Example 5. Solve the equation

$$
\sqrt{11 x+3}-\sqrt{2-x}-\sqrt{9 x+7}+\sqrt{x-2}=0
$$
Solution. Isolate two radicals on each side so that after squaring we get the simplest equation:

$$
\sqrt{11 x+3}-\sqrt{2-x}=\sqrt{9 x+7}-\sqrt{x-2}
$$

Perform a chain of transformations:

$$
\begin{gathered}
11 x+3-2 \sqrt{(11 x+3)(2-x)}+2-x= \\
=9 x+7-2 \sqrt{(9 x+7)(x-2)}+x-2 \\
\sqrt{6+19 x-11 x^{2}}=\sqrt{9 x^{2}-11 x-14} \\
6+19 x-11 x^{2}=9 x^{2}-11 x-14 \\
20 x^{2}-30 x-20=0 \\
2 x^{2}-3 x-2=0
\end{gathered}
$$

The last equation is a consequence of the original equation and has roots $x_{1}=2$ and $x_{2}=-0.5$.

When $x=2$, the original equation turns into a true numerical equality. Therefore, $x_{1}=2$ is its root.

When $x=-0.5$, in the left part of the original equation, we have the expression $\sqrt{-0.5-2}$, which is meaningless; therefore, the number $x_{2}=-0.5$ does not satisfy the original equation.

Thus, $x_{2}=2$ is the only root of the original equation.