How many integers x satisfy |x − 3| + |x + 2| ≤ 9?

The expression |x − 3| + |x + 2| is the sum of distances of x from 3 and from −2.

- For −2 ≤ x ≤ 3 the sum is constant = 5 ≤ 9 ✓.
- For x > 3: (x − 3) + (x + 2) = 2x − 1 ≤ 9 ⇒ x ≤ 5.
- For x < −2: (3 − x) + (−x − 2) = 1 − 2x ≤ 9 ⇒ x ≥ −4.

So −4 ≤ x ≤ 5, giving 10 integers.

Tip: The distance view shows the solution is an interval symmetric about the midpoint 0.5; the extra 9 − 5 = 4 is split as 2 on each side.