1. **State the problem:** Solve the inequality $$\frac{x - 1}{5} > \frac{x + 2}{6} + \frac{7}{15}$$. 2. **Find a common denominator:** The denominators are 5, 6, and 15. The least common denominator (LCD) is 30. 3. **Multiply both sides by 30 to clear denominators:** $$30 \times \frac{x - 1}{5} > 30 \times \left(\frac{x + 2}{6} + \frac{7}{15}\right)$$ 4. **Simplify each term:** $$6(x - 1) > 5(x + 2) + 2 \times 7$$ 5. **Distribute:** $$6x - 6 > 5x + 10 + 14$$ 6. **Combine like terms on the right:** $$6x - 6 > 5x + 24$$ 7. **Subtract 5x from both sides:** $$6x - \cancel{5x} - 6 > \cancel{5x} + 24$$ $$x - 6 > 24$$ 8. **Add 6 to both sides:** $$x - 6 + 6 > 24 + 6$$ $$x > 30$$ **Final answer:** $$x > 30$$