How many integers x satisfy |x − 4| + |x + 3| ≤ 10?

|x − 4| + |x + 3| is the sum of the distances of x from 4 and from −3.

For −3 ≤ x ≤ 4 the sum is constant = 7 ≤ 10, so all of these 8 integers work.
For x > 4: (x − 4) + (x + 3) = 2x − 1 ≤ 10 ⇒ x ≤ 5.5 ⇒ x ≤ 5.
For x < −3: (4 − x) + (−3 − x) = 1 − 2x ≤ 10 ⇒ x ≥ −4.5 ⇒ x ≥ −4.

So −4 ≤ x ≤ 5, which contains 10 integers.