What you really need to internalize is that you can always add parentheses, and that you can treat all the stuff inside a pair of parentheses as just one blob.
Example: (1 + 1/x)/(1 - 1/x)
Let's look at the two blobs, (1 + 1/x) and (1 - 1/x). First things first:
(1 + 1/x)
= 1*(1 + 1/x) Well, multiplying by one doesn't change anything
= x/x * (1 + 1/x) And x/x = 1, so I'm allowed to write it like this
= 1/x * (x + x/x) Here I just pull the upper x inside the parentheses
= (x + 1)/x Because x/x = 1. For convenience I pulled the 1/x to the end,
Similarly we get that (1 - 1/x) = (x - 1)/x. (This you should do as an exercise! It's the same steps as above, but it'll help you to write it out yourself.)
So now we can go back to the original problem:
(1 + 1/x)/(1 - 1/x)
= ((x + 1)/x)/((x - 1)/x) I just wrote the two blobs in a different way, as seen above - this doesn't change anything
= 1*((x + 1)/x)/((x - 1)/x) Multiplying with one doesn't do anything, see above
= x/x*((x + 1)/x)/((x - 1)/x) And we can do this because x/x = 1, as above
= (x*(x + 1)/x))/(x*(x - 1)/x) I just pulled together the two fractions - if you write it down by hand, this is very intuitive
= (x + 1)/(x - 1) Because the x's inside the two blobs cancel out each other
You should write down all that by hand, in proper notation - it's next to impossible to really see anything the way I had to write it here. Once you've done it as above a couple times, you'll notice that you can do most of the steps in your head - so it won't always be as much of a pain in the ass as the stuff above
Any questions? I tried to be fairly explicit and clear, but I have a nasty habit of not living up to these aspirations...