The feasible region of a system of inequalities is that set of values for each variable for which all the inequalities hold.
For example, in the x-y plane, the feasible region for x > 0 is half the plane - everything to the right of the y-axis. The feasible region of x2 + y2 ≤ 1 is a disc. The feasible region of the pair of inequalities x2 + y2 ≤ 1; x + y ≥ 1 is a segment covering a quarter of the circumference of the circle. The feasible region of the three inequalities x2 + y2 ≤ 1; x ≥ 0; y ≥ 0 is a sector, one-quarter of the disc. The feasible region of x > 5; x < 2 is empty. The feasible region for x ≥ 0; x ≤ 0; y ≥ 0; y ≤ 0 is the single point at the origin.
A feasible region is called closed (or fully closed) if all of the borders of the region are included in the region (all the limiting inequalities use ≥ or ≤), it is called open (or fully open) of none of the border is included (all the limiting inequalities use > or <). A region which is not fully open is called partially closed, and a region which is not fully closed is called partially open. Thus, the 'fully open' regions are a subset of 'partially open' regions, and 'fully closed' regions are a subset of 'partially closed' regions. A region which includes part of its border and excludes part is both partially open and partially closed.
The feasible region in one dimension is usually called an interval. Feasible regions can also be described in 3-dimensional, or higher, spaces. (The feasible region for 0D is basically either "yes" or "no".)
It is possible to create feasible regions with disjunctions; in this case, there may be an oddly-shaped region, or two or more disjoint regions. These can be expressed as the union of conjunctive feasible regions.
If the feasible region for conjunctive linear inequalities (in any number of dimensions) is finite and closed, it is called a convex hull.