**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

Go to Solutions.

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