1. The problem is to solve the equation $2\cot 2x = 3$ for $x$. 2. Start by isolating the cotangent term: $$\cot 2x = \frac{3}{2}$$ 3. Recall that $\cot \theta = \frac{\cos \theta}{\sin \theta}$, but here it's easier to solve by rewriting in terms of tangent since $\cot \theta = \frac{1}{\tan \theta}$. 4. So, $$\tan 2x = \frac{2}{3}$$ 5. Now solve for $2x$: $$2x = \arctan\left(\frac{2}{3}\right) + k\pi, \quad k \in \mathbb{Z}$$ because tangent has period $\pi$. 6. Divide both sides by 2 to get $x$: $$x = \frac{1}{2} \arctan\left(\frac{2}{3}\right) + \frac{k\pi}{2}, \quad k \in \mathbb{Z}$$ 7. This is the general solution for $x$. Final answer: $$x = \frac{1}{2} \arctan\left(\frac{2}{3}\right) + \frac{k\pi}{2}, \quad k \in \mathbb{Z}$$