Solving Rational Inequalities
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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 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 "cross-multiply".
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: