1. **State the problem:** We need to solve for $x$ in the equation $$2x = 2n\pi \pm \frac{\pi}{6} - \frac{\pi}{3}$$ where $n$ is an integer. 2. **Rewrite the equation:** The right side has two parts due to the $\pm$ sign. We will consider both cases separately: $$2x = 2n\pi + \frac{\pi}{6} - \frac{\pi}{3}$$ and $$2x = 2n\pi - \frac{\pi}{6} - \frac{\pi}{3}$$ 3. **Simplify each case:** For the first case: $$2x = 2n\pi + \frac{\pi}{6} - \frac{\pi}{3} = 2n\pi + \frac{\pi}{6} - \frac{2\pi}{6} = 2n\pi - \frac{\pi}{6}$$ For the second case: $$2x = 2n\pi - \frac{\pi}{6} - \frac{\pi}{3} = 2n\pi - \frac{\pi}{6} - \frac{2\pi}{6} = 2n\pi - \frac{3\pi}{6} = 2n\pi - \frac{\pi}{2}$$ 4. **Divide both sides by 2 to solve for $x$:** First case: $$x = n\pi - \frac{\pi}{12}$$ Second case: $$x = n\pi - \frac{\pi}{4}$$ 5. **Final solution:** $$x = n\pi - \frac{\pi}{12} \quad \text{or} \quad x = n\pi - \frac{\pi}{4}$$ where $n$ is any integer. This means $x$ takes values shifted by multiples of $\pi$ minus either $\frac{\pi}{12}$ or $\frac{\pi}{4}$.