QUADRATIC INEQUALITIES

Definition
Quadratic inequalities in one variable are inequalities which can be written in one of
the following forms: ax 2 + bx + c > 0 ,
ax 2 + bx + c < 0 ,
ax 2 + bx + c ≥ 0 or
ax 2 + bx + c ≤ 0 where a, b and c are real numbers.
Procedure
Solving Quadratic Inequalities
1. Move all terms to one side.
2. Simplify and factor the quadratic expression.
3. Find the roots of the corresponding quadratic equation.
4. Use the roots to divide the number line into regions.
5. Test each region using theinequality.
Example 1 Solve the inequality, x2 > x + 2 .
Solution x2 > x + 2
x2 − x − 2 > 0
(x - 2)(x + 1) > 0
The corresponding equation is (x - 2)(x + 1) = 0 so…
x - 2 = 0 or x + 1 = 0
x = 2 x = -1
I II III
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Now we test one point in each region.
Region Test Point Inequality Status
I x = -2 (x - 2)(x + 1) = (-2 - 2)(-2 + 1) = 4 > 0 True
II x = 0 (x - 2)(x + 1) = (0 - 2)(0 + 1) = -2 > 0 False
III x = 3 (x - 2)(x + 1) = (3 - 2)(3 + 1) = 4 > 0 True
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
So the solution to this inequality is x < -1 or x > 2.
Copyright©2007 by Lawrence Perez and Patrick Quigley
Example 2 Solve the inequality, (x + 3)2 ≥ 2(x2 + 7).
Solution (x + 3)2 ≥ 2(x2 + 7)
x2 + 6x + 9 ≥ 2x2 +14
− x2 + 6x − 5 ≥ 0
−(x2 − 6x + 5) ≥ 0
−(x2 − 6x + 5)
−1
≤
0
−1
x2 − 6x + 5 ≤ 0
(x - 1)(x - 5) ≤ 0
The corresponding equation is (x - 1)(x - 5) = 0 so…
x - 1 = 0 or x - 5 = 0
x = 1 x = 5
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
I II III
Now we check one point in each region.
Region Test Point Inequality Status
I x = 0 (x - 1)(x - 5) = (0 - 1)(0 - 5) = 5 < 0 False
II x = 2 (x - 1)(x - 5) = (2 - 1)(2 - 5) = -3 < 0 True
III x = 6 (x - 1)(x - 5) = (6 - 1)(6 - 5) = 5 < 0 False
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
So the solution to this inequality is 1 ≤ x ≤ 5.
Copyright©2007 by Lawrence Perez and Patrick Quigley
§4-2 PROBLEM SET
Solve each quadratic inequality, and graph the solution on a number line.
1. y2 −17y + 70 < 0 2. x2 + 9x + 13 > −7
3. x(x + 1) >112 − 5x 4. a2 + 3a + 2 < −3(a + 2)
5. 2x2 ≤ 5x − 2 6. 10 − 9y ≥ −2y2
7. b(b + 3) ≥ −2 8. a2 ≤ 4(2a − 3)
9. y2 −17y + 70 <0 10. x2 + 9x + 13 > −7
11. x(x + 1) >112 − 5x 12. a2 + 25 <10a
13. 2d2 + 5d ≤ 12 14. a2 + 3a + 2 ≥ −3(a + 2)
15. 10 − 9y ≥ −2y2 16. 2x2 ≤ 5x − 2
17. c(c + 4) < 3 + 3(9 + c) 18. 2a(a + 6) > 5 − a(a + 2)
19. b(b + 3) > −2 20. a2 < 4(2a − 3)
21. (x + 3)2 ≤ 6(x + 15) 22. 2x2 + 7 ≥ 9x
23. 7x2 ≥ 4(1+ 3x) 24. 3x 2 + 7x ≤ −2
25. −8 < 4(x − x2 ) 26. x2 − x − 2 > 0
27. 2k2 + 3k − 2 > 0 28. t2 + 2t − 3 < 0
29. 4x 2 + 8 ≤ 33x 30. x2 ≥ 4(x − 5)
31. x2 + 4 ≥ 2x2 − 3x 32. 10 − 3x ≤ x2
33. 4 < 13x − 3x2 34. 6(x2 + 1) > −13
35. 6x − x2 > 8 36. 20a2 <1 − a
37. 8x ≤ −3(1 − x 2) 38. y2 ≥ 25
39. t2 +18 ≥ 11t 40. 3x(x + 1) ≤ x(x + 5)
41. x2 < 8 42. x2 + 3x > 12
43. 2t 2 > 9t +18 44. 4x 2 − 9x + 2 < 0

 PROBLEM SOLUTIONS
1. 7 < y < 10 2. x < -5 or x > -4 3. x < -14 or x > 8
4. -4 < a < -2 5.
1
2
≤ x ≤ 2 6. y ≤ 2 or y ≥ 5
2
7. b ≤ -2 or b ≥ -1 8. 2 ≤ a ≤ 6 9. 7 < y < 10
10. x < -5 or x > - 11. x < -14 or x > 8 12. no solution
13. −4 ≤ d ≤ 3
2
14. x ≤ -4 or x ≥ -2 15. y ≤ 2 or y ≥ 5
2
16.
1
2
≤ x ≤ 2 17. -6 < c < 5 18. a < -5 or a > 1
3
19. b ≤ -2 or b ≥ -1 20. 2 < a < 6 21. -9 ≤ x ≤ 9
22. x < 1 or x ≥ 7
2
23. x ≤ − 2
7
or x ≥ 2 24. −2 ≤ x ≤ − 1
3
25. -1 < x < 2 26. x < -1 or x > 2 27. k < -2 or k > 1
2
28. -3 < t < 1 29.
1
4
≤ x ≤ 8 30. all real numbers
31. -1 < x < 4 32. x ≤ -5 or x ≥ 2 33.
1
3
< x < 4
34. no real solutions 35. 2 < x < 4 36. − 1
4
< x < 1
5
37. x ≤ − 1
3
or x ≥ 3 38. x ≤ -5 or x ≥ 5 39. t ≤ 2 or t ≥ 9
40. 0 ≤ x ≤ 1 41. −2 2 < x <2 2
42. x <
−3 − 57
2
or x >
−3 + 57
2
43. t < − 3
2
or t > 6 44.
1
4
< x < 2