1. **State the problem:** Simplify the expression $$-4 - \left(- \left(\frac{2}{9} - \frac{2}{3}\right)\right) - 1 - \frac{5}{6} + \frac{1}{2}$$. 2. **Recall the rules:** - Subtracting a negative is the same as adding a positive. - Find common denominators to combine fractions. 3. **Simplify inside the parentheses:** $$\frac{2}{9} - \frac{2}{3} = \frac{2}{9} - \frac{6}{9} = -\frac{4}{9}$$ 4. **Apply the negative outside:** $$-\left(-\frac{4}{9}\right) = +\frac{4}{9}$$ 5. **Rewrite the expression:** $$-4 + \frac{4}{9} - 1 - \frac{5}{6} + \frac{1}{2}$$ 6. **Combine the integer terms:** $$-4 - 1 = -5$$ 7. **Find a common denominator for the fractions $\frac{4}{9}$, $\frac{5}{6}$, and $\frac{1}{2}$:** The least common denominator is 18. 8. **Convert fractions:** $$\frac{4}{9} = \frac{8}{18}, \quad \frac{5}{6} = \frac{15}{18}, \quad \frac{1}{2} = \frac{9}{18}$$ 9. **Combine the fractions:** $$\frac{8}{18} - \frac{15}{18} + \frac{9}{18} = \frac{8 - 15 + 9}{18} = \frac{2}{18} = \frac{1}{9}$$ 10. **Add the integer and fraction parts:** $$-5 + \frac{1}{9} = -\frac{45}{9} + \frac{1}{9} = -\frac{44}{9}$$ **Final answer:** $$-\frac{44}{9}$$