How many integers x satisfy |x − 5| + |x + 4| ≤ 13?

The expression is the sum of the distances of x from 5 and from −4 on the number line.

For −4 ≤ x ≤ 5 the sum is always 9 (the distance between −4 and 5), so the inequality holds.

For x > 5: (x − 5) + (x + 4) = 2x − 1 ≤ 13 → x ≤ 7.
For x < −4: (5 − x) + (−x − 4) = 1 − 2x ≤ 13 → x ≥ −6.

So −6 ≤ x ≤ 7, which contains 7 − (−6) + 1 = 14 integers.

Tip: |x − a| + |x − b| ≤ k (with k ≥ |a − b|) is an interval centred at (a + b)/2 with half-width k/2: here 0.5 ± 6.5.