If 500 is the multiple of 100 that is closest to x and 400 is the multiple of 100 that is closest to y, which multiple of 100 is closest to x+y?

(1) x < 500
(2) y < 400

okay, so off the bat we can tell that 450<x<550 and 350<y<450 because in order for x and y to be closest to 500 and 400 respectively, they need to be within those ranges. 450 is equally as close to 400 and 500, and 434 is closer to 400 than 500, for example.

so 800<x+y<1000

now this doesn't solve our answer, but it's a start. Let's correct our formulas for each part.
(1)
450<x<500
800<x+y<950

this tells us it's either 800 or 900, but not which one.

(2)
350<y<400
800<x+y<950
same as (1)

together
800<x+y<900.

it could be either 800 or 900 still, so I'd go with E.

An important take-away here is that if inequalities face the same way, they can be safely added. for example, if a>b and c>d, then it's safe to conclude that a+c>b+d.

However, one way in which inequalities are very different from equations is that you cannot subtract inequalities. If a>b and c>d, it is not safe to subtract one from the other and conclude that a-c>b-d. This may or may not be true.

The question above is GMATPrep question 1064. A detailed video solution of it is available.

-Patrick