Quadratic Inequalities

A quadratic inequality has the form ax + bx + c > 0 (or <, or , or ). The highest exponent is 2. The next few examples show how to solve quadratic inequalities.

Solving Quadratic Inequalities

EXAMPLE

Solve the quadratic inequality x - x < 12.

Solution

Write the inequality with 0 on one side, as x - x - 12 < 0. This inequality is solved with values of x that make x - x - 12 negative (< 0). The quantity x - x - 12 changes from positive to negative or from negative to positive at the points where it equals 0. For this reason, first solve the equation x - x - 12 = 0.

x - x - 12 = 0

(x - 4)(x + 3) = 0

x = 4 or x = -3

Locating -3 and 4 on a number line, as shown in Figure 3, determines three intervals A, B, and C.

Decide which intervals include numbers that make x - x - 12 negative by substituting any number from each interval in the polynomial. For example,

choose -4 from interval A: (-4) - (-4) - 12 = 8 > 0;

choose 0 from interval B: 0 - 0 - 12 = -12 < 0;

choose 5 from interval C: 5 - 5 - 12 = 8 > 0.

Only numbers in interval B satisfy the given inequality, so the solution is (-3, 4). A graph of this solution is shown in Figure 4.

Solving Polynomial Inequalities

EXAMPLE

Solve the inequality x(x-1)(x+3) 0.

Solution

This is not a quadratic inequality. If the three factors are multiplied, the highest-degree term is x. However, it can be solved in the same way as a quadratic inequality because it is in factored form. First solve the corresponding equation.

x(x - 1)(x + 3) = 0

x = 0 or x - 1 = 0 or x + 3 = 0

x = 0 or x = 1 or x = -3

These three solutions determine four intervals on the number line: , (-3, 0), (0, 1) and . Substitute a number from each interval into the original inequality to determine that the solution includes the numbers less than or equal to -3 and the numbers that are equal to or between 0 and 1. See Figure 5.

In interval notation, the solution is

* The symbol indicates the union of two sets, which includes all elements in either set.