3.05   Fractional Equations

from Basic Algebra: One Step at a Time © 2002

P. 281-288

Dr. Robert J. Rapalje

Seminole State College of Florida

 

ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE

 

To see selected solutions in Living C O L O R  click here!

 

It is often necessary to solve equations that involve fractions.  The first step here, as before when simplifying fractional exercises is to find the least common denominator (LCD).  The second step when solving equations is different--you may multiply both sides of the equation by the LCD, and in doing so, you eliminate all denominators from the problem.  By contrast, in working with fractional expressions (i.e., NOT equations!) in the previous section, it was necessary to carry the common denominator all the way through the problem.  It is important to emphasize that the methods of this section are valid only with equations.  It is not correct to say “multiply through by the LCD.”  Rather, you should emphasize the equation involved by saying, “multiply both sides of the equation by the LCD,” as long as you don’t multiply both sides of the equation by zero.   YOU ARE NEVER ALLOWED TO MULTIPLY BOTH SIDES OF AN EQUATION BY ZERO! 

There is one small “fly in the ointment”--at times you will be multiplying both sides of the equation by variables.  This may not appear to be a problem, but it could be!  If you multiply both sides of an equation by a variable, you really do not know what you are multiplying by--that is, not until you finish the problem and find out what “x” (or whatever the variable may be) equals.  The “small fly” is this:  what if, without realizing it, you multiplied both sides of the equation by zero?  It could happen, and multiplying both sides of an equation by zero is not allowed!  Therefore, any time you multiply both sides of an equation by a variable, you must check the answer(s) to make sure that no  denominators ever equal zero.  If any answer that you get ever makes a denominator equal zero, then this answer must be rejected.  The answer that was obtained was not a legal answer.  Like evidence that is illegally obtained and cannot be allowed in a court of law, such answers must be thrown out. If no other solutions can be found, then there is no solution for the problem.  In this case, you should answer “No Solution,” or the empty set, which is written either { } or φ .

 Text Box:    

 

PRINCIPLE

 

          Whenever an equation is solved by multiplying both sides of that equation by a variable, the solution must be checked to be sure no denominators equal zero.

 

   

 

 


 

Before beginning the exercises, one more principle will be useful in this section.  This is the definition of equality of fractions.  Two fractions,  , are equal if and only if   a × d = b × c, assuming no denominators are zero.

 

You can check out this definition to see that it makes sense by reducing a few fractions, as follows. 

EXAMPLE 1.  Reduce the fraction  and check the answer using the above definition.

Solution:         since  4 × 2 = 8 × 1  (which is true because 8 = 8).

 

EXAMPLE 2.  Reduce the fraction  and check the answer using the above definition.

Solution:         because 6 × 3 = 9 × 2  (which is true because 18 = 18).

 

EXAMPLE 3.  Reduce the fraction  and check the answer using the above definition.

Solution:          because 3 × 9 = 27 × 1  (which is true because 27 = 27).

 

EXAMPLE 4.  Reduce the fraction  and check the answer using the above definition.

Solution:         because 12 × 5 = 15 × 4  (which is true because 60 = 60).

 

The main use of the definition of equality of fractions is to solve equations in which you have a fraction equal to another fraction.  Using this definition, in one step, you can completely eliminate the fractions in the problem and have a regular (that is, linear or quadratic!) equation to solve.

EXAMPLE 5.  Solve the equation.  Check to be sure denominators are not zero.

Solution:         By the definition of equality of fractions:

5(x) = 4(x + 2)

 5x   =  4x + 8

-4x     -4x    

   x   =      8

Check:                     

 

EXERCISES.  Solve the equations for x.  Check to be sure denominators are not zero.

 1.                                                            2. 

 

 

 

3.                                                     4. 

                 

 


 

 5.                                              6.  

 

 

 

 

 7.                                                          8.  

 

 

 

 

9.                                                         10.  

 

 

 

EXAMPLE 6.    Solve the equation.  Check that denominators are not zero.

Solution:         By the definition of equality of fractions:

4(x+1) = 6(x + 1)

4x + 4  =  6x + 6

- 6x            -6x    

     -2x + 4   =          6

             - 4            - 4

        -2x    =     2

x    =   -1

Check:                           Division by zero is UNDEFINED.      No Solution!!

11.                                                     12. 

 

 

 

13.                                                14. 

 

 

 

In 15 - 16, these look like quadratic equations, but they are not since the x2  terms subtract out!!

15.                                                  16.  

 

 

 

 

In Example 7 and the following exercises, these ARE quadratic equations!  You know what to do!

EXAMPLE  7.            Solve the equation     

Solution:                                                    x(x - 2) = 8           This is a quadratic equation.

                                                             x2 - 2x - 8 = 0       You must set equal to zero and factor.

                                                         (x - 4)    (x + 2)  =  0

                                                         x = 4         x = -2   

Check:  If x= 4,       If x= -2,     

                                                     1  =   1                          -2  =  -2       Both answers check!  

 

17.                                                      18.  

 

 

19.                                                  20.  

 

  

 

21.                                                       22.  

 

  

 

23.                                              24.  

 

 


 

EXAMPLE  8.   Solve for x:               

Solution:  Multiply both sides of the equation by the LCD which is 6:  

                          

              

          4x + 9x - 36   =   3

           + 36  + 36

      13x       =  39                    

  x   = 3

 

EXERCISES.  In each of the following equations, solve for x.

25.                                               26.            

 

  

 

 

27.                                            28.   

                

 

 

  

29.                                     30. 

 

 

 

 

 

EXTRA CHALLENGE        

EXAMPLE 4.    Solve for x:       

Solution:         Multiply each side of the equation by the LCD which is (x-5)(x-1).  Before you do, however, notice that x cannot equal 5 or 1, since these values of x cause one or more denominators to be zero.  So, x¹5 and x¹1.

This looks pretty scary, but look how it simplifies--EVERY denominator divides out!

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

         x2 - 5x   +     2x - 2       =    - 4

                      + 4            + 4

             x2   -   3x   +   2       =  0

  (x - 2)(x - 1)  = 0

    x = 2   x = 1

However, the answer of x = 1 must be rejected, since x¹1.  The solution is x = 2.

 

EXERCISES.  Solve for x.                             

31.                                        32.         

 

 

 

 

 

  

ANSWERS 3.05

 p. 283 - 288:

 1. -8; 2. 6;  3. -5;  4. 2;  5. 5;  6. 12;  7. -7;  8. 12;  9. 2;  10. 9;  11. No Sol;  12. No Sol;  13. 4;  14. No Sol; 15.  ‑1; 16. 6; 17. -4,2; 18. 12,-2; 19. -12,2;  20. 4,-2;  21. 12, -4;  22. -6,4;  23. -9, -3;  24. 2;  25. 14;  26. -12;  27.-8;  28.-3, 2;  29. 7,-2;  30. 3, -2;  31. 10 (Reject 5);  32. No sol (Reject 3).

Return to main page       Math in Living C O L O R !!

     Return to Basic Algebra page     Return to Intermediate Algebra page  

 Return to College Algebra page

 

 
Dr. Robert J. Rapalje Altamonte Springs Campus
Contact me at:   rapaljer@seminolestate.edu
Phone number:  NONE Retired!!
OFFICE:          NONE  
Copyright © Seminole State College of Florida, 1997