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2016 AMC 10B Problem 3

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2016 AMC 10B Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Absolute Value & Inequalities · difficulty 1 of 5 · about 1 min · position P1-10

Let $x=-2016$ . What is the value of $\bigg| \big||x|-x\big|-|x| \bigg| -x?$

A)
$-2016$
B)
$0$
C)
$2016$
D)
$4032$
E)
$6048$

Answer

Key insight

Because x is negative, |x| = -x = 2016, so every '-x' adds 2016; peel the absolute values from the inside out.

Solution

Since $x = -2016$ is negative, $|x| = 2016$ and $-x = 2016$ . Work from the innermost bars outward.

$|x| - x = 2016 + 2016 = 4032$ .
$\big| 4032 \big| = 4032$ .
$4032 - |x| = 4032 - 2016 = 2016$ .
$\big| 2016 \big| = 2016$ .
Finally, $2016 - x = 2016 + 2016 = 4032$ .

The answer is $\boxed{\textbf{(D)}\ 4032}$ .

Why this works

Once you record the two facts $|x| = 2016$ and $-x = +2016$ , the expression is plain arithmetic. Nested absolute values are evaluated from the inside out, and here every quantity inside the bars is already positive, so the bars change nothing. The only danger is the sign of $-x$ .

Alternative approach

Substitute $x = -y$ with $y = 2016 > 0$ : the expression becomes $\big||y+y| - y\big| + y = |2y - y| + y = 2y = 4032$ .

The trap

Treating the final '-x' as subtracting 2016 instead of adding it, which gives 0 (choice B).

Common mistakes

Treating the final $-x$ as subtracting $2016$ instead of adding it, which gives $0$ (choice B).
Computing $|x| - x$ as $0$ by forgetting that $x$ itself is negative.

Techniques

Set up the equation/formula and compute; no special trick needed