Example 1. Solve the equation: $\frac{3 x}{x^{2}-1}+\frac{x^{2}-1}{3 x}=-\frac{5}{2}$.
Solve the original equation $\stackrel{\text { Proposition } 2 \text { }}{\Longleftrightarrow} \frac{3 x}{x^{2}-1}=2$ or $\frac{3 x}{x^{2}-1}=$ $\frac{1}{2}$, solving it yields $x_{1}=2, x_{2}=-\frac{1}{2}, x_{3}=3+\sqrt{10}$,