Example 4. Solve the equation

$$
\left|\frac{x^{2}-6 \sqrt{x}+7}{x^{2}+6 \sqrt{x}+7}\right|=1
$$
Solution. The given equation is equivalent to the system of equations

$\left[\begin{array}{l}\frac{x^{2}-6 \sqrt{x}+7}{x^{2}+6 \sqrt{x}+7}=1, \\ \frac{x^{2}-6 \sqrt{x}+7}{x^{2}+6 \sqrt{x}+7}=-1\end{array} \Leftrightarrow\right.$

$$
\Leftrightarrow\left[\begin{array}{l}
\frac{-12 \sqrt{x}}{x^{2}+6 \sqrt{x}+7}=0 \\
\frac{2 x^{2}+14}{x^{2}+6 \sqrt{x}+7}=0
\end{array} \Leftrightarrow \sqrt{x}=0 \Leftrightarrow x=0\right.
$$

Thus, the only solution to the original equation is the number 0.