Wikipedia says that any real number $x$ can be written as $|x| = \text{sgn}(x)x$ where $\text{sgn}(x)$ is the sign function. Rearranging, this means that $\frac{\text{sgn}(x)}{|x|} = \frac{1}{x}$ for $x\neq 0$.
On the other hand, WolframAlpha says that $\frac{\text{sgn}(x)}{|x|} = \frac{1}{x^*}$ where $x^*$ is the conjugate of $x$ and that $\frac{\text{sgn}(x)}{|x|}$ is only equal to $\frac{1}{x}$ for $x>0$.
Is this an inconsistency? Why does WolframAlpha omit the $x<0$ case?
That WA is talking about conjugates should be a clear hint that you are thinking about $x$ as a real number and WA is working with complex $x$. If you give WA the hint that $x$ is real, it give you what you expect: WA: sgn(x)/|x| for x in reals