1. **State the problem:** Solve the absolute value equation $$\frac{2|x - 3|}{5} = 6$$ for $x$. 2. **Formula and rules:** To solve equations involving absolute values, recall that if $$|A| = B$$ where $B \geq 0$, then $$A = B$$ or $$A = -B$$. 3. **Isolate the absolute value:** Multiply both sides by 5 to clear the denominator: $$\frac{2|x - 3|}{5} = 6 \implies 2|x - 3| = 6 \times 5 = 30$$ 4. **Divide both sides by 2:** $$|x - 3| = \frac{30}{2} = 15$$ 5. **Apply the absolute value rule:** $$x - 3 = 15 \quad \text{or} \quad x - 3 = -15$$ 6. **Solve each equation:** - For $$x - 3 = 15$$, add 3 to both sides: $$x = 15 + 3 = 18$$ - For $$x - 3 = -15$$, add 3 to both sides: $$x = -15 + 3 = -12$$ 7. **Final answer:** The solutions are $$x = -12$$ and $$x = 18$$. Entering the smallest solution first: $$\boxed{-12, 18}$$