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.


Problem 5.2f:

tex2html_wrap_inline178


Answer: tex2html_wrap_inline180 is the exact answer and tex2html_wrap_inline182 is the approximate answer.


Comment on answers: You may wonder why we give you the answers in two forms: exact and approximate. There is a reason. Students seem perplexed when they think they have worked a problem correctly and yet, their exact answers differ from the exact answers in the book. The student is not necessarily wrong. Depending on the method chosen to work the problem, exact answers have a different look. How do you know whether your exact answer is equivalent to a different looking exact answer in the book? Simplify both. If both exact answers are correct, they will both simplify to the same approximate answer.


Next time your answer differs from the answer in the book, simplify both. If the approximate answers are the same, you are correct. If not, go back to the drawing board and try to find your mistake.



Solution:

Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline184 and the second fraction is valid if tex2html_wrap_inline186 If either tex2html_wrap_inline188 or tex2html_wrap_inline190 turn out to be the solutions, you must discard them as extraneous solutions.

Multiply both sides by the least common multiple tex2html_wrap_inline192 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.


eqnarray51


which is equivalent to


eqnarray66


which can be rewritten as


eqnarray77


which can be rewritten as


eqnarray84


which can be simplified to


eqnarray95


eqnarray104



The answer ise tex2html_wrap_inline194 However, this may or may not be the answer. You must check the solution with the original equation.


Check the solution tex2html_wrap_inline180 by substituting 3.3333333333 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 reasonably close to the right side of the original equation after we substitute the value tex2html_wrap_inline206 for x, then tex2html_wrap_inline180 is a solution.



You can also check your answer by graphing tex2html_wrap_inline210 (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 one place: x=3.3333333333.


We have verified the solution in two ways.




If you would like to review the solution to problem 5.2g, click on Problem

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