More on Inequalities

Focus: Solving Polynomial Inequalities

· Determining solution sets by the graph of an equation

· Determining solution sets by charting the test intervals

Property Summary Key Numbers and Persistence of Sign

Let P and Q be polynomials with no common factors (other than, possibly, constants), and consider the following four inequalities:

< 0 0 > 0 0

The key numbers for each of these inequalities are the real numbers for which P = 0 or Q = 0. (Geometrically speaking, the key numbers are the x-intercepts for the graphs of the equations y = P and y = Q.) It can be proved (using calculus) that the algebraic sign of P/Q does not change within each of the intervals determined by the key numbers.

Steps for solving polynomial inequalities

1. If necessary, rewrite the inequality so that the polynomial is on the left had side and zero is on the right hand side.

2. Find the key numbers for the inequality and locate them on the number line.

3. List the intervals determined by the key numbers.

4. From each interval, choose a convenient test number. Then use the test number to determine the sign of the polynomial throughout the interval.

5. Use the information obtained in the previous step to specify the required solution set. [Don’t forget to take into account whether the original inequality is strict (< or >) or non-strict/weak ( or ).]