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 5.2f:
Answer:
is the exact answer and 
is 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 , 
and the second fraction is valid if 
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
The answer ise 
However, this may or may not be the
answer. You must check the solution with the original equation.
Check the solution by substituting 3.3333333333 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 
for x, then 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 one place: x=3.3333333333.
We have verified the solution in two ways.
If you would like to review the solution to problem 5.2g, click on
Problem
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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