Start with a known result

40. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2

41. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz

42. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x

43. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2

44. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left

45. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left
Show that if a  0, b  0 then ab  a 2  b 2   2a 2b 2

46. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left
Show that if a  0, b  0 then ab  a 2  b 2   2a 2b 2
ab  a 2  b 2   2a 2b 2

47. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left
Show that if a  0, b  0 then ab  a 2  b 2   2a 2b 2
ab  a 2  b 2   2a 2b 2  ab  a 2  2ab  b 2 

48. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left
Show that if a  0, b  0 then ab  a 2  b 2   2a 2b 2
ab  a 2  b 2   2a 2b 2  ab  a 2  2ab  b 2 
 ab  a  b 
2

49. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left
Show that if a  0, b  0 then ab  a 2  b 2   2a 2b 2
ab  a 2  b 2   2a 2b 2  ab  a 2  2ab  b 2 
 ab  a  b 
2
0

50. 3. Proving Inequalities
(I) Start with a known result
xz
If x  y  y  z prove y 
2
x y  yz
2 y  z  x
xz
y
2
(II) Move everything to the left
Show that if a  0, b  0 then ab  a 2  b 2   2a 2b 2
ab  a 2  b 2   2a 2b 2  ab  a 2  2ab  b 2 
 ab  a  b 
2
0
 ab  a 2  b 2   2a 2b 2

51. (III) Squares are positive or zero

52. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac

53. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2

54. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2
a 2  2ab  b 2  0

55. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2
a 2  2ab  b 2  0
a 2  b 2  2ab

56. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2
a 2  2ab  b 2  0
a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc

57. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2
a 2  2ab  b 2  0
a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2bc  2ac

58. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2
a 2  2ab  b 2  0
a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2bc  2ac
a 2  b 2  c 2  ab  bc  ac

59. (III) Squares are positive or zero
Show that if a, b and c are positive, then a 2  b 2  c 2  ab  bc  ac
a  b  0
2
a 2  2ab  b 2  0
a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2bc  2ac
a 2  b 2  c 2  ab  bc  ac
Exercise 3A; 4, 6ace, 7bdf, 8bdf, 9, 11, 12, 13ac, 15, 18bcd, 22, 24