Question 146052
{{{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.
<pre><font size = 5 color = "indigo"><b>

{{{1/2 + 3/x = (x+6)/(2x)}}}

Put all the terms in parentheses:

{{{(1/2) + (3/x) = ((x+6)/(2x))}}}


Look at all the denominators:

{{{2}}},{{{x}}} and {{{2x}}}

The LCD of these is {{{2x}}}, so
we write {{{2x}}} as {{{((2x)/1)}}} and
multiply every term by it

{{{((2x)/1)(1/2) + ((2x)/1)(3/x) = ((2x)/1)((x+6)/(2x))}}}

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:

{{{((cross(2)x)/1)(1/cross(2)) + ((2cross(x))/1)(3/cross(x)) = ((cross(2x))/1)((x+6)/(cross(2x)))}}}

{{{x + 2*3 = x+6}}}

{{{x+6=x+6}}}

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</pre>