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 5.2b:

tex2html_wrap_inline280

Answer:

tex2html_wrap_inline282 are the exact answers and tex2html_wrap_inline284 are the approximate answers.




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:

Rewrite the problem so that every denominator is factored

eqnarray54




Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline286 the second fraction is valid if tex2html_wrap_inline288 and the third fraction is valid is tex2html_wrap_inline290 .If either tex2html_wrap_inline292 or tex2html_wrap_inline294 turn out to be the solutions, you must discard them as extraneous solutions.





Multiply both sides by the least common multiple tex2html_wrap_inline296 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.

eqnarray69




which is equivalent to

eqnarray84




which can be rewritten as

eqnarray99




which can be rewritten as

eqnarray108




which can be simplified to

eqnarray124



eqnarray132



Solve for x using the quadratic formula tex2html_wrap_inline298 tex2html_wrap_inline300

eqnarray141

eqnarray148




The answers are tex2html_wrap_inline302 However, this may or may not be the answer. You must check the solution with the original equation.




Check the solution tex2html_wrap_inline306 by substituting 1.919596 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 1.919596 for x, then x=1.919596 is a solution.




What do we mean by reasonably close? When we calculated the value of exact value tex2html_wrap_inline306 , we rounded to six places. The answer is not exact and therefore the check will not be exact.




Check the solution tex2html_wrap_inline320 by substituting 0.223261 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.

  • Left Side:

    eqnarray164

    eqnarray200

  • Right Side: tex2html_wrap_inline326

    Since the left side of the original equation is reasonably close to the right side of the original equation after we substitute the value 0.223261 for x, then x=0.223261 is a solution.




    What do we mean by reasonably close? When we calculated the value of exact value tex2html_wrap_inline320 , we rounded to six places. The answer is not exact and therefore the check will not be exact.





    You can also check your answer by graphing tex2html_wrap_inline334 (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 two places: tex2html_wrap_inline336 .




    We have verified the solution two ways.




    If you would like to review the solution to problem 5.2c, click on solution.


    If you would like to return to the beginning of this section, click on again.


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


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