4.5 Polynomial and Rational Inequalities

Steps for Solving Polynomial and Rational Inequalities Algebraically Write the inequality in one of the following forms: where f(x) is written as a single quotient. Determine the numbers at which f(x) equals zero and also those numbers at which it is undefined.

Use these numbers to separate the real line into intervals. Select a test number from each interval and evaluate f at the test number. (a) If the value of f is positive, then f(x)> 0 for all numbers x in the interval. (b) If the value of f is negative, then f(x)<0 for all numbers x in the interval. If the inequality is not strict, include the solutions of f(x)=0 in the solution set, but do not include those where f is undefined.

Undefined for Solve the inequality: The inequality is in lowest terms, so we will first find where f(x)=0. And where is it undefined. Undefined for x=-2

The real line is split into:

The solution is all numbers x for which Pick x =-3 Pick x =0 Pick x =2 Pick x =-3/2 f(-3)=-8 f(-3/2)=5/2 f(0)=-1/2 f(2)=3/4 POSITIVE NEGATIVE POSITIVE NEGATIVE The solution is all numbers x for which or