[sfpotter]

I didn't find these examples particularly illuminating, and I'm also a geometry-forward thinker.

Systems of linear inequalities became transparent to me when I took a class on optimization and learned linear programming from the perspective of polytope geometry.

The basic concept is that you can define a halfspace by a linear inequality of the a^T x <= b. This means that taking the intersection of multiple halfspaces is the same as having multiple linear inequalities active simultaneously, which could be rewritten in matrix form as A x <= b. The intersection of two convex sets is again convex, and a halfspace is obvious convex, so it's clear that A x <= b is a convex polytope (polygon in 2D, polyhedron in 3D).

Systems of nonlinear inequalities are more complicated but you can sometimes approach them similarly.

This style of thinking is much more approachable for me because I have an easy time playing with these kinds of geometric objects in my head.