This is a problem from the Manhattan GMAT Ratios book:

2x/y = 4x/8500

If you cross multiply you get:

17,000x = 4xy...the book suggests doing this and then dividing by x to cancel out the x's. My question is why can I divide by x on both sides...How do we know x isn't 0?

I'm traveling and don't have that book available to look the problem up myself but, FYI, the in-action problems are not designed to be GMAT-type problems (eg, they aren't multiple choice). They're just to test your understanding of the math concepts in that chapter. On the official test, you might be told, for example, that the variables don't equal zero, which would preclude the x=0 possibility.

You mentioned that this is in the "Ratios book" - do you mean the Ratios chapter of the Word Translations book? I just want to send along to someone who can look this up (since I can't right now) and need to know where to tell them to look.