2010 AMC 12A Problems/Problem 4: Difference between revisions
Created page with '== Problem 4 == 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…' |
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<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. | |||
Revision as of 16:30, 10 February 2010
Problem 4
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.
