Delta-Epsilon proof

I am having a real tough time with these proofs.

f(x)=2-(1/x)

find delta such that 0<|x-1|<d then |f(x)-1|<0.1

I understand what it means, but I'm having a hard time proving it.

I start out by working the epsilon inequality...

I end up with something like this...
|1/x||x-1|<0.1

Now I'm stuck. But from what I've been reading you would set
|x-1|<1

This is obviously not a good option given 1/x. So I chose 1/2.
I eventually get...
1/2<x<3/2

Now I don't understand the next step. This should be so easy.

Let and .
From what you have already done this means that .
Thus