2010 AMC 12A Problems/Problem 4

Problem

If $x<0$, then which of the following must be positive?

$\textbf{(A)}\ \frac{x}{\left|x\right|} \qquad \textbf{(B)}\ -x^2 \qquad \textbf{(C)}\ -2^x \qquad \textbf{(D)}\ -x^{-1} \qquad \textbf{(E)}\ \sqrt[3]{x}$

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 $-1$.

$\textbf{(A)} \Rightarrow \frac{-1}{|-1|} = -1$

$\textbf{(B)} \Rightarrow -(-1)^2 = -1$

$\textbf{(C)} \Rightarrow -2^{(-1)} = -\frac{1}{2}$

$\textbf{(D)} \Rightarrow -(-1)^{(-1)} = 1$

$\textbf{(E)} \Rightarrow \sqrt[3]{-1} = -1$


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

See also

2010 AMC 12A (ProblemsAnswer KeyResources)
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. AMC logo.png

Invalid username
Login to AoPS