Show all work to simplify (2)/(x)+(2)/(x+1)-(2)/(x+2) . Use complete sentences to explain how to simplify this expression. Remember to list all restrictions.

Question

Show all work to simplify (2)/(x)+(2)/(x+1)-(2)/(x+2)  . Use complete sentences to explain how to simplify this expression. Remember to list all restrictions.

e.mccormick · Answer

OK, what part of this are you having trouble with?

bahrom7893 · Answer

find the common denominator

anonymous · Answer

I know the steps but I don't know how to do them I know I have to 
1. factor the denominators.
2. find the common denominators.
3. combine the rational expressions. 
4. simplify
I just don't know how to get there.

e.mccormick · Answer

So lets lookat them in order:
1, factor
$$\frac{2}{x}+\frac{2}{x+1}-\frac{2}{x+2}$$
Anything there look like it needs to be factored?

anonymous · Answer

no

e.mccormick · Answer

So that step was easy.  Now the fun one, common denominator.  Do you understand what that means in gneral?

anonymous · Answer

not really

e.mccormick · Answer

So lets review a little addition of fracions, because that is where it comes from.  If I have to add these:$$\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$$I need a common denominator.

anonymous · Answer

1/2

anonymous · Answer

sorry didn't mean to write that

e.mccormick · Answer

No problem.

OK, each of those fractions can be represented many ways.  1/2 is also 2/4, and so on.  So I need to find the combination that has the same thing on the bottom of all three fractions.  In this case, because all three are different, it would need to be a number that can be divided by 2, 3, and 4!  Well, 2*3=6... not much help.  2*4=8, anagin not much help.  3*4=12.... Ah ha!  12 is dividable by 2 as well!  So I can eep going with my example.  I need to make the bottom of each fracion into 12.

e.mccormick · Answer

$$
\frac{1}{2}\cdot \frac{6}{6}
+\frac{1}{3}\cdot \frac{4}{4}
+\frac{1}{4}\cdot \frac{3}{3}
$$

And all of that becomes:
$$\frac{6}{12}+\frac{4}{12}+\frac{3}{12}$$

e.mccormick · Answer

Now that is something that can be added because the bottoms are all the same.
$$\frac{6+4+3}{12}=\frac{13}{12}$$
The goal here is to do something similar, but with the algabraic expressions.

e.mccormick · Answer

Now, instead of 12... you have three factors.  $x$, $x+1$, and $x+2$.  So you must multiply each fraction by something to add the missing parts.

e.mccormick · Answer

Does all of that make sense to this point?

e.mccormick · Answer

Well, looks like you lost connection.  Anyhow, after you multiply things by fancy versions of 1, you can add them,  combine like terms, possibly factor, simplify, and basically be done.  However, remember that if any of the factors on the bottom were ever 0, this would all fall apart.  That is what the restrictions are about.  So you need to make a list of when different parts of the bottom would be 0 and call that the domain restrictions.  

I said fancy versions of 1 there.  See, you can't just multiply by any old number.  That changes things.  However, you can multiply anything by 1 and it is the same.  But the great thing is, 1 can take on disguises and look very different.  All of the following are examples of the number 1:
$$1
=\frac{1}{1}
=\frac{a}{a}
=\frac{(a-4)(a+2)}{(a-4)(a+2)}
=\frac{x+1}{x+1}
=\frac{x}{x}
=\frac{-1}{-1}
$$

So when you get the common denominator, keep that in mind