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# 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.

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

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

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

 -2m ( m + 1 ) + (m + 3 ) = m 2 - 1 Multiply -2m 2 - 2m + m + 3 = m 2 - 1 Collect like terms. -2m 2 - m + 3 = m 2 - 1 This is a quadratic equation so we need to set it equal to zero. Let's move all terms to the right side. 0 = 3m 2 + m - 4

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.

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