siddhans wrote:

How do we know we need to take \(3.5<x+y<4.5\) or \(3.5<=x+y<=4.5\) ??????

4 is the integer that is closest to x+y i.e. there is a single integer that is closest to (x+y)

If (x+y) = 3.5, which integer is closest to it? Both 3 and 4 are at equal distance i.e. they are both 0.5 away from (x+y). But then, we cannot say that 4 is the integer closest to x+y.

Hence, x+y must be greater than 3.5. It must also be less than 4.5 due to the same reason.

Note: 3.5 is rounded up to 4 instead of 3 only because we generally follow round up convention. If we follow 'round down' convention, 3.5 will be rounded off to 3.

3.5 is equidistant from both 3 and 4.

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