Polynomial Inequalities

Example of Solving a Polynomial Inequality

Rational Inequalities

Unlike polynomial functions, rational functions are not necessarily continuous. The vertical asymptotes of a rational function r break up the graph into separate “branches.” So the intervals on which r does not change sign are determined by the vertical asymptotes as well as the zeros of r. This is the reason for the following definition: If r(x) = P(x)/Q(x) is a rational function, the cut points of r are the values of x at which either P(x) = 0 or Q(x)= 0. In other words, the cut points of r are the zeros of the numerator and the zeros of the denominator (see Figure). So to solve a rational inequality like r(x) = 0, we use test points between successive cut points to determine the intervals that satisfy the inequality. We use the following guidelines.

Example of Solving a Rational Inequality

Go back to   Rational Functions    Go forward to Exponential and Logarithmic Functions