1. **State the problem:** Solve the equation $|x-2|=9$ for $x$. 2. **Recall the definition of absolute value:** For any real number $a$, $|a|=b$ means $a=b$ or $a=-b$ where $b\geq0$. 3. **Apply the definition:** Here, $|x-2|=9$ implies two cases: - Case 1: $x-2=9$ - Case 2: $x-2=-9$ 4. **Solve Case 1:** $$x-2=9$$ Add 2 to both sides: $$x=9+2$$ $$x=11$$ 5. **Solve Case 2:** $$x-2=-9$$ Add 2 to both sides: $$x=-9+2$$ $$x=-7$$ 6. **Final answer:** The solutions to $|x-2|=9$ are $$x=11 \text{ or } x=-7$$