Find the solution set of the in equation

$\frac{|x-2|}{(x-2)}<0$ means we have to find values of x for which $\frac{|x-2|}{(x-2)}$ is negative

Observe that the numerator $|x-2|$ is always positive because of mod, hence for $^{\frac{|x-2|}{(x-2)}}$ to be a negative quantity the denominator $(x-2)$ has to be negative

That is $x-2$ should be less than 0

$\Rightarrow x-2<0$

$\Rightarrow x<2$

Hence $x$ should be less than 2 for $\frac{|x-2|}{(x-2)}<0$

$x<2$ means $x$ can take values from $-\infty$ to 2 hence $x \in(-\infty, 2)$

Hence the solution set for $\frac{|x-2|}{(x-2)}<0$ is $(-\infty, 2)$