## 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.

Example 1:

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.

Rewrite the problem so that every denominator is factored

Multiply both sides 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

No Solution

Since we stated at the beginning of the problem that , the only conclusion is that there is no solution. How can that be?. When we transformed the original equation into an equation without denominators, the new equation was not equivalent to the original equation. You can verify this by graphing both equations. Why did we do it? Because the second equation is easier to solve. The down side is that we might turn up a wrong answer (extraneous solution). So CHECK YOUR ANSWERS!

Check this answer in the original equation. Verify that it will not check.

Check the solution x=-2 by substituting -2 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 division by zero is undefined, -2 is not a solution.

Since the left side of the original equation is not equal to the right side of the original equation after we substitute the value -2 for x, then x=-2 is a not 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 never crosses the x-axi. This means that there are no real solutions.

If you would like to work another example, click on Example

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

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Nancy Marcus