According to the inequality of Clausius, $$S\ge \dfrac {q_{rev}}T$$ , where $S$ is the entropy of the system, $q$ the heat absorbed by the system during a reversible process and $T$ is the temperature in kelvin.

Now, how can we calculate the entropy change during an irreversible process? I am confused about this since a long time.

This pdf made it clear to me that we cannot define the entropy change for an irreversible process as $\dfrac {q_{rev}}T$ because the entropy of a system consists of "produced entropy" (which is zero in a reversible process but not in an irreversible process) and entropy due to exchange.

So I searched more about the actual calculation, becuase I could easily calculate the change for an isothermal process but not for an adiabatic process.

I found this NASA article which clearly defines entropy as simply heat change (and not reversible heat change) upon temperature i.e. $$\Delta S = \dfrac{\Delta q}{T}$$

I was quite satisfied with this definition until I ran across this on MIT's website which made me confused again. According to MP $6..9$ on the site, for an irreversible process "we need to define a reversible process between the two states in order to calculate the entropy". I don't get what this means. How can we define a reversible process between two states connected by an irreversible process? Wouldn't it lead to erroneous results?

Tl;dr: Is entropy $\dfrac{q_{rev}}{T}$ or is it $\dfrac{q}{T}$? How can we calculate the entropy change for an adiabatic irreversible expansion?