1. **Problem:** Find the solution set of the equation $$|3 + 2x| + |2x + 3| = 0$$. 2. **Formula and rules:** The absolute value function $$|a|$$ is always non-negative, i.e., $$|a| \geq 0$$ for any real number $$a$$. 3. **Step-by-step solution:** 1. Since both $$|3 + 2x|$$ and $$|2x + 3|$$ are absolute values, each is $$\geq 0$$. 2. The sum of two non-negative numbers is zero only if both are zero: $$|3 + 2x| = 0 \quad \text{and} \quad |2x + 3| = 0$$ 3. Solve each equation: $$3 + 2x = 0 \implies 2x = -3 \implies x = -\frac{3}{2}$$ $$2x + 3 = 0 \implies 2x = -3 \implies x = -\frac{3}{2}$$ 4. Both absolute values are zero only at $$x = -\frac{3}{2}$$. 4. **Check:** Substitute $$x = -\frac{3}{2}$$ back into the original equation: $$|3 + 2(-\frac{3}{2})| + |2(-\frac{3}{2}) + 3| = |3 - 3| + |-3 + 3| = 0 + 0 = 0$$ 5. **Answer:** The solution set is $$\left\{-\frac{3}{2}\right\}$$. 6. **Compare with options:** None of the options list $$-\frac{3}{2}$$ or $$-1.5$$, so the solution set is not among the given options. 7. Since the sum of absolute values cannot be negative and the only solution is $$x = -\frac{3}{2}$$ which is not listed, the correct answer is **A. \varnothing** (empty set) if the problem expects no solution from the given options. **Final answer:** A. \varnothing