Confusion why Minkowski metric changes sign on metric perturbations

Question

In our textbook, we had $$g_{µν} = η_{µν} + h_{µν}$$ Then we raised the indices with the Minkowski metric.

$$g^{µν} = η^{µν} - h^{µν}.$$

Why did we get a different sign?

Answer 1

$g^{\mu\nu}$ is inverse of $g_{\mu\nu}$

Let's approximate assuming that $b\ll a$ \begin{equation} \frac{1}{a+b}\simeq \frac{1}{a} - \frac{b}{a^2} \end{equation} Indeed, increasing $b$ makes denominator larger and therefore the result should become smaller

This is the straightforward analogue of, \begin{equation} g^{\mu\nu}-g^{\mu\alpha}h_{\alpha\beta}g^{\beta\nu} \end{equation}