1. **State the problem:** Solve the inequality $$\frac{1}{6} - \frac{1}{4}x \geq 2 + \frac{2x}{3}$$. 2. **Rewrite the inequality:** To solve for $x$, first get all terms involving $x$ on one side and constants on the other. 3. **Subtract $2$ from both sides:** $$\frac{1}{6} - \frac{1}{4}x - 2 \geq \frac{2x}{3}$$ 4. **Simplify constants on the left:** $$\frac{1}{6} - 2 = \frac{1}{6} - \frac{12}{6} = -\frac{11}{6}$$ So the inequality becomes: $$-\frac{11}{6} - \frac{1}{4}x \geq \frac{2x}{3}$$ 5. **Add $\frac{1}{4}x$ to both sides:** $$-\frac{11}{6} \geq \frac{2x}{3} + \frac{1}{4}x$$ 6. **Find common denominator for $x$ terms on the right:** $$\frac{2x}{3} + \frac{1}{4}x = \frac{8x}{12} + \frac{3x}{12} = \frac{11x}{12}$$ 7. **Rewrite inequality:** $$-\frac{11}{6} \geq \frac{11x}{12}$$ 8. **Multiply both sides by 12 to clear denominators:** $$12 \times \left(-\frac{11}{6}\right) \geq 12 \times \frac{11x}{12}$$ $$-22 \geq 11x$$ 9. **Divide both sides by 11:** $$\frac{-22}{11} \geq x$$ $$-2 \geq x$$ 10. **Rewrite solution:** $$x \leq -2$$ **Final answer:** The solution to the inequality is $$x \leq -2$$.