How do I find the supremum and infimum of $\left\{x\in\mathbb{R} \setminus 0: \frac{2x}{3}-\frac{x^2-3}{2x} + 0.5 < \frac{x}{6} \right\}$?

$A=\{x \in \mathbb{R}: \frac{2x}{3}-\frac{x^2-3}{2x} + 0.5 < \frac{x}{6}, x \neq 0\}$

I tried simplifying this inequality and got $x+3<0$, which is $x<-3$. But that means that either $x>0$ and $x<-3$ or $x<0$ and $x<-3$. The first case can be disregarded (I think) because it is not bounded on either sides, so that leaves us with the second case ($x<0$, $x<-3$), where $0$ is the supremum, but I don't know how to find the infimum.