Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 10A Problems/Problem 6"

(Created page with "==Problem== Which expression is equal to <cmath>\left|a-2-\sqrt{(a-1)^2}\right|</cmath> for <math>a<0?</math> <math>\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textb...")
 
(→‎Solution)
Line 14: Line 14:
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
 +
 +
== Solution 2 ==
 +
WLOG, assume <math>a=-1.</math> Then, the given expression simplifies to <math>5</math>:
 +
<cmath>\left|a-2-\sqrt{(a-1)^2}\right| = \left|-1-2-\sqrt{(-1-1)^2}\right|
 +
= \left|-1-2-\sqrt{4}\right|
 +
= \left|-1-2-2\right|
 +
= 5.</cmath>
 +
 +
Then, we test each of the answer choices to see which one is equal to <math>5</math>:
 +
 +
<math>A:</math> <math>3-2a = 3-2\cdot(-1) = 3+2 = 5.</math>
 +
 +
<math>B:</math> <math>1-a = 1-(-1) = 2 \neq 5.</math>
 +
 +
<math>C:</math> <math>1 \neq 5.</math>
 +
 +
<math>D:</math> <math>a+1 = -1+1 = 0 \neq 5.</math>
 +
 +
<math>E:</math> <math>3 \neq 5.</math>
 +
 +
The only answer choice equal to <math>5</math> for <math>a=-1</math> is <math>A</math>, so the answer is <math>\boxed{\textbf{(A) } 3-2a}.</math>
 +
-MathWizard09
  
 
== See Also ==
 
== See Also ==

Revision as of 01:47, 12 November 2022

Problem

Which expression is equal to \[\left|a-2-\sqrt{(a-1)^2}\right|\] for $a<0?$

$\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

Solution

We have \begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|a-2-|a-1|\right| \\ &=\left|a-2-(-a+1)\right| \\ &=\left|2a-3\right| \\ &=\boxed{\textbf{(A) } 3-2a}. \end{align*} ~MRENTHUSIASM

Solution 2

WLOG, assume $a=-1.$ Then, the given expression simplifies to $5$: \[\left|a-2-\sqrt{(a-1)^2}\right| = \left|-1-2-\sqrt{(-1-1)^2}\right| = \left|-1-2-\sqrt{4}\right| = \left|-1-2-2\right| = 5.\]

Then, we test each of the answer choices to see which one is equal to $5$:

$A:$ $3-2a = 3-2\cdot(-1) = 3+2 = 5.$

$B:$ $1-a = 1-(-1) = 2 \neq 5.$

$C:$ $1 \neq 5.$

$D:$ $a+1 = -1+1 = 0 \neq 5.$

$E:$ $3 \neq 5.$

The only answer choice equal to $5$ for $a=-1$ is $A$, so the answer is $\boxed{\textbf{(A) } 3-2a}.$ -MathWizard09

See Also

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_10A_Problems/Problem_6&oldid=180892"