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 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 work another example, click on Example

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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