How many integer values of x satisfy |x − 3| + |x + 2| ≤ 7?

|x−3| + |x+2| is the sum of distances of x from 3 and from −2, whose minimum value is the distance between 3 and −2, i.e., 5 (attained for all x in [−2,3]). The 'slack' available is 7 − 5 = 2, split equally on each side, so x ranges over [−2−1, 3+1] = [−3, 4]. Integers from −3 to 4 inclusive: −3,−2,−1,0,1,2,3,4 — that's 8 values.