1. **State the problem:** Solve the inequality $1 + 2(x - 3) > 0$. 2. **Apply the distributive property:** $$1 + 2x - 6 > 0$$ 3. **Simplify the expression:** $$2x - 5 > 0$$ 4. **Isolate the variable term:** $$2x > 5$$ 5. **Divide both sides by 2 (positive number, inequality direction stays the same):** $$x > \frac{5}{2}$$ 6. **Final answer:** The solution to the inequality is all $x$ such that $$x > 2.5$$. This means any number greater than 2.5 satisfies the inequality.