We need to solve the inequality $\frac{|x-1|}{x} < 0$.

1. Problem Description

We need to solve the inequality $\frac{|x-1|}{x} < 0$.

2. Solution Steps

First, we observe that the absolute value $|x-1|$ is always non-negative, i.e., $|x-1| \ge 0$ for all $x$. For the fraction $\frac{|x-1|}{x}$ to be negative, since the numerator is non-negative, we must have:

$|x-1| \ge 0$

For the fraction to be negative, we need the numerator to be positive and the denominator to be negative.

However, if $|x-1|=0$, the fraction is equal to 0, which does not satisfy the strict inequality $\frac{|x-1|}{x} < 0$. So, we must have $|x-1| > 0$.

$|x-1| > 0$ means $x-1 \ne 0$, thus $x \ne 1$.

We also require the denominator to be negative, i.e., $x < 0$.

Therefore, we need $x<0$ and $x \ne 1$. Since $x<0$, $x$ is already not equal to

1.

3. Final Answer

The solution to the inequality $\frac{|x-1|}{x} < 0$ is $x < 0$. In interval notation, this is $(-\infty, 0)$.