1. **State the problem:** Solve the equation $|x - 3| + 1 = 4$ for $x$. 2. **Isolate the absolute value:** Subtract 1 from both sides: $$|x - 3| = 4 - 1 = 3$$ 3. **Definition of absolute value:** The equation $|A|=B$ where $B \geq 0$ implies $A = B$ or $A = -B$. Here, $A = x - 3$ and $B=3$. 4. **Set up two equations:** - $x - 3 = 3$ - $x - 3 = -3$ 5. **Solve each:** - For $x - 3 = 3$, add 3 to both sides: $x = 6$ - For $x - 3 = -3$, add 3 to both sides: $x = 0$ 6. **Final answers:** $x = 6$ or $x = 0$. These values satisfy the original equation.