Solving Polynomial and Rational Inequalities

To solve an inequality such as or where and are polynomials,

1. Factor and completely over the real numbers.

2. Mark the zeros of and on a number line.

3. Determine the sign of on each of the resulting intervals.

4. Select the intervals corresponding to the sign of the original inequality. (If the inequality is not a strict inequality, include the zeros of in the solution.)

In determining the sign of on each interval, we can use the following:

If is the highest power of which is a factor of or , then

A. the sign of changes at if is odd; and

B. the sign of does not change at if is even.
Ex 1 Solve the inequality .

Sol Factoring gives ; so marking off 3 and -1 on a number line and using the facts that and that the exponents of and are both odd, we get the sign chart shown below:

Therefore the solution is given by .

Ex 2 Solve the inequality

Sol Factoring gives

Marking off 6,-4,10, and -2 on a number line and using the facts that and that the exponents of all the factors are odd, we get the sign chart shown below:

Since the inequality is not strict, we can include the zeros of the numerator; so the solution is given by .

Pr 1 Solve the inequality .

Pr 2 Solve the inequality .

Pr 3 Solve the inequality

Pr 4 Solve the inequality

Pr 5 Solve the inequality

Pr 6 Solve the inequality

Pr 7 Solve the inequality

Pr 8 Find all values of for which .

Pr 9 Find all values of for which