**
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 1:
**
**

**
**

**
**

Rewrite the equation such that all the denominators are factored:

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 either or turn out to be the solutions, you must discard them as extraneous solutions.

Multiply both sides of the equation by an expression that represents the lowest common denominator. The expression is the smallest expression because it is the smallest expression that is divisible by all three denominators.

This equation can be written as

Multiply the fractions where indicated.

Rearrange the factors in the numerators and rewrite the equations as

Rewrite the equation once again as

Simplify the last equation and solve for x.

The original equation has two real solutions: The exact answers are and the approximate answers are and -3.4244289009. In most cases it is easier to check your solutions with the approximate answers.

Check the two answers in the original equation.

Check the solution
by substituting
6.4244289009 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:
0.892684537537+0.237404816816 = 1.13008935435

- Right Side:
.

**
**

Since the left side of the original equation is not equal to the right side
of the original equation after we substitute the value
6.4244289009 for x,
then
*x*=6.4244289009 is a solution.**
**

**
**

**
**

**
**

**
**

Check the solution
by substituting
-3.4244289009 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:
&=&0.482315462463-1.40407148348=-0.92175602102

- Right Side:
.

**
**

Since the left side of the original equation is not equal to the right side
of the original equation after we substitute the value
-3.4244289009 for
x, then
*x*=-3.4244289009 is a solution.**
**

**
**

**
**

**
**

**
**

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; that will be the real
solution. Note that the graph crosses the x-axis in two places:
-3.4244289009 and
6.4244289009.**
**

**
**

**
**

This means that there are two real solutions and the solutions are
*x*=-3.4244289009 and
6.4244289009.**
**

**
**

**
**

**
**

**
**

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

Do you need more help? Please post your question on our
S.O.S. Mathematics CyberBoard.

Contact us

Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA

users online during the last hour