Example 3 Solve the equation
$$
\frac{1}{6 x-5}-\frac{1}{5 x+4}=\frac{1}{5 x-4}-\frac{1}{4 x+5} .
$$
Solution: By differentiating both sides of the equation, we get
$$
\frac{9-x}{30 x^{2}-x-20}=\frac{9-x}{20 x^{2}+9 x-20}
$$

By the property of fractions, when two fractions are equal, if the numerators are equal, then the fractions are zero or the denominators are equal.
$$
\therefore 9-x=0
$$

or $30 x^{2}-x-20=20 x^{2}+9 x-20$.
Solving, we get $x_{1}=9, x_{2}=0, x_{3}=1$.