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2016 AMC 10B Problem 3

Problem 3 of 25EasierAlgebra

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

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Solution

Since x=−2016<0,x=-2016<0, we have ∣x∣=2016.|x|=2016. Thus ∣x∣−x=2016−(−2016)=4032, \begin{aligned} |x|-x&=2016-(-2016) \\ &=4032, \end{aligned} so the innermost absolute value is 4032.4032. Therefore the full expression is ∣4032−2016∣+2016=4032.|4032-2016|+2016=4032. Thus, the correct answer is D .

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Concepts: absolute value

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.