1. **State the problem:** Solve the equation $$3 |3x - 7| - 1 = 6x + 20$$ for $x$. 2. **Isolate the absolute value:** Add 1 to both sides: $$3 |3x - 7| = 6x + 21$$ 3. **Divide both sides by 3:** $$|3x - 7| = 2x + 7$$ 4. **Consider the definition of absolute value:** The equation $$|A| = B$$ implies two cases: - Case 1: $$3x - 7 = 2x + 7$$ - Case 2: $$3x - 7 = -(2x + 7)$$ 5. **Solve Case 1:** $$3x - 7 = 2x + 7$$ Subtract $2x$ from both sides: $$x - 7 = 7$$ Add 7 to both sides: $$x = 14$$ 6. **Check Case 1 solution validity:** Since $$|3x - 7| = 2x + 7$$, the right side must be non-negative: $$2(14) + 7 = 28 + 7 = 35 \\ > 0$$ So, $x=14$ is valid. 7. **Solve Case 2:** $$3x - 7 = -2x - 7$$ Add $2x$ to both sides: $$5x - 7 = -7$$ Add 7 to both sides: $$5x = 0$$ Divide both sides by 5: $$x = 0$$ 8. **Check Case 2 solution validity:** Check right side: $$2(0) + 7 = 7 > 0$$ So, $x=0$ is valid. 9. **Final solutions:** $$\boxed{x = 0 \text{ or } x = 14}$$