The reason why your post is not getting answers and is instead getting downvotes is that this seems like a homework, quiz, or test question you want us to solve. That is not the purpose of this site; we are here to help you learn. Perhaps if you edit your posts in the future to include context as well as what you have already tried you will get help faster.
That being said, I do want to help you. So lets consider this inequality. We want $x^2-1+\lfloor x\rfloor \geq 0$. Lets instead write this as $\lfloor x\rfloor \geq 1-x^2$. What can you say about the right hand side (RHS)? This is a function you can easily understand. What can you say about the left hand side (LHS)? The floor function function is a bit awkward to visualize if you are not used to it, so you should draw the graph for values of $x$ between, say, $-2$ and $2$.
Since we want the LHS to be greater than or equal to the RHS, and since the behavior of the sides should be easy to understand for most values of $x$, all you will have to do is treat a very small domain on a case by case basis.
As per your comments posted above, you should avoid (when possible) attempting to deal with integer and fractional parts separately. This can be useful in some contexts, but here I do not think it is helpful. And while we all want to be able to solve all problems with the raw power of analysis, do not look down on graphical tools.