1. **State the problem:** Simplify the expression $1 \frac{1}{2} \div \left( \frac{1}{3} + \frac{5}{6} \right) + 2$. 2. **Convert mixed number to improper fraction:** $$1 \frac{1}{2} = \frac{3}{2}$$ 3. **Add the fractions inside the parentheses:** $$\frac{1}{3} + \frac{5}{6} = \frac{2}{6} + \frac{5}{6} = \frac{7}{6}$$ 4. **Rewrite the expression:** $$\frac{3}{2} \div \frac{7}{6} + 2$$ 5. **Division of fractions rule:** $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 6. **Apply the division rule:** $$\frac{3}{2} \times \frac{6}{7} + 2$$ 7. **Multiply the fractions:** $$\frac{3 \times 6}{2 \times 7} + 2 = \frac{18}{14} + 2$$ 8. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{18}^{9}}{\cancel{14}^{7}} + 2 = \frac{9}{7} + 2$$ 9. **Convert 2 to fraction with denominator 7:** $$2 = \frac{14}{7}$$ 10. **Add the fractions:** $$\frac{9}{7} + \frac{14}{7} = \frac{23}{7}$$ 11. **Final answer:** $$\frac{23}{7}$$ or approximately 3.2857.