1. **State the problem:** Solve the equation $$|3x - 45| = 12x$$ and check the solutions. 2. **Recall the definition of absolute value:** For any expression $A$, $$|A| = \begin{cases} A & \text{if } A \geq 0 \\ -A & \text{if } A < 0 \end{cases}$$ 3. **Set up two cases based on the absolute value:** **Case 1:** When $$3x - 45 \geq 0$$, then $$|3x - 45| = 3x - 45$$. Solve: $$3x - 45 = 12x$$ Subtract $3x$ from both sides: $$3x - 45 - 3x = 12x - 3x$$ $$-45 = 9x$$ Divide both sides by 9: $$\frac{-45}{\cancel{9}} = \frac{9x}{\cancel{9}}$$ $$-5 = x$$ Check if this solution satisfies the case condition $3x - 45 \geq 0$: $$3(-5) - 45 = -15 - 45 = -60 < 0$$ It does not satisfy the condition, so discard $x = -5$. 4. **Case 2:** When $$3x - 45 < 0$$, then $$|3x - 45| = -(3x - 45) = -3x + 45$$. Solve: $$-3x + 45 = 12x$$ Add $3x$ to both sides: $$-3x + 45 + 3x = 12x + 3x$$ $$45 = 15x$$ Divide both sides by 15: $$\frac{45}{\cancel{15}} = \frac{15x}{\cancel{15}}$$ $$3 = x$$ Check if this solution satisfies the case condition $3x - 45 < 0$: $$3(3) - 45 = 9 - 45 = -36 < 0$$ It satisfies the condition, so $x = 3$ is a valid solution. 5. **Final answer:** The solution to the equation is $$\boxed{3}$$. 6. **Check the solution:** Substitute $x=3$ into the original equation: $$|3(3) - 45| = |9 - 45| = |-36| = 36$$ $$12(3) = 36$$ Both sides equal 36, so the solution is correct.