EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:




For an in-depth review on fractions, click on Fractions.



Solve for x in the following equation.


Problem 5.1d: tex2html_wrap_inline155tex2html_wrap_inline216


Answer: tex2html_wrap_inline155 tex2html_wrap_inline218


Solution:


Rewrite the problem so that every denominator is fully factored.


eqnarray42



Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline220 the second fraction is valid if tex2html_wrap_inline222 , and the third fraction is valid if tex2html_wrap_inline224 or tex2html_wrap_inline226 If either 8 or -1 turn out to be solutions, you must discard them as extraneous solutions.

The least least common multiple (the smallest expression that all the denominators will divide into evenly) is tex2html_wrap_inline232 . Multiply both sides of the equation by the least common multiple


eqnarray53


eqnarray62


which is equivalent to


eqnarray72


which can be rewritten as


eqnarray88


which can be rewritten as


eqnarray105


which can be rewritten as

eqnarray110

eqnarray114

The answers are tex2html_wrap_inline234



Check this answer in the original equation.



Check the solution tex2html_wrap_inline238 by substituting tex2html_wrap_inline240 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_inline248 for x, then tex2html_wrap_inline238 is a solution.



Check the solution tex2html_wrap_inline252 by substituting tex2html_wrap_inline254 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_inline260 for x, then tex2html_wrap_inline252 is a solution.


You can also check your answer by graphing tex2html_wrap_inline264 (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_inline266 . This means that the real solutions are tex2html_wrap_inline268 .






If you would like to review the solution to problem 5.1e, click on Problem.


If you would like to go back to the problem page, click on Problem


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Author: Nancy Marcus

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