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2016 AMC 12B Problem 3

Problem 3 of 25EasierAlgebraArithmetic

Let x=−2016.x=-2016. What is the value of ∣  ∣ ∣x∣−x ∣−∣x∣  ∣−x? \Big|\;\big|\,|x|-x\,\big|-|x|\;\Big|-x?

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Solution

Since x=−2016,x=-2016, ∣x∣=2016.|x|=2016. The innermost expression is ∣x∣−x=2016+2016=4032.|x|-x=2016+2016=4032. Then ∣4032∣−∣x∣=4032−2016\big|4032\big|-|x|=4032-2016 =2016,=2016, and the outer absolute value leaves 2016.2016. Finally subtracting xx gives 2016−(−2016)=4032.2016-(-2016)=4032. Thus, the correct answer is D.
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