1. **State the problem:** Solve the inequality $$4(2x - 3) - 2(5 - 3x) > \frac{x + 4}{2}$$. 2. **Expand and simplify both sides:** $$4(2x - 3) = 8x - 12$$ $$-2(5 - 3x) = -10 + 6x$$ So the left side becomes: $$8x - 12 - 10 + 6x = 14x - 22$$ 3. **Rewrite the inequality:** $$14x - 22 > \frac{x + 4}{2}$$ 4. **Eliminate the fraction by multiplying both sides by 2:** $$2(14x - 22) > x + 4$$ $$28x - 44 > x + 4$$ 5. **Bring all terms to one side:** $$28x - 44 - x - 4 > 0$$ $$27x - 48 > 0$$ 6. **Isolate $x$:** $$27x > 48$$ 7. **Divide both sides by 27:** $$\cancel{27}x > \cancel{27}\frac{48}{27}$$ $$x > \frac{48}{27}$$ 8. **Simplify the fraction:** $$x > \frac{16}{9}$$ **Final answer:** $$x > \frac{16}{9}$$