6. The solution set of the inequality $\frac{1}{x-1}+\frac{2}{x-2} \geqslant \frac{3}{2}$ is the union of some non-overlapping intervals with a total length of $\qquad$.
6. 2 .

The original inequality is transformed into $\frac{3 x^{2}-15 x+14}{(x-1)(x-2)} \leqslant 0$, yielding $1<x \leqslant \frac{15-\sqrt{57}}{6}$ or $2<x \leqslant \frac{15+\sqrt{57}}{6}$.