Difference between revisions of "2010 AMC 12A Problems/Problem 4"
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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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
