Are These Relations Symmetric?

[Shark1]

Determine which of these relations are symmetric

1) x~y if and only if x-y is positive
2) x~y if and only if xy >= 0
3) x~y if and only if x+2y is positive
4) x~y if and only if x+y is positive
5) x~y if and only if x+y is odd

I thought all but 1) but this was wrong.
The only one I am sure that is symmetric is 2)

 

[HallsofIvy]

It is certainly true that if x+ y is positive, so is y+ x.

It is certainly true that if x+ y is odd, so is y+ x.

What about "x+ 2y is positive"? Can you find x such that x+ 2y is positive but y+ 2x is not?