jmc

Subtracting $\frac{1}{30}$ from both sides, we get $$\frac{1}{x-1}-\frac{4}{x-2}+\frac{4}{x-3}-\frac{1}{x-4}-\frac{1}{30}<0.$$Putting everything over a common denominator, we get $$\frac{-x^4+10x^3-5x^2-100x-84}{30(x-1)(x-2)(x-3)(x-4)}<0,$$which factors as $$-\frac{(x+2)(x+1)(x-6)(x-7)}{(x-1)(x-2)(x-3)(x-4)}<0.$$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 $x=-3,$ the expression is negative. As $x$ increases past $-2,$ the expression becomes positive. As $x$ increases past $-1,$ the expression becomes negative, and so on. Thus, the solution is $$x\in \boxed{(-\infty ,-2)\cup (-1,1)\cup (2,3)\cup (4,6)\cup (7,\infty )}.$$