The numbers 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

it shouldn't take too long for you to conclude that the individual statements are insufficient.

statement 1 means that 3.5 < x + y < 4.5
this of course doesn't tell us anything about the sizes of x and y.
for instance, x and y could be 1.5 and 2.5. or, they could be -999.5 and 1003.5.
insufficient by itself.

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statement 2 means that 0.5 < x - y < 1.5
this likewise tells us nothing about the individual values of x and y.
for instance, x and y could be 2.5 and 1.5. or, they could be 1001.5 and 1000.5.
insufficient by itself.

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together, you can ADD THE INEQUALITIES, so that 'y' cancels out.
(TAKEAWAY: you can add inequalities whenever the 'alligators' - i.e., the "<" or ">" - face the SAME WAY.)
this gives
3.5 < x + y < 4.5
0.5 < x - y < 1.5
add
4 < 2x < 6
therefore
2 < x < 3

this is still insufficient, because x could be closer to 2, closer to 3, or neither (if it's exactly in the middle, at 2.5).

ans (e).

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you could also just LOOK FOR SPECIFIC NUMBERS that satisfy the criteria in the problem. for instance, x = 2.7 and y = 1.5 satisfy both statements, making x closest to 3. also, x = 2.3 and y = 1.5 satisfy both statements, making x closest to 2.
so, both statements together are still insufficient, so, (e).