Question 81347
<pre><font size = 2><b>
Solve {{{1/(x+2)- 2/x=3}}}
I used LCD and got (1x/(x^2 +2x))- (2x +4)/(x^2 + 2x) = 3
Then multiplied bith sides by x^2 -2x
3x^2 - 5x -4

You need to multiply on both sides to begin with, not
multiply on the left and fail to on the right.  You also
should not multiply the LCD out as you did.  Here is
step by step, what you should have:

         1       2
      ------- - --- = 3
       x + 2     x 

Put the 3 over 1 so everything will be a fraction:

         1       2     3
      ------- - --- = ---
       x + 2     x     1

Now get the LCD = <font color = "red">x(x + 2)</font> but DON'T multiply it out!
That was one thing you did wrong above.  Again, don't multiply the LCD out!

Instead put it over 1 like this, with a multiplication dot 
symbol · beside it:

<font color = "red"> x(x + 2)    </font>
<font color = "red">---------- · </font>
<font color = "red">     1       </font>

Now place that whole thing before each terms on the left side AND the right side:

      <font color = "red"> x(x + 2)    </font>   1      <font color = "red"> x(x + 2)    </font> 2    <font color = "red"> x(x + 2)    </font> 3 
      <font color = "red">---------- · </font>------- - <font color = "red">---------- · </font>--- = <font color = "red">---------- · </font>---
      <font color = "red">     1       </font> x + 2    <font color = "red">     1       </font> x    <font color = "red">     1       </font> 1 
 
Now you're ready to do some canceling. 

1. Cancel the black x + 2 into the red <font color = "red">(x + 2)</font>
2. Cancel the black x into the red <font color = "red">x</font>

           1                  1
      <font color = "red"> x<s>(x + 2)</s>    </font>   1      <font color = "red"> <s>x</s>(x + 2)    </font> 2    <font color = "red"> x(x + 2)    </font> 3 
      <font color = "red">---------- · </font>------- - <font color = "red">---------- · </font>--- = <font color = "red">---------- · </font>---
      <font color = "red">     1       </font> <s>x + 2</s>    <font color = "red">     1       </font> <s>x</s>    <font color = "red">     1       </font> 1
                      1                    1

Now notice that after canceling, you have only 1's in the denominators.
So you can erase all the 1's and like magic all the fractions are gone:

1. All that's left in the first term is the red <font color = "red">x</font>
2. All the's left in the second term is - <font color = "red">(x+2)2</font>
3. All that's left on the right side is <font color = "red">x(x+2)</font>3

                                 <font color = "red">x<font><font color = "black"> - <font color = "red">(x + 2)2</font> = <font color = "red">x(x + 2)</font>3
<font color = "black">
It's customary to write numbers before parenthese, not after them,
so we write that as

                                 <font color = "red">x</font> - <font color = "red">2(x + 2)</font> = 3<font color = "red">x(x + 2)</font>


I'll now dispense with the colors and make everything black:

                                 x - 2(x + 2) = 3x(x + 2)

Now we're finally ready to multiply things out, but not until now!

                                   x - 2x - 4 = 3x² + 6x

Can you finish solving that?  If not post again asking how.
I'll assume you can.   You get two solutions x = -4/3 and x = -1

[Neither of these answers cause any denominators to become zero
when substituted in the original equation, so they are both
solutions.  Had we gotten x = -2 we would have had to discard
it because it would have made the denominator x - 2 become 0.
Sometimes this is the case when we multiply through by the LCD,
so we should always look to make sure we aren't including an 
extraneous solution. I thought I'd better warn you about
extraneous solutions because you do get them sometimes, but
not in this example.]

Edwin</pre>