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 6: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 if tex2html_wrap_inline181 If either 1 or tex2html_wrap_inline185 turn out to be the solutions, you must discard them as extraneous solutions.



Multiply both sides by the least common multiple (x-1)(3x+1) (the smallest expression that all the denominators will divide into evenly).


eqnarray42


eqnarray51



which is equivalent to


eqnarray61



which can be rewritten as


eqnarray77



which can be rewritten as


eqnarray87



which can be rewritten again as


eqnarray103


eqnarray106


eqnarray109



The answer is tex2html_wrap_inline187



Check this answer in the original equation.



Check the solution tex2html_wrap_inline191 by substituting tex2html_wrap_inline193 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_inline199 for x, then tex2html_wrap_inline201 is a solution.


You can also check your answer by graphing tex2html_wrap_inline203 (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 -4.333333. This means that the real solution is -4.333333.








If you would like to test yourself by working some problems similar to this example, click on Problem


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Author: Nancy Marcus

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