Call Now: (800) 537-1660

June 19th

June 19th

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.

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.

Free Tutoring
• Get 30 Minutes of FREE Live Tutoring from Tutor.com if you

Testimonials
"I ordered the Algebra Buster late one night when my daughter was having problems in her honors algebra class. After we ordered your software she was able to see step by step how to solve the problems. Algebra Buster definitely saved the day."
Tami Garleff

Home    Why Algebra Buster?    Guarantee    Testimonials    Ordering    FAQ    About Us
What's new?    Resources    Animated demo    Algebra lessons    Bibliography of     textbooks