Example 27. Solve the equation

$$
\sqrt{5 x+7}-\sqrt{2 x+3}=\sqrt{3 x+4}
$$
Solution. Solving the system of inequalities

$$
\left\{\begin{array}{l}
5 x+7 \geqslant 0 \\
2 x+3 \geqslant 0 \\
3 x+4 \geqslant 0
\end{array}\right.
$$

we find the domain of the equation: $x \geqslant -4 / 3$. We have:

$$
\begin{aligned}
& (13) \Leftrightarrow\left\{\begin{array}{l}
x \geqslant -4 / 3 \\
\sqrt{5 x+7}=\sqrt{2 x+3}+\sqrt{3 x+4}
\end{array} \Leftrightarrow\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
x \geqslant -4 / 3 \\
5 x+7=2 x+3+2 \sqrt{(2 x+3)(3 x+4)}+3 x+4
\end{array} \Leftrightarrow\right. \\
& \Leftrightarrow\left\{\begin{array}{l}
x \geqslant -4 / 3 \\
\sqrt{(2 x+3)(3 x+4)}=0
\end{array}\right.
\end{aligned}
$$

From this, we find that $x_{1}=-4 / 3$ is the only root of equation (13).

A mixed system consisting of an equation and inequalities is rationally solved as follows: without solving the system of inequalities, solve the equation of this system, and then investigate which of the roots satisfy all the conditions of the system and which do not.