1. **State the problem:** Solve the equation $\sin 2x = \sin x$ for $x$.\n\n2. **Recall the formula and rules:** The equation $\sin A = \sin B$ implies that either:\n\n$$A = B + 2k\pi \quad \text{or} \quad A = \pi - B + 2k\pi$$\n\nwhere $k$ is any integer.\n\n3. **Apply the formula:** Here, $A = 2x$ and $B = x$. So we have two cases:\n\nCase 1: $$2x = x + 2k\pi$$\nCase 2: $$2x = \pi - x + 2k\pi$$\n\n4. **Solve Case 1:**\n$$2x = x + 2k\pi \implies 2x - x = 2k\pi \implies x = 2k\pi$$\n\n5. **Solve Case 2:**\n$$2x = \pi - x + 2k\pi \implies 2x + x = \pi + 2k\pi \implies 3x = \pi + 2k\pi \implies x = \frac{\pi + 2k\pi}{3}$$\n\n6. **Final solution:**\n$$x = 2k\pi \quad \text{or} \quad x = \frac{\pi + 2k\pi}{3} \quad \text{for any integer } k.$$