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 5:tex2html_wrap_inline155tex2html_wrap_inline185


Rewrite the equation so that every denominator is factored.


eqnarray37



Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline187 the second fraction is valid if tex2html_wrap_inline189 and the third fraction is valid only if tex2html_wrap_inline191 If either tex2html_wrap_inline193 or tex2html_wrap_inline195 turn out to be the solutions, you must discard them as extraneous solutions.


Multiply both sides by the least common multiple tex2html_wrap_inline197 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators. This step will eliminate all the denominators.


eqnarray48


eqnarray62



which is equivalent to


eqnarray71



which can be rewritten as


eqnarray86



which can be rewritten as


eqnarray95



which can be simplified to


eqnarray105


eqnarray107


eqnarray110



The answer is tex2html_wrap_inline199 However, this may or may not be the answer. You must check the solution with the original equation.


Check the solution tex2html_wrap_inline203 by substituting 1.33333333 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_inline211 for x, then tex2html_wrap_inline203 is a solution.


You can also check your answer by graphing tex2html_wrap_inline215 (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_inline217 .


We have verified the solution two ways.








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Author: Nancy Marcus

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