1. **State the problem:** Solve the inequality $$\frac{x-9}{3} - \frac{x+1}{2} < \frac{5x}{6}$$. 2. **Find a common denominator:** The denominators are 3, 2, and 6. The least common denominator (LCD) is 6. 3. **Rewrite each term with denominator 6:** $$\frac{x-9}{3} = \frac{2(x-9)}{6} = \frac{2x - 18}{6}$$ $$\frac{x+1}{2} = \frac{3(x+1)}{6} = \frac{3x + 3}{6}$$ 4. **Rewrite the inequality:** $$\frac{2x - 18}{6} - \frac{3x + 3}{6} < \frac{5x}{6}$$ 5. **Combine the left side over common denominator:** $$\frac{2x - 18 - (3x + 3)}{6} < \frac{5x}{6}$$ 6. **Simplify numerator:** $$2x - 18 - 3x - 3 = -x - 21$$ 7. **Inequality becomes:** $$\frac{-x - 21}{6} < \frac{5x}{6}$$ 8. **Multiply both sides by 6 to clear denominators:** $$\cancel{6} \times \frac{-x - 21}{\cancel{6}} < \cancel{6} \times \frac{5x}{\cancel{6}}$$ $$-x - 21 < 5x$$ 9. **Add $x$ to both sides:** $$-x - 21 + x < 5x + x$$ $$-21 < 6x$$ 10. **Divide both sides by 6:** $$\frac{-21}{6} < x$$ $$-\frac{7}{2} < x$$ 11. **Final solution:** $$x > -\frac{7}{2}$$ This means all values of $x$ greater than $-3.5$ satisfy the inequality.