1. **State the problem:** Solve the inequality $$-\left|x-\frac{1}{4}\right| \ge 2 - \frac{1}{2}x$$. 2. **Rewrite the inequality:** Multiply both sides by -1 to remove the negative sign in front of the absolute value. Remember, multiplying an inequality by a negative number reverses the inequality sign. $$-\left|x-\frac{1}{4}\right| \ge 2 - \frac{1}{2}x \implies \left|x-\frac{1}{4}\right| \le -2 + \frac{1}{2}x$$ 3. **Analyze the right side:** The right side is $$-2 + \frac{1}{2}x$$. For the inequality $$\left|x-\frac{1}{4}\right| \le -2 + \frac{1}{2}x$$ to hold, the right side must be non-negative because absolute values are always $$\geq 0$$. So, set: $$-2 + \frac{1}{2}x \ge 0$$ Multiply both sides by 2: $$-4 + x \ge 0$$ $$x \ge 4$$ 4. **Rewrite the inequality for $$x \ge 4$$:** $$\left|x-\frac{1}{4}\right| \le -2 + \frac{1}{2}x$$ 5. **Solve the absolute value inequality:** Recall that $$|A| \le B$$ means $$-B \le A \le B$$ when $$B \ge 0$$. So, $$-\left(-2 + \frac{1}{2}x\right) \le x - \frac{1}{4} \le -2 + \frac{1}{2}x$$ Simplify the left inequality: $$2 - \frac{1}{2}x \le x - \frac{1}{4}$$ Add $$\frac{1}{2}x$$ to both sides: $$2 \le x - \frac{1}{4} + \frac{1}{2}x$$ Combine like terms: $$2 \le \frac{3}{2}x - \frac{1}{4}$$ Add $$\frac{1}{4}$$ to both sides: $$2 + \frac{1}{4} \le \frac{3}{2}x$$ $$\frac{9}{4} \le \frac{3}{2}x$$ Divide both sides by $$\frac{3}{2}$$: $$\frac{9}{4} \times \frac{2}{3} \le x$$ $$\frac{18}{12} \le x$$ $$\frac{3}{2} \le x$$ 6. **Simplify the right inequality:** $$x - \frac{1}{4} \le -2 + \frac{1}{2}x$$ Subtract $$\frac{1}{2}x$$ from both sides: $$x - \frac{1}{2}x - \frac{1}{4} \le -2$$ $$\frac{1}{2}x - \frac{1}{4} \le -2$$ Add $$\frac{1}{4}$$ to both sides: $$\frac{1}{2}x \le -2 + \frac{1}{4}$$ $$\frac{1}{2}x \le -\frac{7}{4}$$ Multiply both sides by 2: $$x \le -\frac{7}{2}$$ 7. **Combine the inequalities:** From step 3, $$x \ge 4$$. From step 5, $$x \ge \frac{3}{2}$$. From step 6, $$x \le -\frac{7}{2}$$. The right inequality $$x \le -\frac{7}{2}$$ contradicts $$x \ge 4$$, so no $$x$$ satisfies both. Therefore, the solution must satisfy $$x \ge 4$$ and $$x \ge \frac{3}{2}$$, which simplifies to $$x \ge 4$$. 8. **Check the domain:** Since the right side must be non-negative, $$x \ge 4$$. **Final solution:** $$\boxed{x \ge 4}$$