Note:
If you would like an in-depth review of fractions, click on Fractions.
Solve for x in the following equation.
Problem 5.2b:
Answer:
are the exact
answers and 
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
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 1.919596
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 1.919596 for x, then x=1.919596 is a solution.
What do we mean by reasonably close? When we calculated the value of exact
value , we rounded to six places. The answer is
not exact and therefore the check will not be exact.
Check the solution 
by substituting 0.223261
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 0.223261 for x, then x=0.223261 is a solution.
What do we mean by reasonably close? When we calculated the value of exact
value , we rounded to six places. The answer is
not exact and therefore the check will not be exact.
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.
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.
This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site. If you feel that some of the material in this section is ambiguous or needs more clarification, please let us know by e-mail.
