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A rational inequality is an inequality
which contains a rational expression.
When solving
these rational inequalities, there are steps that lead us to the
solution.
| To solve Rational Inequalities: |
| (1) Write the inequality as an
equation, and solve the equation. |
| (2) Determine values that make any
denominator equal to 0. |
(3) On a number line, mark each of
the critical values from steps 1 and 2.
These values
will create intervals on the number line. |
(4) Select test points in each
interval, and check to see which of the
intervals satisfy
the inequality. |
(5) Mark the number line to reflect
the values and intervals that satisfy
the inequality. |
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Example
1: Solve 
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Change < to =,
and solve |
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Notice that if
 , the
denominator is 0. |
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Critical values are
 and
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On the number line, plot -5
and -8. Since -5 cannot be
used, it is an open circle. |
The inequality is strictly
less
than, so the -8 is also
an open circle. |
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In algebraic notation,
the solution is:
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In interval notation,
the solution is:
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When the numerator of the inequality is a quadratic expression, then we
combine the
Quadratic Inequality method of solution with this Rational
Inequality method.
Check out this example.
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Example 2: Solve
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Separate the
numerator, and solve
as in a quadratic equation. |
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Factor, and find the solutions
or critical values. |
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These are the
critical values for the numerator. The
denominator
has
as a critical value as well.
Since x = 2 creates an undefined
expression, it an open circle on the
number line. |
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Place the
critical values on a number line. Since the
inequality is greater than or
equal to, the
go on the number line as a solid circle, meaning
to include them as part of the answer.
Test the intervals next.
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In algebraic
notation,
the solution is:
 |
In interval notation,
the solution is:
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