**
Note:
**

**
**

- A rational equation is an equation where at least one denominator
contains a variable.
- When a denominator contains a variable, there is usually a
restriction on the domain. The variable cannot take on any number that would
cause any denominator to be zero.
- The first step is solving a rational equation is to convert the
equation to an equation without denominators. This new equation may be
equivalent (same solutions as the original equation) or it may not be
equivalent (extraneous solutions).
- The next step is to set the equation equal to zero and solve.
- Remember that you are trying to isolate the variable.
- Depending on the problem, there are several methods available to help you solve the problem.

If you would like an in-depth review of fractions, click on Fractions.

**Solve for x in the following equation.**

Example 4:

Recall that you cannot divide by zero. Therefore, the first fraction is
valid if ,
the second fraction is valid if
and the third fraction is valid is
. If -7, 2, or
turn out to be the solutions, you
must discard them as extraneous solutions.

Multiply both sides of the equation by the least common multiple
(the smallest
number that all the denominators will divide into evenly). This step will
eliminate all the denominators in the equation. The resulting equation may
be equivalent (same solutions as the original equation) or it may not be
equivalent (extraneous solutions),

which is equivalent to

which can be rewritten as

which can be rewritten again as

which can be rewritten yet again as

Solve for x by factoring:

The only way that a product can have a value of zero is if at least one of
the factors is equal to zero. The factor 3 can never be zero. The factor
is zero when *x*=3, and the factor
is zero when

The exact answers are
and 3.

Check the answer in the original equation.

Check the solution
by
substituting the approximate solution
*x*=-0.152873 in the original equation
for x. If the left side of the equation equals the right side of the
equation after the substitution, you have found the correct answer.

- Left Side:

- Right Side:

It does not check exactly because we rounded the answer. However, it checks
enough to tell us that the answer
is
a solution to the original problem.

Check the solution
by substituting
the approximate solution
*x*=-2.18046 in the original equation for x. If the
left side of the equation equals the right side of the equation after the
substitution, you have found the correct answer.

- Left Side:

- Right Side:

It does not check exactly because we rounded the answer. However, it checks
close enough to tell us that the answer
is a solution to the original problem.

Check the solution *x*=3 by substituting the 3 in the original equation
for x. If the left side of the equation equals the right side of the
equation after the substitution, you have found the correct answer.

- Left Side:

- Right Side:

You can also check your answer by graphing
(formed by subtracting
the right side of the original equation from the left side). Look to see
where the graph crosses the x-axis; the intercept(s) will be the real
solution(s). Note that the graph crosses the x-axis in two places: -0.152873 and
-2.180460.

If you would like to test yourself by working some problems similar to this
example, click on Problem

If you would like to go back to the equation table of contents, click on
Contents

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