Is the sum of two integers always greater than the difference between them? why or why not? give examples to support your answer.

Let x and y be the two integers. 

The sum of the integers is x+y while the difference is x-y assuming x is larger than y. 

If x+y > x-y, then 

x+y > x-y
x+y-x > x-y-x
y > -y
y+y > -y+y
2y > 0
2y/2 > 0/2
y > 0

So as long as y is positive, this makes the sum greater than the difference
For example, if x = 10 and y = 2, then
x+y = 10+2 = 12
x-y = 10-2 = 8
clearly 12 > 8 is true

If y is some negative number (say y = -4), then
x+y = 10+(-4) = 10-4 = 6
x-y = 10-(-4) = 10+4 = 16
and things flip around

Saying a blanket statement "the sum of two integers is always greater than their difference" is false overall. If you require y to be positive, then it works but as that last example shows, it doesn't always work. 

So to summarize things up, I'd say the answer is "no, the statement isn't true overall"