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 1:tex2html_wrap_inline155tex2html_wrap_inline186


Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline188 the second fraction is valid if tex2html_wrap_inline190 and the third fraction is valid is tex2html_wrap_inline192 . If either 3 or-3 turn out to be the solutions, you must discard them as extraneous solutions.


Rewrite the problem so that every denominator is factored


eqnarray36



Multiply both sides by the least common (x-3)(x+3) multiple (the smallest expression that all the denominators will divide into evenly).


eqnarray46


eqnarray55



which is equivalent to


eqnarray65



which can be rewritten


eqnarray81



which can be rewritten


eqnarray91


which can be rewritten


eqnarray107


eqnarray110



The answer is tex2html_wrap_inline194



Check this answer in the original equation.



Check the solution x=5 by substituting 5 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 5 for x, then x=5 is a solution.


You can also check your answer by graphing tex2html_wrap_inline206 (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 5. This means that the real solution is 5.








If you would like to work another example, click on Example


If you would like to test yourself by working some problems similar to this example, click on Problem


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Author: Nancy Marcus

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