1. **State the problem:** Solve the inequality $$\frac{1}{2}(3x - 4) \leq \frac{3}{4}(x - 6) + 7$$. 2. **Write the inequality clearly:** $$\frac{1}{2}(3x - 4) \leq \frac{3}{4}(x - 6) + 7$$ 3. **Distribute the fractions:** $$\frac{1}{2} \times 3x - \frac{1}{2} \times 4 \leq \frac{3}{4} \times x - \frac{3}{4} \times 6 + 7$$ which simplifies to $$\frac{3x}{2} - 2 \leq \frac{3x}{4} - \frac{18}{4} + 7$$ 4. **Simplify constants on the right side:** $$\frac{3x}{2} - 2 \leq \frac{3x}{4} - \frac{9}{2} + 7$$ 5. **Convert 7 to halves to combine:** $$7 = \frac{14}{2}$$ So, $$\frac{3x}{2} - 2 \leq \frac{3x}{4} - \frac{9}{2} + \frac{14}{2}$$ 6. **Combine constants on the right:** $$- \frac{9}{2} + \frac{14}{2} = \frac{5}{2}$$ So, $$\frac{3x}{2} - 2 \leq \frac{3x}{4} + \frac{5}{2}$$ 7. **Bring all terms to one side:** $$\frac{3x}{2} - \frac{3x}{4} - 2 - \frac{5}{2} \leq 0$$ 8. **Find common denominator for x terms:** $$\frac{3x}{2} = \frac{6x}{4}$$ So, $$\frac{6x}{4} - \frac{3x}{4} = \frac{3x}{4}$$ 9. **Combine constants:** $$-2 - \frac{5}{2} = -\frac{4}{2} - \frac{5}{2} = -\frac{9}{2}$$ 10. **Rewrite inequality:** $$\frac{3x}{4} - \frac{9}{2} \leq 0$$ 11. **Add $$\frac{9}{2}$$ to both sides:** $$\frac{3x}{4} \leq \frac{9}{2}$$ 12. **Multiply both sides by 4 to clear denominator:** $$\cancel{4} \times \frac{3x}{\cancel{4}} \leq \frac{9}{2} \times 4$$ which simplifies to $$3x \leq 18$$ 13. **Divide both sides by 3:** $$\frac{\cancel{3}x}{\cancel{3}} \leq \frac{18}{3}$$ which simplifies to $$x \leq 6$$ **Final answer:** $$x \leq 6$$