2010 AMC 12A Problems/Problem 4: Difference between revisions
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== Problem | == Problem == | ||
If <math>x<0</math>, then which of the following must be positive? | If <math>x<0</math>, then which of the following must be positive? | ||
<math>\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}</math> | <math>\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}</math> | ||
[[ | == Solution == | ||
<math>x</math> is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use <math>-1</math>. | |||
<math>\textbf{(A)} \Rightarrow \frac{-1}{|-1|} = -1</math> | |||
<math>\textbf{(B)} \Rightarrow -(-1)^2 = -1</math> | |||
<math>\textbf{(C)} \Rightarrow -2^{(-1)} = -\frac{1}{2}</math> | |||
<math>\textbf{(D)} \Rightarrow -(-1)^{(-1)} = 1</math> | |||
<math>\textbf{(E)} \Rightarrow \sqrt[3]{-1} = -1</math> | |||
Obviously only <math>\boxed{\textbf{(D)}}</math> is positive. | |||
==Video Solution== | |||
https://youtu.be/TKbHkw07O34?si=Lb-qc11KHuUooO22 | |||
~Charles3829 | |||
==Video Solution== | |||
https://youtu.be/13Hp_RPhX4Q | |||
~Education, the Study of Everything | |||
== See also == | |||
{{AMC12 box|year=2010|num-b=3|num-a=5|ab=A}} | |||
[[Category:Introductory Algebra Problems]] | |||
{{MAA Notice}} | |||
Latest revision as of 09:56, 18 September 2024
Problem
If 
, then which of the following must be positive?
![$\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}$](http://latex.artofproblemsolving.com/8/a/6/8a6b210377d3c8faaed5b00e17cbd935d20a0295.png)
Solution
is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use 
.
![$\textbf{(E)} \Rightarrow \sqrt[3]{-1} = -1$](http://latex.artofproblemsolving.com/5/d/a/5da4b91ca769554cfbff384adaa5aa432a2ef63a.png)
Obviously only 
is positive.
Video Solution
https://youtu.be/TKbHkw07O34?si=Lb-qc11KHuUooO22
~Charles3829
Video Solution
~Education, the Study of Everything
See also
| 2010 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 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 | |
These problems are copyrighted © by the Mathematical Association of America. 
