Numerical answer
How many integers x satisfy |x − 6| + |x + 5| ≤ 19?
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Answer: 20
|x − 6| + |x + 5| is the sum of distances of x from 6 and from −5.
- For −5 ≤ x ≤ 6 the sum is constant = 11 ≤ 19, so all of these work.
- For x > 6: (x − 6) + (x + 5) = 2x − 1 ≤ 19 ⇒ x ≤ 10.
- For x < −5: (6 − x) + (−x − 5) = 1 − 2x ≤ 19 ⇒ x ≥ −9.
So −9 ≤ x ≤ 10, which contains 10 − (−9) + 1 = 20 integers.
Tip: The sum of distances equals 19 at points (19 − 11)/2 = 4 units outside the interval [−5, 6], i.e. at −9 and 10.
- For −5 ≤ x ≤ 6 the sum is constant = 11 ≤ 19, so all of these work.
- For x > 6: (x − 6) + (x + 5) = 2x − 1 ≤ 19 ⇒ x ≤ 10.
- For x < −5: (6 − x) + (−x − 5) = 1 − 2x ≤ 19 ⇒ x ≥ −9.
So −9 ≤ x ≤ 10, which contains 10 − (−9) + 1 = 20 integers.
Tip: The sum of distances equals 19 at points (19 − 11)/2 = 4 units outside the interval [−5, 6], i.e. at −9 and 10.
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