Answer
False.
Work Step by Step
Assume that for $y=\frac{x}{|x|}$, $y=1$ for all $x\ne0$ in the real numbers. Therefore, $y$ should equal $1$ for $x=-1$. At $x=-1$, $$y=\frac{-1}{|-1|}=\frac{-1}{1}=-1$$ This y-value isn't equal to $1$, though it is a real number not equal to zero. Therefore, by contradiction, the statement is false.
