Difference between revisions of "2016 AMC 12B Problems/Problem 3"

Latest revision as of 21:20, 14 October 2020

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

$\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

By: dragonfly

First of all, lets plug in all of the $x$ 's into the equation.

$\bigg|$ $||-2016|-(-2016)|-|-2016|$ $\bigg|$ $-(-2016)$

Then we simplify to get

$\bigg|$ $|2016+2016|-2016$ $\bigg|$ $+2016$

which simplifies into

$\bigg|$ $2016$ $\bigg|$ $+2016$

and finally we get $\boxed{\textbf{(D)}\ 4032}$

Consider $x$ is negative.

We replace all instances of $x$ with $|x|$ :

$\bigg|$ $||x|+|x||-|x|$ $\bigg|$ $+|x|$

$=$ $\bigg|$ $|2x|-|x|$ $\bigg|$ $+|x|$

$=$ $|x|$ $+|x|$

$=|2x|$

$=4032 \implies \boxed{\textbf{(D)}}$

omerselim1

2016 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

@@ Line 34: / Line 34: @@
 <math>=</math> <math>|x|</math> <math>+|x|</math>
-<math>=|2x|
+<math>=|2x|</math>
-</math>=4032 \boxed{\textbf{(D)}\}$
+<math>=4032 \implies \boxed{\textbf{(D)}}</math>
 omerselim1