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 any values that make
the
denominator equal 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 a test point in each
interval, and check to see if that test point
satisfies the inequality. (Find the
intervals which satisfy
the inequality). 
(5) Mark the number line to reflect
the values and intervals that satisfy
the inequality. 
(6) State
your answer using the desired form of notation. 
Example
1: Solve


Create an equation. Change < to =,
and solve. Notice that if x = 5, the
denominator is 0. 

Multiply both sides by (x
+ 5)
to eliminate the fraction.
You could also "crossmultiply".
In a proportion the product of the
means
equals the product of the extremes. 

Critical values are
and

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. 
Test a point in each of the three intervals
formed:

Stated as an inequality,
the solution is:

Stated in interval notation,
the solution is:

When the numerator of the inequality is a quadratic expression,
combine the
Quadratic Inequality method of solution with this Rational
Inequality method.
Check out this example.
Example 2: Solve



Separate the
numerator,
form an equation,
and solve this quadratic equation.
Factor, and find the solutions
or critical values for the numerator.
Also keep in mind that the
denominator
has
as a critical value as well.
Since x = 2 creates an undefined
expression, it is drawn as an
open circle on the
number line. 


Place the
critical values on a number line. Since the
inequality is greater than or
equal to, the
are drawn on the number line as a solid circle,
which means
to include them as part of the answer.
Test the intervals that are formed.

The solution is:

In interval notation,
the solution is:


