Q.Find the feasible solution of the following inequations graphically : $3x + 4y \ge 12,\ 4x + 7y \le 28,\ y \ge 1,\ x \ge 0$

A solution of an L.P.P. is any set of values of the decision variables. A feasible solution is a solution that satisfies ALL of the given constraints (including non-negativity) simultaneously — it is a point that could actually be carried out in the real situation the problem describes. The feasible region (also written as the set $S$) is the set of ALL feasible solutions taken together — graphically, it is the region of the plane common to every constraint's half-plane, found by shading each constraint's half-plane separately and then keeping only the portion shaded by every one of them. Because each individual constraint's half-plane is convex, and the intersection of convex sets is itself convex, the feasible region of an L.P.P. is always a convex set (Theorem 1 of section 7.2.3) — usually a convex polygon when the region is bounded, but it can also be an unbounded convex region (for example, when every constraint only imposes a lower, $\ge$, bound with no upper cap). The corners of this polygon (or of the bounded part of an unbounded region) are called its vertices or corner points, and they are exactly the points where two of the boundary constraint-lines cross while still satisfying every OTHER constraint too — a point where two lines cross but which fails some other constraint is NOT a vertex of the feasible region.