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 5.2a:

tex2html_wrap_inline276

Answer :

tex2html_wrap_inline278 are the exact answers and tex2html_wrap_inline280 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

eqnarray53



Recall that you cannot divide by zero. Therefore, the first fraction is valid if , tex2html_wrap_inline282 or tex2html_wrap_inline284 the second fraction is valid if tex2html_wrap_inline286 and the third fraction is valid is tex2html_wrap_inline288 .If either tex2html_wrap_inline290 or tex2html_wrap_inline292 turn out to be the solutions, you must discard them as extraneous solutions.




Multiply both sides by the least common multiple tex2html_wrap_inline294 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.

eqnarray68




which is equivalent to

eqnarray83




which can be rewritten as

eqnarray98




which can be rewritten as

eqnarray107




which can be simplified to

eqnarray123



eqnarray133




Solve for x using the quadratic formula tex2html_wrap_inline296 tex2html_wrap_inline298

eqnarray141

eqnarray150

eqnarray157




The answers are tex2html_wrap_inline300 However, this may or may not be the answer. You must check the solution with the original equation.





Check the solution tex2html_wrap_inline304 by substituting 1.1150692933 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 1.1150692933 for x, then x=1.1150692933 is a solution.





Check the solution tex2html_wrap_inline316 by substituting -0.448402626637 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 -0.448402626637 for x, then x=-0.448402626637 is a solution.




You can also check your answer by graphing tex2html_wrap_inline328 (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_inline330 .




We have verified the solution two ways.



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


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


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