AMC 10 · 2016 · #3

Grade 7 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
absolute-valueorder-of-operations identify-subproblems ↑ Prerequisites: absolute-value
📏 Medium solution 💡 2 insights
Problem
With x=-2016, evaluate the nested absolute-value expression | | |x|-x |-|x| |-x.

Pick an answer.

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

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The expression looks tangled because three absolute-value bars are nested inside one another, but each bar is just a grouping symbol. Tool #7 (Identify Subproblems) lets us peel the bars one at a time, innermost first, so the scary stack becomes a short list of easy subtractions. Tool #4 (Introduce a Variable) is already half-done for us — x stands for -2016, and substituting that value turns every bar into plain arithmetic. Tool #3 (Eliminate Possibilities) then matches the final number to the one answer choice that fits.

1STEP 1

Read the bars as grouping

Bars act like parentheses: finish the innermost group first and work outward — |x|-x, then subtract |x|, then subtract x.

| | |x|-x¹st |₂nd-|x| |-x
2STEP 2

Substitute the value of x

Replace x with -2016, so |x|=|-2016|=2016 (its distance from 0). From now every x is -2016 and every |x| is 2016.

x=-2016, |x|=|-2016|=2016
3STEP 3

Innermost bar: |x| - x

The innermost group |x|-x=2016-(-2016)=4032; subtracting a negative adds. Since it is already positive, |4032|=4032.

|x|-x = 2016-(-2016) = 4032, |4032| = 4032
4STEP 4

Middle bar: subtract |x|

Subtract |x|=2016 from that: 4032-2016=2016; the middle bar changes nothing since 2016 is positive, so |2016|=2016.

| 4032-|x| | = | 4032-2016 | = |2016| = 2016
5STEP 5

Outer step: subtract x

The last step subtracts x: 2016-(-2016) adds one more 2016, giving 4032. Only that choice fits, so the answer is (D).

2016-x = 2016-(-2016) = 4032 = (D)
Answer
4032
Track the value outward: |x|-x=4032, then 4032-2016=2016, then 2016-x=4032. The two subtractions of a negative each add 2016, so a final value of 4032=2×2016 is exactly what we'd expect, and it matches choice (D). The trap choices 2016 and 6048 come from forgetting the final -x or double-counting it.
💡Key takeaway

Absolute-value bars are just grouping boxes — open them from the inside out, and remember that subtracting a negative means adding.

  • Read the bars as grouping
  • Substitute the value of x
  • Innermost bar: |x| - x
  • Middle bar: subtract |x|
  • Outer step: subtract x

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