SOLUTION: the problem: 1/2 + 3/x = x+6/2x How does the x get from denominator to numerator? I'm not sure how to work the problem to get the x on top. Please help explain this. Tha

Algebra ->  Algebra  -> Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: the problem: 1/2 + 3/x = x+6/2x How does the x get from denominator to numerator? I'm not sure how to work the problem to get the x on top. Please help explain this. Tha      Log On

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Question 146052: the problem:
1/2 + 3/x = x+6/2x
How does the x get from denominator to numerator? I'm not sure how to work the problem to get the x on top.
Please help explain this. Thank you.
Lorraine
Found 2 solutions by Edwin McCravy, BrittanyM:
Answer by Edwin McCravy(6937) About Me  (Show Source):
You can put this solution on YOUR website!
1%2F2+%2B+3%2Fx+=+%28x%2B6%29%2F%282x%29
How does the x get from denominator to numerator?
I'm not sure how to work the problem to get the x on top.
Please help explain this. Thank you.


1%2F2+%2B+3%2Fx+=+%28x%2B6%29%2F%282x%29

Put all the terms in parentheses:

%281%2F2%29+%2B+%283%2Fx%29+=+%28%28x%2B6%29%2F%282x%29%29


Look at all the denominators:

2,x and 2x

The LCD of these is 2x, so
we write 2x as %28%282x%29%2F1%29 and
multiply every term by it

%28%282x%29%2F1%29%281%2F2%29+%2B+%28%282x%29%2F1%29%283%2Fx%29+=+%28%282x%29%2F1%29%28%28x%2B6%29%2F%282x%29%29

Cancel the 2's in the first term on the left.
Cancel the x's in the second term on the left.
Cancel the 2x's in the term on the right:

%28%28cross%282%29x%29%2F1%29%281%2Fcross%282%29%29+%2B+%28%282cross%28x%29%29%2F1%29%283%2Fcross%28x%29%29+=+%28%28cross%282x%29%29%2F1%29%28%28x%2B6%29%2F%28cross%282x%29%29%29

x+%2B+2%2A3+=+x%2B6

x%2B6=x%2B6

adding -x to both sides

6=6

This is a true numerical identity.
Therefore the answer would be
{all real numbers}, however we can see
that x cannot be 0 since it would
cause two denominators in the original
equation to be 0.  Therefore any value 
of x except 0 is a solution.  So the
solution set is graphed as:

<=====================o====================>
 -5  -4  -3  -2  -1   0   1   2   3   4   5    

or in interval notation:   (-infinity,0)U(0,infinity)

Edwin

Answer by BrittanyM(80) About Me  (Show Source):
You can put this solution on YOUR website!
Lorraine,

For the equation we've been given, the best way to solve is to find a common denomoiatior and cancel.

We have:

1%2F2 + 3%2Fx = x + 6%2F2x

In order to get rid of all of these fractions and get all of the x's to the top, we need to find a common denominator and then we can canel all of the denominators out.

for this problem, our best bet for the common denominator is 2x.

x%2F2x + 6%2F2x = 2x%5E2%2F2x +6%2F2x

Now, because this is an equation and each side is equal, we can cancel out all of the denominators.

x + 6 = 2x%5E2 + 6

And we can finish solving for x.

2x%5E2 - x = 0
x(2x-1) = 0

So our answers are 0 and 1%2F2