How many integers x satisfy |x − 3| + |x + 5| ≤ 12?

|x − 3| + |x + 5| is the sum of distances of x from 3 and from −5.

For −5 ≤ x ≤ 3 the sum is always 8 (the distance between −5 and 3), which is ≤ 12.
For x > 3: (x − 3) + (x + 5) = 2x + 2 ≤ 12 → x ≤ 5.
For x < −5: (3 − x) + (−x − 5) = −2x − 2 ≤ 12 → x ≥ −7.

So −7 ≤ x ≤ 5, which contains 13 integers.

Tip: The extra allowance 12 − 8 = 4 is split equally, 2 units beyond each end: [−7, 5].