1. **State the problem:** Solve the inequality $$\frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10)$$ and graph the solution on a number line. 2. **Write the inequality:** $$\frac{3}{4}(x + 8) > \frac{1}{2}(2x + 10)$$ 3. **Distribute the fractions:** $$\frac{3}{4}x + \frac{3}{4} \times 8 > \frac{1}{2} \times 2x + \frac{1}{2} \times 10$$ $$\frac{3}{4}x + 6 > x + 5$$ 4. **Bring all terms involving $x$ to one side and constants to the other:** $$\frac{3}{4}x - x > 5 - 6$$ 5. **Simplify the left side:** $$\frac{3}{4}x - \frac{4}{4}x = -\frac{1}{4}x$$ So the inequality becomes: $$-\frac{1}{4}x > -1$$ 6. **Multiply both sides by $-4$ to solve for $x$.** Remember, multiplying by a negative number reverses the inequality sign: $$\cancel{-\frac{1}{4}}x \times \cancel{-4} < -1 \times -4$$ $$x < 4$$ 7. **Interpretation:** The solution is all $x$ values less than 4. 8. **Graph:** On the number line, draw an open circle at 4 and shade all values to the left (less than 4). **Note:** The user mentioned an open circle at -2 and shading to the left, but the solution to the inequality is $x < 4$, so the open circle should be at 4, not -2. **Final answer:** $$x < 4$$