# If dS ≥ dq/T for a spontaneous change is dS ≤ dq/T for a non-spontaneous change and is that even possible?

I am struggling with the concept of entropy but might be on the verge of understanding it. The Clausius inequality, $$\mathrm{d}S \geq \mathrm{d}q/T,$$ is true for all spontaneous changes with the equality for reversible changes. But what if a change is not spontaneous or is that impossible i.e a change cannot be forced. Any help with answering this question or to help conceptualize and understand the entropy would be greatly appreciated.

As you perfectly say, Clausius inequality states that $$\mathrm{d}s\geq \mathrm{d}q/T.$$ This has to be true for any process.
For reversible processes, what happens is that $\mathrm{d}s= \mathrm{d}q/T$, and for non-reversible processes, $\mathrm{d}s > \mathrm{d}q/T$. If for a given process $\mathrm{d}s< \mathrm{d}q/T$, it would not fulfill Clausius inequality.
Consider the situation where the system is isolated from its surroundings. In this case, $\mathrm{d}q_\mathrm{(sys)} = 0$ , and thus the Clausius relation implies $\mathrm{d}S\geq0$, so this tells us that the entropy of an isolated system cannot decrease in the course of a spontaneous change.