How many integers x satisfy |x − 3| + |x + 2| ≤ 11?

The critical points are x = −2 and x = 3.

- For −2 ≤ x ≤ 3: the sum is (3 − x) + (x + 2) = 5 ≤ 11, so every x in this range works.
- For x > 3: 2x − 1 ≤ 11, so x ≤ 6.
- For x < −2: −2x + 1 ≤ 11, so x ≥ −5.

So −5 ≤ x ≤ 6, which gives 6 − (−5) + 1 = 12 integers.

Tip: |x − a| + |x − b| is the total distance from x to a and to b. It is constant between a and b and grows by 2 per unit outside that range.