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?
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
x3
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
x3
x3 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
x3 5
x3
x3 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
x3 5
x3
x3 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
x3 5
x3
x3 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
x3 5
x3
x3 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
x3 5
x3
x3 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
x3 5
x3
x3 0 2 5 x 15 –3 13
x 3 5 x 13 5
13
x
13 x 3 or x
5 5
42. 3. Proving Inequalities
(I) Start with a known result
xz
If x y y z prove y
2
x y yz
2 y z x
43. 3. Proving Inequalities
(I) Start with a known result
xz
If x y y z prove y
2
x y yz
2 y z x
xz
y
2
44. 3. Proving Inequalities
(I) Start with a known result
xz
If x y y z prove y
2
x y yz
2 y z x
xz
y
2
(II) Move everything to the left
45. 3. Proving Inequalities
(I) Start with a known result
xz
If x y y z prove y
2
x y yz
2 y z x
xz
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
xz
If x y y z prove y
2
x y yz
2 y z x
xz
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
xz
If x y y z prove y
2
x y yz
2 y z x
xz
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
xz
If x y y z prove y
2
x y yz
2 y z x
xz
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
xz
If x y y z prove y
2
x y yz
2 y z x
xz
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
xz
If x y y z prove y
2
x y yz
2 y z x
xz
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
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