Solution for the inequality: $(x-1)(x-2)(x-3)>0$ Recently I came across an inequality like this:
$$(x-1)(x-2)(x-3)>0$$
The question was: Which solution for this in inequality is correct?


*

*$x>3$ or $x<1$ 

*$x>3$ or $1<x<2$ 

*$x<1$ or $2<x<3$ 

*There are no solutions


There was a timelimit of 3 minutes for solving this. I freaked out a  bit as I haven't worked that much with these kind of inequalities yet. I thought about multiplying out the terms on the left side, and then checking the solutions, but even after I tried that on paper, there is no way I could have done that in 3 minutes under pressure. 
What is the expected, fast way to solve this inequality? Or was it just expected to "see" it, based on the given solutions?

For a multiple-choice problem, some simple reasoning can narrow down the options quickly.
If $x$ is a large positive number, all three terms in the product are positive.  This eliminates options 4 and 3.
If $x$ is a large negative number. all three terms in the product are negative.  This eliminates option 1.
That leaves option 2 as the only remaining possibility.  
