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2022 AMC 12B Problem 1

Problem 1 of 25EasierAlgebraArithmetic

Define x⋄yx \diamond y to be ∣x−y∣|x - y| for all real numbers xx and y.y. What is the value of (1⋄(2⋄3))−((1⋄2)⋄3)?(1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3)?

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Solution

Since 2⋄3=∣2−3∣=1,2 \diamond 3 = |2-3| = 1, we get 1⋄(2⋄3)=1⋄1=0.1 \diamond (2 \diamond 3) = 1 \diamond 1 = 0. Since 1⋄2=∣1−2∣=1,1 \diamond 2 = |1-2| = 1, we get (1⋄2)⋄3=1⋄3=∣1−3∣=2.(1 \diamond 2) \diamond 3 = 1 \diamond 3 = |1-3| = 2. The value is 0−2=−2.0 - 2 = -2. Thus, the correct answer is A.
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Tagged: custom operation · absolute value · order of operations

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