Question

Simplify LHS: Let $y=x-4$, then equation becomes easier. Or directly: $\frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3}$. Finding common denominator $(x-3)(x-5)$: $(x-2)(x-5)+(x-4)(x-3)=\frac{10}{3}(x-3)(x-5)$. Expanding and simplifying leads to a quadratic equation. Solving gives the roots.