1. **State the problem:** Solve the equation $-18 = 2a - 2|1 - 3a|$ for $a$. 2. **Understand the absolute value:** The absolute value expression $|1 - 3a|$ splits the problem into two cases: - Case 1: $1 - 3a \geq 0 \Rightarrow a \leq \frac{1}{3}$ - Case 2: $1 - 3a < 0 \Rightarrow a > \frac{1}{3}$ 3. **Case 1: $a \leq \frac{1}{3}$** Replace $|1 - 3a|$ with $(1 - 3a)$: $$-18 = 2a - 2(1 - 3a)$$ Simplify: $$-18 = 2a - 2 + 6a$$ $$-18 = 8a - 2$$ Add 2 to both sides: $$-18 + 2 = 8a$$ $$-16 = 8a$$ Divide both sides by 8: $$\cancel{8}a = \frac{-16}{\cancel{8}}$$ $$a = -2$$ Check if $a = -2$ satisfies $a \leq \frac{1}{3}$: Yes, since $-2 \leq \frac{1}{3}$. 4. **Case 2: $a > \frac{1}{3}$** Replace $|1 - 3a|$ with $-(1 - 3a) = 3a - 1$: $$-18 = 2a - 2(3a - 1)$$ Simplify: $$-18 = 2a - 6a + 2$$ $$-18 = -4a + 2$$ Subtract 2 from both sides: $$-18 - 2 = -4a$$ $$-20 = -4a$$ Divide both sides by $-4$: $$\cancel{-4}a = \frac{-20}{\cancel{-4}}$$ $$a = 5$$ Check if $a = 5$ satisfies $a > \frac{1}{3}$: Yes, since $5 > \frac{1}{3}$. 5. **Final solution:** The solutions are $a = -2$ and $a = 5$. 6. **Answer set:** $\{-2, 5\}$ which corresponds to option D.