You could also think about this conceptually. |a - b| is the distance between a and b, and |a + b| is the distance between a and -b. So
|x + 5| = distance between x and -5
|x - 5| = distance between x and 5
We want the difference in those distances, or |x + 5| - |x - 5|. Since the difference between 5 and -5 is 10, the difference between |x + 5| and |x - 5| can be anything from -10 to 10, but nothing beyond that. (This isn't simply true for integers: we can get any real number from -10 to 10, but no real number outside of that range.)
To see why, imagine x at either end of the range. If x = -5, then we have 0 - 10, or -10. If x = 5, then we have 10 - 0, or 10. With 5 > x > -5, we can get any value between -10 and 10. If x > 5, then the difference simply becomes the difference between 5 and -5, or 10, and if x < -5, then the difference is simply -5 - 5, or -10, since the only relevant distance now is the distance from -5 to 5.
There are 21 integers in the set {-10, -9, ..., 0, ..., 9, 10}, so that's our pick.
