EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:


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Solve for x in the following equation.


Problem 5.2g:

tex2html_wrap_inline269


Answer:
tex2html_wrap_inline271 are the exact answers and tex2html_wrap_inline273 1.08452405258 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_inline277 the second fraction is valid if tex2html_wrap_inline279 and the third fraction is valid is tex2html_wrap_inline281 .If either tex2html_wrap_inline283 or tex2html_wrap_inline285 turn out to be the solutions, you must discard them as extraneous solutions.


Multiply both sides by the least common multiple tex2html_wrap_inline287 (the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.


eqnarray54


eqnarray62



which is equivalent to


eqnarray77



which can be rewritten as


eqnarray92



which can be rewritten as


eqnarray101



which can be simplified to


eqnarray116


eqnarray125



Solve for x using the quadratic formula tex2html_wrap_inline289 tex2html_wrap_inline291


eqnarray133


eqnarray142


eqnarray153



The answers are tex2html_wrap_inline293 However, this may or may not be the answer. You must check the solution with the original equation.


Check the solution tex2html_wrap_inline297 by substituting 6.91547594742 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 6.91547594742 for x, then x=6.91547594742 is a solution.


Check the solution tex2html_wrap_inline309 by substituting 1.08452405258 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 1.08452405258 for x, then x=1.08452405258 is a solution.


You can also check your answer by graphing tex2html_wrap_inline321 (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_inline323 .


We have verified the solution two ways.








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