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_inline175


Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline177 the second fraction is valid if tex2html_wrap_inline179 and the third fraction is valid is tex2html_wrap_inline181 . If either tex2html_wrap_inline183 or tex2html_wrap_inline185 turn out to be the solutions, you must discard them as extraneous solutions.


Rewrite the problem so that every denominator is factored


eqnarray39



Multiply both sides by the least common multiple tex2html_wrap_inline187 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators in the equation. The resulting equation may be equivalent (same solutions as the original equation) or it may not be equivalent (extraneous solutions),


eqnarray48



which is equivalent to


eqnarray63



which can be rewritten as


eqnarray78



which can be rewritten again as


eqnarray87



which can be rewritten yet again as


eqnarray97


eqnarray100



No Solution



Since we stated at the beginning of the problem that tex2html_wrap_inline189 , the only conclusion is that there is no solution. How can that be?. When we transformed the original equation into an equation without denominators, the new equation was not equivalent to the original equation. You can verify this by graphing both equations. Why did we do it? Because the second equation is easier to solve. The down side is that we might turn up a wrong answer (extraneous solution). So CHECK YOUR ANSWERS!


Check this answer in the original equation. Verify that it will not check.


Check the solution x=-2 by substituting -2 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 division by zero is undefined, -2 is not a solution.


Since the left side of the original equation is not equal to the right side of the original equation after we substitute the value -2 for x, then x=-2 is a not solution.


You can also check your answer by graphing tex2html_wrap_inline213 (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 never crosses the x-axi. This means that there are no real solutions.








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Author: Nancy Marcus

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