Polynomial and Rational Inequalities

Up to this point we have been solving quadratic inequalities. The technique involving sign charts extends to solving polynomial inequalities of higher degree.

Solve.

Step 1: Determine the critical numbers, which are the roots or zeros in the case of a polynomial inequality.

Step 2: Create a sign chart.

Step 3: Use the sign chart to answer the question.

The last problem shows that not all sign charts will alternate. Do not take any shortcuts and test each interval.

Rational inequalities are solved using the same technique. The only difference is in the critical numbers. It turns out that the y-values may change from positive to negative at a restriction. So we will include the zeros of the denominator in our list of critical numbers.

Solve.

Tip: Always use open dots for critical numbers that are also zeros of the denominator, or restrictions. This reminds us that they are restrictions and should not be included in the solution set even if the inequality is inclusive.