1975 AHSME Problem 7

For which non-zero real numbers x is \frac{|x-|x||}{x} a positive integer?

\begin{array}{l} \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 \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.