Given a hyperplane in $n$ dimensions $$H(\vec{a} \in \mathbb{R}^{n}, b \in \mathbb{R}) = \{\vec{x}\in\mathbb{R}^n \;|\; \vec{a} \cdot \vec{x} = b\}\;,$$how can one intuitively understand on "which side" of the hyperplane (i.e. on which halfspace that it defines) the inequality $\vec{a} \cdot \vec{x} > b$ holds—and on which side $\vec{a} \cdot \vec{x} < b$ holds?
Conversely, if we know "which is which", how can one see on what halfspace the normal vector $\vec{a}$ lies—i.e. the direction it points to?
You can use the origin as a reference of whether it is the side with origin.
If $0>b$, the side with the origin is $\vec{a} \cdot \vec{x} > b$.
Now, what if $b=0$. you can use $\vec{x}=\vec{a}$ to help decide. The side with $\vec{x}=\vec{a}$ satisfies $\vec{a} \cdot \vec{x} \ge b$.