Complex Fractions and Equations with Rational Expressions

Complex Fractions

First we begin with a complex fraction that contains no variables.

Example

       1         5                          1                  5
             -                                     12   -            12    
       2         6                          2                  6       
                                    =                                               Multiply Numerator and 
       1          2                           1                 2                 Denominator by 12
             +                                      12  +           12 
       4          3                           4                 3

 

            6 - 10                    4  
    =                      =     -          
            3 + 8                    11 

 

Notice we first multiplied by the total Least Common Denominator, then we simplified.

 

Complex Fractions Involving Expressions

When we have a complex fraction with rational expressions as the numerator and denominator, we follow similar steps, except, of course factoring plays a key role.  

 

  • Step 1  Factor everything.

  • Step 2  Determine the total least common denominator, using the maximum power of each factor.

  • Step 3  Multiply all terms by the LCD.

  • Step 4  Combine like terms.

  • Step 5  Factor and Cancel.

 

Note:  Usually you will not have to do all of the steps.

 

Example:  

               7                                                7
      1 -                                   1(x + 1)  -             (x + 1)    
            x + 1                                           x + 1       
                                    =                                                   Multiply Numerator and 
         4                                    4                                         Denominator by (x+1)
                +   1                               (x + 1)  + 1(x + 1) 
      x + 1                              x + 1            

 

            x + 1 - 7              x - 6 
    =                        =                 
           4 + x + 1              x + 5 

 

Cross Multiplication

Recall that if 

         a          c
               =        
         b          d

then 

        ad  =  bc  

The same hold true for functions:

         f           g
               =        
         h           k

then 

        fg   =   hk 

 

Example

Solve.

         3x - 1           x + 2
                      =                
         5x - 2          3x + 4

 

Solution

We cross multiply

        (3x - 1)(3x + 4)  =  (5x - 2)(x + 2)

        9x2 + 12x - 3x - 4  =  5x2 + 10x - 2x - 4

        9x2 + 9x - 4  =  5x2 + 8x - 4

        4x2 + x  =  0

        x (4x + 1)  =  0

                                       1
        x = 0     or     x = -      
                                       4

 

Caution:  Always check and see that the solution works by plugging back into the original equation!

 

Equations with Rational Expressions

To solve equations that involve rational expressions, we following the following steps:

 

  • Step 1  Factor if possible.

  • Step 2  Multiply the left hand and right hand sides by the LCD.

  • Step 3  Combine like terms.

  • Step 4  Bring everything to the left side of the equation.

  • Step 5  Solve by the zero product method or by basic algebra.

  • Step 6  Plug back in to the original equation to check for extraneous solutions.

 

Example

Solve

           3             4                 48
                   -              =               
        x - 6        x + 6          x2 - 36

Solution

First factor.

           3               4                   48
                    -              =                       
         x - 6         x + 6        (x - 6)(x + 6)

Then multiply by the LCD (x - 6)(x+ 6).

           3                                4                                        48
                 (x - 6)(x + 6)  -             (x - 6)(x + 6)  =                        (x - 6)(x + 6)
        x - 6                           x + 6                             (x - 6)(x + 6)

 

        3(x + 6) - 4(x - 6)  =  48

        3x + 18 - 4x + 24  =  48

        -x + 42  =  48

        -x = 6

        x = -6

Notice that -6 cannot be put back into the original equation, since there would be a zero in the denominator. We can conclude that this equation has no solution.

 

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