Solving rational equations

I'm losing the will... I simply can't get my head around these rational equations:

I've got the answers using MathCad, but I want to understand how they work... If someone wouldn't mind walking me through the process I would be very grateful. Please treat my like a child, as I can't for the life of me seem to grasp this concept. Also pretend that I don't understand LCM or GCF (I do for normal equations, but I don't seem able to transfer this knowledge to algebra).

Here are a few examples I know the answers to, just not the workings:

1) x+4/x - 14/2x-1 = 0. Answers are x=-1/2, x=4

2) x+1/x - 4/2x-3 = 0. Answers are x=-1/2, x=3

3) x/1-2x - 2/3x+1 = 0. Ansers are x=1/3, x=-2

I would be insanely grateful if anyone could point me to some great algebra resources on the net, which have highly detailed step x step tutorials.

Thanks a lot for your time,

Mez