AMC10 2022 B

AMC10 2022 B · Q1

AMC10 2022 B · Q1. It mainly tests Absolute value, Algebra misc.

Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of \[(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?\]

定义对于所有实数$x$和$y$，$x\diamond y=|x-y|$。$(1\diamond(2\diamond3))-((1\diamond2)\diamond3)$的值是多少？

(A) {-}2 {-}2

(B) {-}1 {-}1

(D) 1 1

(E) 2 2

Answer

Correct choice: (A)

正确答案：（A）

Solution

We have \begin{align*} (1\diamond(2\diamond3))-((1\diamond2)\diamond3) &= |1-|2-3|| - ||1-2|-3| \\ &= |1-1| - |1-3| \\ &= 0-2 \\ &= \boxed{\textbf{(A)}\ {-}2}. \end{align*}

我们有\begin{align*} (1\diamond(2\diamond3))-((1\diamond2)\diamond3) &= |1-|2-3|| - ||1-2|-3| \\ &= |1-1| - |1-3| \\ &= 0-2 \\ &= \boxed{\textbf{(A)}\ {-}2}. \end{align*}

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