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.
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