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 5.2h:

tex2html_wrap_inline268

Answer: tex2html_wrap_inline270 are the exact answers and tex2html_wrap_inline272 0.765331812733 are 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_inline276 the second fraction is valid if tex2html_wrap_inline278 , and the third fraction is valid if tex2html_wrap_inline280 If either tex2html_wrap_inline282 or tex2html_wrap_inline284 or0 turn out to be the solutions, you must discard them as extraneous solutions.



Multiply both sides by the least common multiple tex2html_wrap_inline288 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.


eqnarray51


eqnarray59


which is equivalent to


eqnarray74


which can be rewritten as


eqnarray89


which can be rewritten as


eqnarray98


which can be simplified to


eqnarray113


eqnarray124


> Solve for x using the quadratic formula tex2html_wrap_inline290 tex2html_wrap_inline292


eqnarray138


eqnarray146


eqnarray155



The answers are tex2html_wrap_inline294 However, this may or may not be the answer. You must check the solution with the original equation.





Check the solution tex2html_wrap_inline298 by substituting 3.40858123075 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 3.40858123075 for x, then x=3.40858123075 is a solution.




Check the solution tex2html_wrap_inline310 by substituting 0.765331812733 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 0.765331812733 for x, then x=0.765331812733 is a solution.




You can also check your answer by graphing tex2html_wrap_inline322 (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_inline324 .


We have verified the solution two ways.



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