Question 69365
<pre><font size = 4><b>{{{2/(x+1)-3/(x-3)}}}

The LCD is the product of the denominators {{{(x+1)(x-3)}}}:

We ask ourselves:

What factor does the denominator of the first fraction
{{{2/(x+1)}}}, which is {{{(x+1)}}}, lack which the LCD contains?

The answer is that it lacks the factor {{{(x-3)}}}. 

Therefore we multiply the first fraction by {{{(x-3)/(x-3)}}}
which:

(1) Does not change its value because {{{(x-3)/(x-3)}}} equals
    1 and multiplying by 1 does not change any value.
(2) It will cause the denominator of the first fraction to be
    the LCD {{{(x+1)(x-3)}}}

So {{{2/(x+1)-3/(x-3)}}}

becomes:

{{{(2/(x+1))((x-3)/(x-3))-3/(x-3)}}}

Now we do the same with the second fraction. We ask ourselves:

What factor does the denominator of the second fraction
{{{-3/(x-3)}}}, which is {{{(x-3)}}}, lack which the LCD contains?

The answer is that it lacks the factor {{{(x+1)}}}. 

Therefore we multiply the second fraction by {{{(x+1)/(x+1)}}}
which:

(1) Does not change its value because {{{(x+1)/(x+1)}}} equals
    1 and multiplying by 1 does not change any value.
(2) It will cause the denominator of the second fraction to be
    the LCD {{{(x+1)(x-3)}}}

So  {{{(2/(x+1))((x-3)/(x-3))-3/(x-3)}}}

becomes

    {{{(2/(x+1))((x-3)/(x-3))-(3/(x-3))((x+1)/(x+1))}}}

or

    {{{ (2(x-3))/((x+1)(x-3)) - (3(x+1))/((x-3)(x+1))}}}

Now the denominators are the same, i.e., both denominators
are now equal to the LCD {{{(x+1)(x-3)}}}

Multiply the tops out but do not multiply out the bottoms:

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

Since the denominators are the same we indicate the
subtraction of the numerators over the LCD:

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

Remove the parentheses in the top but not in the bottom:

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

Combine terms in the numerator:

{{{ (-x-9)/((x+1)(x-3))}}}

I am sure your teacher would accept that answer. But
if you like you can factor -1 out of the numerator:

{{{ (-1)(x+9)/((x+1)(x-3))}}}

and then eliminate the {{{(-1)}}} by putting a negative 
sign out in front of the fraction:

{{{ -(x+9)/((x+1)(x-3))}}}
 
</pre>Edwin