Answer
$$x=-\frac{5}{3}$$
Work Step by Step
In order to solve a problem like this where the variable is in the exponent on both sides of the equation, we try to get the same base on both sides of the equation. After we do this, we set the numbers in the exponent equal to each other to find:
$$\left(6^2\right)^{5x+2}=\left(\frac{1}{6}\right)^{11-x} \\ 6^{2\left(5x+2\right)}=6^{-1\cdot \left(11-x\right)} \\ 2\left(5x+2\right)=-1\cdot \left(11-x\right)\\ x=-\frac{5}{3}$$
