Can we divide two terms in finding solution for Linear Inequalities?

Question

Solve the following inequality.$$\frac{|x-2|}{x-2}>0$$

If $x$ is greater than or equal to $2$, it becomes: $\frac{x-2}{x-2}>0$, which can become $1>0$, which is true for all $x>2$ for when $x=2$,its value will become undefined. Is my solution correct?

Answer 1

Just three cases:

$x>2$ gives $1>0$, which is true.

$x<2$ gives $-1>0$, which is wrong and

$x=2$, which is impossible.

After all these things we got the answer: $(2,+\infty)$.

Answer 2

Yes, you solution is OK.

You could also notice that $|x-2| \ge 0$ and $x \ne 2$ as denominator cannot be $0$.

Thus in order for your whole equation to be greater than $0$, since the numerator is positive, it has to be the case that the denominator is also positive, which gives us the answer: