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:** $$\frac{3x}{2} - 2 \leq \frac{3x}{4} - 4.5 + 7$$ $$\frac{3x}{2} - 2 \leq \frac{3x}{4} + 2.5$$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$\frac{3x}{2} - \frac{3x}{4} \leq 2.5 + 2$$ 6. **Find common denominator and subtract:** $$\frac{3x}{2} - \frac{3x}{4} = \frac{6x}{4} - \frac{3x}{4} = \frac{3x}{4}$$ So, $$\frac{3x}{4} \leq 4.5$$ 7. **Isolate $x$ by dividing both sides by $\frac{3}{4}$:** $$x \leq \frac{4.5}{\frac{3}{4}}$$ Show cancellation: $$x \leq 4.5 \times \frac{\cancel{4}}{\cancel{3}} = 4.5 \times \frac{4}{3}$$ 8. **Calculate the right side:** $$4.5 \times \frac{4}{3} = \frac{4.5 \times 4}{3} = \frac{18}{3} = 6$$ 9. **Final solution:** $$x \leq 6$$ 10. **Interpretation:** The solution set is all $x$ values less than or equal to 6, which is $$(-\infty, 6]$$. **Note:** The user mentioned the solution as $(6, \infty]$, but the correct solution from the inequality is $$(-\infty, 6]$$.