EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:




If you would like an in-depth review of fractions, click on Fractions.



Solve for x in the following equation.


Example 2:tex2html_wrap_inline155tex2html_wrap_inline176


Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline178 the second fraction is valid if tex2html_wrap_inline180 and the third fraction is valid is tex2html_wrap_inline182 . If either 5 or-5 turn out to be the solutions, you must discard them as extraneous solutions.


Rewrite the problem so that every denominator is factored


eqnarray41



Multiply both sides by the least common multiple (x-5)(x+5) (the smallest expression that all the denominators will divide into evenly) .


eqnarray51


eqnarray60



which is equivalent to


eqnarray70



which can be rewritten


eqnarray86



which can be rewritten


eqnarray96



which can be rewritten


eqnarray112


eqnarray115



The answer is 21.



Check this answer in the original equation.



Check the solution x=21 by substituting 21 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.


Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 21 for x, then x=21 is a solution.


You can also check your answer by graphing tex2html_wrap_inline200 (formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at 21. This means that the real solution is 21.








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Author: Nancy Marcus

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