Topic 8.3 (Equations with Fractions)

To solve an equation containing fractions (rational expressions) complete the following.

         1.   Determine the LCD.

         2.   Multiply the LCD by each fraction, eliminating all denominators.

         3.   Solve for the variable as usual.

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Sample:     3/x + 2/(x - 2) = 1    Solve for x.    The LCD is x(x - 2).  Multiply this by each

                                              fraction on both sides of the equation.

   3(x - 2)   +   2x   =   x (x - 2)     Solve for x

3x   -   6   +   2x   =   x²  -   2x    All answers must be checked. If any one

  5x   -   6   =   x²  -   2x                 of them makes a denominator 0, they are

    0   =   x²  -   7x   +   6                called extraneous roots and thrown out.

    0   =   (x   -   1)(x   -   6)

               {1, 6}                               Both of these answers work.

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Sample:    4y   -   9    =    6   -   8y      This equation has two terms, one on

                          8                     16            the left and one on the right.  This is

                                                                  proportion and can be solved by

                                                                  cross-multiplication.

   16(4y  -   9)    =    8(6   -   8y)           Divide both sides of the equation by 8.

     2(4y   -   9)   =    6 - 8y                     In this problem you don’t have to

                                                                  worry about a variable making a

                                                                  denominator 0, because the

       8y   -   18    =    6   -   8y                   denominators are numbers.

       8y   +   8y   =    6   +   18

                 16y   =   24

               y = 24/16   which reduced to   3/2       {3/2}      Ans.