1. **State the problem:** Solve the inequality $$-x + 2 + 3x \geq \frac{1}{2}$$. 2. **Combine like terms:** Combine the terms with $x$ on the left side. $$-x + 3x = 2x$$ So the inequality becomes: $$2x + 2 \geq \frac{1}{2}$$ 3. **Isolate the variable term:** Subtract 2 from both sides. $$2x + 2 - 2 \geq \frac{1}{2} - 2$$ $$2x \geq \frac{1}{2} - 2$$ Calculate the right side: $$\frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2}$$ So: $$2x \geq -\frac{3}{2}$$ 4. **Divide both sides by 2:** $$x \geq \frac{-\frac{3}{2}}{2}$$ Show cancellation: $$x \geq \frac{-\frac{3}{2}}{\cancel{2}} \times \frac{1}{\cancel{2}} = -\frac{3}{4}$$ 5. **Final answer:** $$x \geq -\frac{3}{4}$$ This means $x$ can be any number greater than or equal to $-\frac{3}{4}$.