The document discusses solving inequalities involving quadratic and rational expressions. For quadratic inequalities, it explains how to factorize, set each factor equal to zero to find critical values, and use these to determine intervals where the parabola is above or below the x-axis. For rational inequalities, it outlines steps to find where the denominator is zero, solve the resulting equality, plot critical values on a number line, and test intervals to determine the solution set. The document provides examples demonstrating these techniques.

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