Solving this question requires us to first recognize the 4 situations in which two absolute values can equal each other. I'll show this with 4 examples:
|4|=|4| both values inside are already positive
|4|=|4| both values inside are negative
|-4|=|4| one value is negative and the other is positive
|4|=|-4| one value is positive and the other is negative

So, (1) gives us 4 possible situations:

a. x+2 = x-3 (both sides already yield positive outcomes)
b. -(x+2) = -(x-3) (both sides yield negative outcomes, which we "undo" or make positive, by negating both sides, which is the same as finding the abs value of each side)
c. -(x+2) = x-3 (the left side yields a negative outcome, which we "undo" by negating)
d. x+2 = -(x-3) (the right side yields a negative outcome, which we "undo" by negating)

Note: c is actually impossible since it could never be the case that x+2 is negative while x-3 is positive, but it doesn't really matter here.

If we solve each of the 4 equations we find that a and b cannot be solved. c and d yield x=1/2

Since x=1/2, we can conclude that (1) is sufficient.

(2) is not sufficient.

Answer: A

You're absolutely right, vittalgmat.
I did more work than was necessary.
Good catch!

joanjgonzalez wrote:Is x an integer?

1. |x+2| = |x-3|
2. x>0

answer is [spoiler]A

Thanks![/spoiler]

Or, using a distance approach, from Statement 1 we have |x - (-2)| = |x - 3|, which says, in words, "the distance between x and -2 is equal to the distance between x and 3". So x must be halfway between -2 and 3, and x = 1/2.