Solve the inequality
\[\frac{(x - 2)(x - 3)(x - 4)}{(x - 1)(x - 5)(x - 6)} > 0.\]
We can build a sign chart, but since all of the factors are linear, we can track what happens to the expression as $x$ increases.  At $x = 0,$ the expression is positive.  As $x$ increases past 1, the expression becomes negative.  As $x$ increases past 2, the expression becomes positive, and so on.  Thus, the solution is
\[x \in \boxed{(-\infty,1) \cup (2,3) \cup (4,5) \cup (6,\infty)}.\]