1. 4.5 Polynomial and Rational Inequalities

2. 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.

3. 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.

4. 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

5. The real line is split into:

6. 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