Numerical answer
Find the sum of the squares of all real numbers x that satisfy |x − 2| + |x + 3| = 7.
Show answer and solution
Answer: 25
The critical points are x = −3 and x = 2.
For −3 ≤ x ≤ 2: |x − 2| + |x + 3| = (2 − x) + (x + 3) = 5 ≠ 7 → no solution.
For x > 2: (x − 2) + (x + 3) = 2x + 1 = 7 → x = 3.
For x < −3: (2 − x) − (x + 3) = −2x − 1 = 7 → x = −4.
Sum of squares = 9 + 16 = 25.
Tip: the sum of distances from 2 and −3 is at least 5 (the gap between them); a value 7 is achieved at points 1 unit outside each end.
For −3 ≤ x ≤ 2: |x − 2| + |x + 3| = (2 − x) + (x + 3) = 5 ≠ 7 → no solution.
For x > 2: (x − 2) + (x + 3) = 2x + 1 = 7 → x = 3.
For x < −3: (2 − x) − (x + 3) = −2x − 1 = 7 → x = −4.
Sum of squares = 9 + 16 = 25.
Tip: the sum of distances from 2 and −3 is at least 5 (the gap between them); a value 7 is achieved at points 1 unit outside each end.
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