7. The solution to the equation $\sqrt{\frac{2 x-6}{x-11}}=\frac{3 x-7}{x+6}$ is
7. $x_{1}=19, x_{2}=\frac{13+5 \sqrt{2}}{7}$.

Squaring both sides of the equation and simplifying, we get
$$
\begin{array}{l}
7 x^{3}-159 x^{2}+511 x-323=0, \\
(x-19)\left(7 x^{2}-26 x+17\right)=0 .
\end{array}
$$

Solving, we find $x_{1}=19, x_{2,3}=\frac{13 \pm 5 \sqrt{2}}{7}$.
By the definition of the equation, $x>11$ or $\frac{7}{3} \leqslant x \leqslant 3$ or $x<-6$.
Therefore, the solutions to the equation are $x_{1}=19, x_{2}=\frac{13+5 \sqrt{2}}{7}$.