EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:




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



Solve for x in the following equation.


Problem 5.1c:tex2html_wrap_inline155tex2html_wrap_inline217


Answer:tex2html_wrap_inline155tex2html_wrap_inline219


Solution:


Rewrite the problem so that every denominator is fully factored.


eqnarray45



Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline221 the second fraction is valid if tex2html_wrap_inline223 , and the third fraction is valid if tex2html_wrap_inline225 . If either -8 or 2 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_inline231 . Multiply both sides of the equation by the least common multiple.


eqnarray56


eqnarray65



which is equivalent to


eqnarray75



which can be rewritten as


eqnarray91



which can be rewritten as


eqnarray101



which can be rewritten again as


eqnarray115


eqnarray118


eqnarray121

The answer is tex2html_wrap_inline233



Check this answer in the original equation.



Check the solution tex2html_wrap_inline219 by substituting tex2html_wrap_inline239 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_inline239 for x, then tex2html_wrap_inline219 is a solution.


You can also check your answer by graphing tex2html_wrap_inline249 (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_inline239 . This means that the real solutions is tex2html_wrap_inline239 .








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Author: Nancy Marcus

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