EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:


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



Solve for x in the following equation.

Problem 5.1e:

tex2html_wrap_inline242

Answer: tex2html_wrap_inline244

Solution:

Whenever the numerator or the denominator is itself a fraction, you have a compound fraction. The first step is to convert the compound fractions to simple fractions.

Multiply both sides of the equation by 15 so that the fractions are written as simple fractions.

eqnarray45

eqnarray54

eqnarray62

Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline246 and the second fraction is valid if tex2html_wrap_inline248 If either 2 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_inline254 . Multiply both sides of the equation tex2html_wrap_inline256 by the least common multiple


eqnarray75


eqnarray82


which is equivalent to


eqnarray90


which can be rewritten as


eqnarray103


which can be rewritten as


eqnarray116


which can be rewritten as


eqnarray121


eqnarray125


eqnarray130


eqnarray135


eqnarray142


The answers are tex2html_wrap_inline258 and tex2html_wrap_inline260



Check this answer in the original equation.


Check the solution x=2.13165177 by substituting 2.13165177 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 2.13165177 for x, then x=2.13165177 is a solution.


Check the solution x=-1.03831843666 by substituting -1.03831843666 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 -1.03831843666 for x, then x=-1.03831843666 is a solution.


You can also check your answer by graphing tex2html_wrap_inline288 (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_inline258 and tex2html_wrap_inline292 . This means that the real solutions are tex2html_wrap_inline258 and tex2html_wrap_inline292 .







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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