jmc

To clear denominators, we multiply both sides by $(x-2)(x-4)(x-3)$: $$(x-4)(x-3)+(x-2)(x-3)=3(x-2)(x-4),$$or $$(x^2-7x+12)+(x^2-5x+6)=3(x^2-6x+8).$$Moving all the terms to the right-hand side, we get $$x^2-6x+6=0.$$By the quadratic formula, $$x=\frac{6\pm \sqrt{6^2-4\cdot 6}}{2}=3\pm \sqrt{3}.$$Therefore, the smallest solution is $x=\boxed{3-\sqrt{3}}.$