The number x and y are not integers. The value of x is closest to which integer?
(1) 4 is the integer that is closest to x+y.
(2) 1 is the integer that is closest to x-y.

The correct answer is E, statements (1) and (2) together are not sufficient. But i think that both (1) and (2) together are sufficient because let simplify the statements as follows:
(1) x+y = 4
(2) x-y = 1

Let (1)+(2)=2x=5 => x = 5/2 (to the nearest)

no! no! no no no!

these aren't rephrasings at all; they are completely different statements.
the given information tells you only that (x + y) is closer to 4 than to any other integer. this means that 3.5 < x + y < 4.5, and that's all it means; you cannot infer that x + y = 4.
likewise, the second statement means only that 0.5 < x - y < 1.5; you don't know that x - y is 1, nor do you have any reason to suspect that that is the case.
in fact, since you are specifically told that x and y are not integers, you should be automatically suspicious of integer sums and differences (though they are not impossible in this case, viz., 2.8 + 1.2 = 4). the irony is that this problem is being nice to you: it actually comes out and tells you that x and y are not integers. even if no such conditions were stated, you would still have to consider non-integer values of x and y.

now, since the inequalities "line up", you can just add them together:
0.5 < x - y < 1.5
3.5 < x + y < 4.5
4 < 2x < 6
therefore 2 < x < 3. this means that the value of x could be closest to 2, but could also be closest to 3.
insufficient

answer = e.