1. **State the problem:** Solve the inequality $$\frac{x - 2}{2} + \frac{x + 3}{5} < \frac{1}{10}$$. 2. **Identify the formula and rules:** To solve inequalities involving fractions, find a common denominator to combine terms, then isolate $x$. 3. **Find the common denominator:** The denominators are 2, 5, and 10. The least common denominator (LCD) is 10. 4. **Multiply each term by 10 to clear denominators:** $$10 \times \frac{x - 2}{2} + 10 \times \frac{x + 3}{5} < 10 \times \frac{1}{10}$$ 5. **Simplify each term:** $$5(x - 2) + 2(x + 3) < 1$$ 6. **Distribute:** $$5x - 10 + 2x + 6 < 1$$ 7. **Combine like terms:** $$7x - 4 < 1$$ 8. **Add 4 to both sides:** $$7x - 4 + 4 < 1 + 4$$ $$7x < 5$$ 9. **Divide both sides by 7:** $$\frac{\cancel{7}x}{\cancel{7}} < \frac{5}{7}$$ $$x < \frac{5}{7}$$ **Final answer:** $$x < \frac{5}{7}$$