1. **State the problem:** We have a binary operation defined as $a * b = a + b - 2$. We are asked to find the inverse of 5 under this operation, given that the identity element is 0. 2. **Recall the identity element:** The identity element $e$ satisfies $a * e = a$ for any $a$. Given $e = 0$, check: $$a * 0 = a + 0 - 2 = a - 2$$ This does not equal $a$, so $0$ is not the identity under the operation as defined. However, the problem states the identity is 0, so we accept that. 3. **Find the inverse:** The inverse of 5, denoted $x$, satisfies: $$5 * x = 0$$ Using the operation definition: $$5 + x - 2 = 0$$ 4. **Solve for $x$:** $$5 + x - 2 = 0 \implies x + 3 = 0 \implies x = -3$$ 5. **Conclusion:** The inverse of 5 under the operation $*$ with identity 0 is $-3$. **Final answer:** $\boxed{-3}$