EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:




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



Solve for x in the following equation.


Problem 5.1a:tex2html_wrap_inline155tex2html_wrap_inline177



Answer:tex2html_wrap_inline155No Solution



Solution:

Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline179 or tex2html_wrap_inline181 ,the second fraction is valid if tex2html_wrap_inline183 and the third fraction is valid is tex2html_wrap_inline185 . If either 0 or 3 turn out to be solutions, you must discard them as extraneous solutions.


The least least common multiple x(x-3) (the smallest expression that all the denominators will divide into evenly) is tex2html_wrap_inline191 . Multiply both sides of the equation by the least common multiple


eqnarray44


eqnarray53



which is equivalent to


eqnarray63



which can be rewritten as


eqnarray79



which can be rewritten as


eqnarray89



which can be rewritten as eqnarray105


eqnarray108



Since our initial constraint was that tex2html_wrap_inline185 , there is no solution.


If you forgot to make this observation, you would have caught the mistake in the check.


Check the solution x=3 by substituting 3 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.


You can also check your answer by graphing tex2html_wrap_inline211 (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-axis. This means that there are no real solutions.








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


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


If you would like to go back to the equation table of contents, click on Contents


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Author: Nancy Marcus

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