1. **State the problem:** Solve for $x$ in the equation $-\frac{1}{2} = \frac{4}{9}x + \frac{5}{6}$. 2. **Isolate the term with $x$:** Subtract $\frac{5}{6}$ from both sides: $$-\frac{1}{2} - \frac{5}{6} = \frac{4}{9}x$$ 3. **Find a common denominator to combine the left side:** The common denominator of 2 and 6 is 6. $$-\frac{3}{6} - \frac{5}{6} = \frac{4}{9}x$$ $$-\frac{8}{6} = \frac{4}{9}x$$ 4. **Simplify the fraction on the left:** $$-\frac{4}{3} = \frac{4}{9}x$$ 5. **Solve for $x$ by dividing both sides by $\frac{4}{9}$:** $$x = \frac{-\frac{4}{3}}{\frac{4}{9}}$$ 6. **Rewrite division as multiplication by reciprocal:** $$x = -\frac{4}{3} \times \frac{9}{4}$$ 7. **Cancel common factors:** $$x = -\cancel{\frac{4}{3}} \times \cancel{\frac{9}{4}} = -\frac{3}{1} = -3$$ **Final answer:** $$x = -3$$