Let x=-2016. What is the value of \vert\vert |x|-x\vert-|x|\vert-x?

2016 AMC 10B Problem 3 — Solution

2016 AMC 10B Problem 3

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Concepts:absolute value

Difficulty rating: 960

Let $x=-2016.$ What is the value of $\Bigg\vert\Big\vert |x|-x\Big\vert-|x|\Bigg\vert-x?$

$\ -2016$

$\ 0$

$\ 2016$

$\ 4032$

$\ 6048$

Solution:

Observe that: $\begin{align*} &\Bigg\vert\Big\vert |x|-x\Big\vert-|x|\Bigg\vert-x\\ &=\Bigg\vert \Big\vert -x-x\Big\vert-|x|\Bigg\vert-x \\ &=\Bigg\vert\Big\vert -2x\Big\vert-|x|\Bigg\vert-x \end{align*}$ since $x < 0$ implies that $-x = |x|.$ Substituting values, we can see that $|4032-2016|-(-2016)$ $= 2016 +2016$ $= 4032.$

Thus, the correct answer is D .

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