AMC12 2022 B

AMC12 2022 B · Q1

AMC12 2022 B · Q1. It mainly tests Absolute value.

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 $x\diamond y$ 为所有实数 $x$ 和 $y$ 的 $|x-y|$。$(1\diamond(2\diamond3))-((1\diamond2)\diamond3)$ 的值为多少？

(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*}

Topics

AL.004 Absolute value