1. **State the problem:** Solve the inequality $$5 + 9x < - \frac{1}{2}x - \left( \frac{3}{2}x + 1 \right)$$. 2. **Distribute the negative sign on the right side:** $$5 + 9x < - \frac{1}{2}x - \frac{3}{2}x - 1$$ 3. **Combine like terms on the right side:** $$5 + 9x < - \left( \frac{1}{2} + \frac{3}{2} \right)x - 1$$ $$5 + 9x < - 2x - 1$$ 4. **Add $2x$ to both sides to get all $x$ terms on the left:** $$5 + 9x + 2x < - 2x + 2x - 1$$ $$5 + 11x < -1$$ 5. **Subtract 5 from both sides to isolate the $x$ term:** $$5 - \cancel{5} + 11x < -1 - \cancel{5}$$ $$11x < -6$$ 6. **Divide both sides by 11 to solve for $x$:** $$\frac{11x}{\cancel{11}} < \frac{-6}{\cancel{11}}$$ $$x < -\frac{6}{11}$$ **Final answer:** $$x < -\frac{6}{11}$$ This means all values of $x$ less than $-\frac{6}{11}$ satisfy the inequality.