# 2010 AMC 12A Problems/Problem 4

## Problem

If $x<0$, then which of the following must be positive? $\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt{x}$

## Solution $x$ is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use $-1$. $\textbf{(A)} \Rightarrow \frac{-1}{|-1|} = -1$ $\textbf{(B)} \Rightarrow -(-1)^2 = -1$ $\textbf{(C)} \Rightarrow -2^{(-1)} = -\frac{1}{2}$ $\textbf{(D)} \Rightarrow -(-1)^{(-1)} = 1$ $\textbf{(E)} \Rightarrow \sqrt{-1} = -1$

Obviously only $\boxed{\textbf{(D)}}$ is positive.

## See also

 2010 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 3 Followed byProblem 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

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