1. **State the problem:** Solve the inequality $$6x \geq 3 + 4(2x - 1)$$ and identify the correct representations. 2. **Expand the right side:** $$6x \geq 3 + 8x - 4$$ Simplify the constants: $$6x \geq 8x - 1$$ 3. **Isolate variable terms:** Subtract $$8x$$ from both sides: $$6x - 8x \geq 8x - 1 - 8x$$ $$\cancel{6x} - \cancel{8x} \geq \cancel{8x} - 1 - \cancel{8x}$$ $$-2x \geq -1$$ 4. **Divide both sides by -2:** Remember, dividing by a negative number reverses the inequality: $$\frac{-2x}{-2} \leq \frac{-1}{-2}$$ $$x \leq \frac{1}{2}$$ 5. **Interpret the solution:** The solution is $$x \leq 0.5$$. 6. **Check the number line options:** - Closed circle at 0.5 with shading to the left represents $$x \leq 0.5$$ (correct). - Closed circle at 0.5 with shading to the right represents $$x \geq 0.5$$ (incorrect). - Closed circle at -0.5 with shading to the right represents $$x \geq -0.5$$ (incorrect). 7. **Check the inequality $$1 \geq 2x$$:** Divide both sides by 2: $$\frac{1}{2} \geq x$$ or $$x \leq 0.5$$ (correct). **Correct representations:** - $$6x \geq 3 + 8x - 4$$ (equivalent inequality) - Number line with closed circle at 0.5 and shading to the left - Inequality $$1 \geq 2x$$ **Final answer:** $$x \leq 0.5$$