## EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)

Note:

• A rational equation is an equation where at least one 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 in solving a rational equation is to convert the equation to an equivalent equation without denominators.

• Then 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 help you solve the problem.

For an in-depth review on fractions, click on Fractions.

Solve for x in the following equation.

Problem 5.1a:

Solution:

Recall that you cannot divide by zero. Therefore, the first fraction is valid if , or ,the second fraction is valid if and the third fraction is valid is . If either 0 or 3 turn out to be solutions, you must discard them as extraneous solutions.

The least least common multiple x(x-3) (the smallest expression that all the denominators will divide into evenly) is . Multiply both sides of the equation by the least common multiple

which is equivalent to

which can be rewritten as

which can be rewritten as

which can be rewritten as

Since our initial constraint was that , there is no solution.

If you forgot to make this observation, you would have caught the mistake in the check.

Check the solution x=3 by substituting 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: The fraction is undefined because you cannot divide by zero. You might say ''well what about the fraction If you are adding 100 fractions and 99 are good and one is undefined, the entire expression is undefined.

• Right Side: is also undefined.

Undefined means there is no 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-axis. This means that there are no real solutions.

If you would like to review the solution to problem 5.1b, click on Problem

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