Example 2 When does $a$ satisfy the condition that the equation
$$
\left(a^{2}-1\right) x^{2}-6(3 a-1) x+72=0
$$

has roots of opposite signs and the absolute value of the negative root is larger?
The equation $\left(a^{2}-1\right) x^{2}-6(3 a-1) x+72=0$ has roots of opposite signs with the negative root having a larger absolute value under the conditions:
$$
\left\{\begin{array}{l}
a^{2}-1 \neq 0, \\
\Delta=36(3 a-1)^{2}-4\left(a^{2}-1\right) \times 72>0, \\
x_{1}+x_{2}=\frac{6(3 a-1)}{a^{2}-1}<0, \\
x_{1} x_{2}=\frac{72}{a^{2}-1}<0 .
\end{array}\right.
$$

Solving the system of inequalities yields $\frac{1}{3}<a<1$.