1. We are given the equation $$\cos\left(\frac{\pi}{5} - \frac{1}{2}x\right) = -\frac{\sqrt{2}}{2}$$. We need to solve for $x$. 2. Recall that $$\cos(\theta) = -\frac{\sqrt{2}}{2}$$ implies $$\theta = \pm \frac{3\pi}{4} + 2k\pi$$ for integers $k$ because $$\cos\left(\frac{3\pi}{4}\right) = \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$. 3. Set $$\frac{\pi}{5} - \frac{1}{2}x = \frac{3\pi}{4} + 2k\pi$$ or $$\frac{\pi}{5} - \frac{1}{2}x = -\frac{3\pi}{4} + 2k\pi$$. 4. Solve the first equation: $$\frac{\pi}{5} - \frac{1}{2}x = \frac{3\pi}{4} + 2k\pi$$ Subtract $$\frac{\pi}{5}$$: $$-\frac{1}{2}x = \frac{3\pi}{4} - \frac{\pi}{5} + 2k\pi = \frac{15\pi}{20} - \frac{4\pi}{20} + 2k\pi = \frac{11\pi}{20} + 2k\pi$$ Multiply both sides by $$-2$$: $$x = -2\left(\frac{11\pi}{20} + 2k\pi\right) = -\frac{11\pi}{10} - 4k\pi$$. 5. Solve the second equation: $$\frac{\pi}{5} - \frac{1}{2}x = -\frac{3\pi}{4} + 2k\pi$$ Subtract $$\frac{\pi}{5}$$: $$-\frac{1}{2}x = -\frac{3\pi}{4} - \frac{\pi}{5} + 2k\pi = -\frac{15\pi}{20} - \frac{4\pi}{20} + 2k\pi = -\frac{19\pi}{20} + 2k\pi$$ Multiply both sides by $$-2$$: $$x = -2\left(-\frac{19\pi}{20} + 2k\pi\right) = \frac{19\pi}{10} - 4k\pi$$. 6. Therefore, the general solutions are: $$x = -\frac{11\pi}{10} - 4k\pi\quad \text{and} \quad x = \frac{19\pi}{10} - 4k\pi$$ for any integer $k$. These are all solutions to the given equation.