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Problem #6 · August 2, 2026

Absolute value

How many integers satisfy the equation ∣x−2∣+∣x+3∣=5|x - 2| + |x + 3| = 5?
  1. A2
  2. B4
  3. C5
  4. D6
  5. EInfinitely many

WORKED EXAMPLE

Solution

∣x−2∣+∣x+3∣|x-2| + |x+3| is the sum of the distances from xx to the points 22 and −3-3; its minimum is 2−(−3)=52 - (-3) = 5, and precisely on the interval −3≤x≤2-3 \le x \le 2 it equals 55. Thus, the equation holds at every point in this interval. The integer solutions are −3,−2,−1,0,1,2-3, -2, -1, 0, 1, 2 — a total of 6.
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