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 3:tex2html_wrap_inline155tex2html_wrap_inline200


Rewrite the problem so that the denominator of every fraction is factored.


eqnarray37



Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline202 or tex2html_wrap_inline204 the second fraction is valid if tex2html_wrap_inline206 or tex2html_wrap_inline208 and the third fraction is valid is tex2html_wrap_inline210 .If either tex2html_wrap_inline212 5, or tex2html_wrap_inline216 turn out to be the solutions, you must discard them as extraneous solutions.


Multiply both sides by the least common multiple tex2html_wrap_inline218 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.


eqnarray47


eqnarray56


eqnarray63



which is equivalent to


eqnarray76


eqnarray68



which can be rewritten as


eqnarray86



which can be rewritten as


eqnarray96


eqnarray102



which can be simplified to


eqnarray108


eqnarray110



The answer is tex2html_wrap_inline220 However, this may or may not be the answer. You must check the solution with the original equation.


Check the solution tex2html_wrap_inline224 by substituting tex2html_wrap_inline226 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 tex2html_wrap_inline232 for x, then tex2html_wrap_inline224 is a solution.


You can also check your answer by graphing tex2html_wrap_inline236 (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 tex2html_wrap_inline238 .


We have verified the solution two ways.








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Author: Nancy Marcus

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