Numerical answer
How many integer values of x satisfy |x − 3| + |x + 2| ≤ 9?
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Answer: 10
|x − 3| + |x + 2| is the sum of the distances of x from 3 and from −2.
- For −2 ≤ x ≤ 3, the sum is always 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 the integers −4, −3, …, 5: 10 values.
Tip: The excess 9 − 5 = 4 is split equally, extending the interval [−2, 3] by 2 on each side.
- For −2 ≤ x ≤ 3, the sum is always 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 the integers −4, −3, …, 5: 10 values.
Tip: The excess 9 − 5 = 4 is split equally, extending the interval [−2, 3] by 2 on each side.
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