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
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.
Return to Precalculus Home Page.