managing and visualising sets of solutions to linear inequalities

Question

I am interested in eight objects in $\mathbb{R}^3$, each the solution set of two or three linear inequalities. The solutions range from points to line segments to non-degenerate convex polyhedra.

I want to:

1. demonstrate that these eight objects partition the space; and
2. visualise them in a way that 'makes sense' of their interactions (e.g. as $x_1$ increases, do we pass from a polyhedron through a line segment at a threshold value?) - maybe allowing me to 'zoom' through the space.

I can hand crank this, but - given how standard these objects are - I wondered if there was a convenient tool for doing this.