Example 3. Solve the equation for $x$: $\frac{a-x}{b+x}=5-\frac{4(b+x)}{a-x}$. $(a+b \neq 0)$ (Exercise 3 (2), page 117)
Solve the original equation $\stackrel{\text { transformation }}{\Longleftrightarrow} \frac{a-x}{b+x}+\frac{4(b+x)}{a-x}=5 \stackrel{\text { Proposition 1 }}{\Longleftrightarrow}$ $\frac{a-x}{b+x}=1$ or $\frac{a-x}{b+x}=4$, solving these gives $x_{1}=\frac{a-b}{2}, x_{2}=\frac{a-4b}{5}$ as the roots of the original equation.