Definition

Quadratic inequalities in one variable are inequalities which can be written in one of
the following forms:

where a, b and c are real numbers.

Procedure

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, x^2 > x + 2 .

Solution

The corresponding equation is (x - 2)(x + 1) = 0 so…

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

So the solution to this inequality is x < -1 or x > 2.

Example 2 Solve the inequality,

Solution

The corresponding equation is (x - 1)(x - 5) = 0 so…

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

So the solution to this inequality is 1 ≤ x ≤ 5.

## §4-2 PROBLEM SET

Solve each quadratic inequality, and graph the solution on a number line.