EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:




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



Solve for x in the following equation.


Problem 5.1b:tex2html_wrap_inline155tex2html_wrap_inline173


Answer:tex2html_wrap_inline155x = -1, 4


Solution:tex2html_wrap_inline155

Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline177 and the second fraction is valid if tex2html_wrap_inline179 If either 0 or 2 turn out to be solutions, you must discard them as extraneous solutions.


The least least common multiple x(x-2) (the smallest expression that all the denominators will divide into evenly) is tex2html_wrap_inline185 . Multiply both sides of the equation by the least common multiple


eqnarray40


eqnarray47



which is equivalent to


eqnarray55



which can be rewritten as


eqnarray69



which can be rewritten again as


eqnarray77



which can be rewritten yet again as


eqnarray90


eqnarray95



eqnarray99

The answers are tex2html_wrap_inline187 and tex2html_wrap_inline189



Check this answers in the original equation.



Check the solution x=-1 by substituting -1 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 -1 for x, then x=-1 is a solution.


Check the solution x=4 by substituting 4 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 4 for x, then x=4 is a solution.


You can also check your answer by graphing tex2html_wrap_inline217 (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 -1 and 4. This means that the real solutions is -1and 4.








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


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Author: Nancy Marcus

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