Example 21. Solve the equation

$$
\sqrt{x^{2}-1}=(x+5) \sqrt{\frac{x+1}{x-1}}
$$
Solution. Using formulas 4 and 5, we obtain the equation

$$
\sqrt{|x+1|}\left(\sqrt{|x-1|}-(x+5) \frac{1}{\sqrt{|x-1|}}\right)=0
$$

which is a consequence of the original equation. Solving this equation reduces to solving the system of equations

$$
\sqrt{|x+1|}=0, \quad \sqrt{|x-1|}-\frac{x+5}{\sqrt{|x-1|}}=0
$$

From the first equation of this system, we find $x_{1}=-1$. From the second, it follows that $|x-1|=x+5$, from which we find $x_{2}=-2$.

Thus, the roots of the given equation can only be the numbers (-1) and (-2). Verification shows that both found roots satisfy the given equation.