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 5.2e:
Answer:
are the exact
answers and 
-5.69157361327 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 , 
the second fraction is valid if 
and the third fraction is valid is 
.If either 
or 
turn out to be the solutions, you must discard them as extraneous
solutions.
Multiply both sides by the least common multiple 
(the smallest number that all the denominators
will divide into evenly). This step will eliminate all the
denominators.
which is equivalent to
which can be rewritten as
which can be rewritten as
which can be simplified to
Solve for x using the quadratic formula 
The answers are 
However, this may or may
not be the answer. You must check the solution with the original equation.
Check the solution 
by substituting
1.0249069466 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
1.0249069466 for x, then x=1.0249069466 is a solution.
Check the solution 
by substituting
-5.69157361327 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
0.60996277864 for x, then x=0.60996277864 is a solution.
You can also check your answer by graphing 
(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: 
.
We have verified the solution two ways.
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