Solve the inequality

$7 \leq \frac{(3 x+11)}{2} \leq 11$

$\Rightarrow 14 \leq 3 x+11 \leq 22$

$\Rightarrow 14-11 \leq 3 x \leq 22-11$

$\Rightarrow 3 \leq 3 x \leq 11$

$\Rightarrow 1 \leq x \leq \frac{11}{3}$

Thus, the solution set for the given inequalityis $\left[1, \frac{11}{3}\right]$.