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A shipping clerk flags a package for re-weighing whenever the absolute difference between the package's measured weight w, in pounds, and the target weight of 50 pounds is exactly 8 pounds. What is the sum of all weights w, in pounds, that would be flagged for re-weighing?
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Correct answer
E
Translate the rule into an absolute-value equation: the absolute difference between w and 50 is exactly 8, so |w − 50| = 8. An absolute value equal to a positive number splits into two cases, and both must be kept. Case 1: w − 50 = 8, so w = 58. Case 2: w − 50 = −8, so w = 42. Both 58 and 42 are valid positive weights, so exactly two readings get flagged. Their sum is 58 + 42 = 100, which is choice E.
A useful check: solutions of an absolute-value equation centered at 50 are symmetric about 50 (one is 8 above, one is 8 below), so their sum is twice 50, again 100.
Choice D (58) drops the negative case and reports only the higher reading. Choice B (42) drops the positive case and reports only the lower reading. Choice C (50) is the target weight itself, which is also the average of the two flagged weights, so it tempts anyone who computes the midpoint instead of the sum. Choice A (16) comes from adding the 8-pound tolerance to itself rather than adding the two actual weights.
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