Example 14. Solve the equation

$$
\sqrt{4 x+9}-\sqrt{11 x+1}-\sqrt{7 x+4}=0
$$
Solution. Solving the system of inequalities

$$
\left\{\begin{array}{r}
4 x+9 \geqslant 0 \\
11 x+1 \geqslant 0 \\
7 x+4 \geqslant 0
\end{array}\right.
$$

we find the domain of the equation: $x \geqslant-1 / 11$.

In this equation, any of the three radicals can be isolated. It is reasonable to isolate the radical $\sqrt{11 x+1}$, as the equation obtained after squaring and simplification will be simpler than after isolating any other radical.

Let's construct a chain of equations that are consequences of the original equation:

$$
\begin{gathered}
\sqrt{4 x+9}-\sqrt{7 x+4}=\sqrt{11 x+1} \\
4 x+9-2 \sqrt{(4 x+9)(7 x+4)}+7 x+4=11 x+1 \\
\sqrt{28 x^{2}+79 x+36}=6 \\
28 x^{2}+79 x+36=36 \\
28 x^{2}+79 x=0
\end{gathered}
$$

The roots of the last equation are: $x_{1}=0$ and $x_{2}=-2 \frac{23}{28}$.

The root $x_{2}=-2 \frac{23}{28}$ does not belong to the domain of the original equation and therefore is not its solution.

The root $x_{1}=0$ belongs to the domain of the original equation, and substituting it into the original equation, we get a true numerical equality; therefore, $x_{1}=0$ is a root of the original equation.

Thus, $x_{1}=0$ is the only root of the original equation.