##
EQUATIONS INVOLVING
FRACTIONS (RATIONAL EQUATIONS)

Note:

- A rational equation is an equation where at least one denominator
contains a variable.

- When a denominator contains a variable, there is 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.**

**Problem 5.5 a: **
**
**

**
**

**
****Answer: **

**
**

**
****Simplify the equation by subtracting **
** from both sides of the
equation.
**

**
**

**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 the denominators.**

**The only way a product can equal zero is if at least one of the factors
equals zero.
**

**
**
**
**

**
**** are not real numbers. Therefore, there is
only one real answer ***x*=4.

**Check the answer in the original equation.**

**Check the solution ***x*=4** by substituting **4** 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:**
**.**

**
**

**
**

**
**

**
**

**
****
**

**Since the left side of the original equation is not equal to the right side
of the original equation after we substitute the value **4** for x, then ***x*=4**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 at one spot, **

**This means that there is one real solution and the solution is ***x*=4.

**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**

*
*

*
[Algebra]
[Trigonometry]
**
*
[Geometry]
[Differential Equations]
[Calculus]
[Complex Variables]
[Matrix Algebra]
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Author:
Nancy Marcus

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