Solving Rational Equations

After studying this lesson, you will be able to:

• Solve rational equations.

To solve an equation with fractions (rational equations), multiply each term by a common denominator. This will eliminate the fractions. Then, solve the equation that remains.

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Example 1

The common denominator is 6x so we multiply each term by 6x:

	Now, we have 3 multiplications to do - this will eliminate the variables
	Collect like terms
29 = x

**remember, the solution cannot cause any denominator to equal zero - if it does, it is an extraneous solution

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Example 2

The common denominator is 60 so we multiply each term by 60:

	Now, we have 3 multiplications to do - this will eliminate the variables
30 - 15r + 36r = 20r + 40	Collect like terms
30 + 21r = 20r + 40	Get the variables together by subtracting 20r from each side
30 + r = 40	Subtract 30 from each side
r = 10

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Example 3

To find the common denominator we have to factor the second denominator:

The common denominator is (m + 1) (m - 1) so we multiply each term by (m + 1) (m - 1):

Now, we have 3 multiplications to do - cancel out identical binomials

Now, factor and set each factor equal to zero and solve.

0 = (3m + 4) ( m - 1 )

3m + 4 = 0	m - 1 = 0
	m = 1

m = 1 is an extraneous solution because it will cause a denominator to equal zero.