1. **State the problem:** Solve the equation $|x-1|=3$ for $x$. 2. **Recall the definition of absolute value:** For any real number $a$, $|a|=b$ means $a=b$ or $a=-b$ if $b\geq0$. 3. **Apply the definition:** Since $|x-1|=3$, we have two cases: - Case 1: $x-1=3$ - Case 2: $x-1=-3$ 4. **Solve Case 1:** $$x-1=3$$ Add 1 to both sides: $$\cancel{x-1}+1=\cancel{3}+1$$ $$x=4$$ 5. **Solve Case 2:** $$x-1=-3$$ Add 1 to both sides: $$\cancel{x-1}+1=\cancel{-3}+1$$ $$x=-2$$ 6. **Final answer:** The solutions to $|x-1|=3$ are $$x=4 \text{ or } x=-2$$