What is the definition of the solution of linear inequalities?

Answer 1

Each linear equation is a line that divides the coordinate plane into three regions: one "above" the line, one "below" and the line itself. For a linear inequality, the corresponding equality divides the plane into two, with the line itself belonging to one or the other region depending on the nature of the inequality.

A system of linear inequalities may define a polygonal region (a simplex) that satisfies ALL the inequalities. This area, if it exists, is called the feasible region and comprises all possible solutions of the linear inequalities.

In linear programming, there will be an objective function which will restrict the feasible region to a vertex or an edge of simplex. There may also be a further constraint - integer programming - where the solution must comprise integers. In this case, the feasible region will comprise all the integer grid-ponits with the simplex.