AMC10 2022 A

AMC10 2022 A · Q6

AMC10 2022 A · Q6. It mainly tests Absolute value, Manipulating equations.

Which expression is equal to \[\left|a-2-\sqrt{(a-1)^2}\right|\] for $a<0?$

对于 $a<0$，下式等于多少？\[\left|a-2-\sqrt{(a-1)^2}\right|\]

(A) 3-2a 3-2a

(B) 1-a 1-a

(D) a+1 a+1

(E) 3 3

Answer

Correct choice: (A)

正确答案：（A）

Solution

We have \begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|a-2-|a-1|\right| \\ &=\left|a-2-(1-a)\right| \\ &=\left|2a-3\right| \\ &=\boxed{\textbf{(A) } 3-2a}. \end{align*}

我们有 \begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|a-2-|a-1|\right| \\ &=\left|a-2-(1-a)\right| \\ &=\left|2a-3\right| \\ &=\boxed{\textbf{(A) } 3-2a}. \end{align*}

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