For which non-zero real numbers $x$ is $\frac{|x-|x|\-|}{x}$ a positive integer? 
$\textbf{(A)}\ \text{for negative } x \text{ only} \qquad \\ \textbf{(B)}\ \text{for positive } x \text{ only} \qquad \\ \textbf{(C)}\ \text{only for } x \text{ an even integer} \qquad \\ \textbf{(D)}\ \text{for all non-zero real numbers } x \\ \textbf{(E)}\ \text{for no non-zero real numbers } x$
Solution by e_power_pi_times_i

Notice that if $x$ is negative, then the whole thing would amount to a negative number. Also notice that if $x$ is positive, then $|x-|x|\-|$ would be $0$, hence the whole thing would amount to $0$. Therefore, $\frac{|x-|x|\-|}{x}$ is positive $\boxed{\textbf{(E)}\ \text{for no non-zero real numbers } x}$.