What is the correct way to divide a equation or inequality?

Sure, it's fine.  Most people don't write it that way, but it's not wrong.  Usually people would divide each side separately by 2, but the effect is the same, as long as you communicate what you're doing, it shouldn't matter.

When you mutiply or divide an equation (because they're basically the same operation) and you just do the same thing to both sides.

So, you'd set it out something like this.

2x = 4.  
x = 2 (Divide both sides by two)

If it's obvious what you're doing, you can omit the brackets, but if you're not sure, leaving them in probably doesn't hurt.

As for ineqations, it's pretty much the same.

2x > 4
x > 2 (Divide both sides by 2)

etc.

Just be careful when dividing (or multiplying) by negates, be sure to swap the sign around.

You could also do something like this:

2x = 4.  
x = 2 (÷2)

I would avoid trying to preform some operation on a whole equation like that. People often talk about multiplying equations, etc, but It's not technically something you're supposed to do, it's more of a short hand for saying that something will be done to both sides of an equation.

Also, I've never seen A fraction live (a=b)/2 so you're better off just writing a/2 = b/2.
You can never really be "wrong" in aesthetic cases like this, as long as you arrive at a correct answer in the end, but there are preferred ways to set things out, and I would reccomend abiding by those standards, in the long run it;ll make things a lot easier.