Systems of linear inequalities
Several limits must hold at once, each on two quantities. Which points survive all of them, and which of those is best?
Where you are 4 inside
What this area is about
A linear inequality in x and y, such as y > 2x - 1, is true on one side of a line (and on the line itself for \le or \ge). A system of linear inequalities asks for the points that make every inequality true at once. Its answer is a region of the plane, drawn rather than listed, and often a choice of the best point in it.
How these fit together
Procedures for drawing. Graphing a linear inequality in two variables draws the boundary line, solid for \le or \ge and dashed for < or >, and shades the side that works. The overlap of two shaded regions puts several on one grid: the solution set is where all the shadings agree, and it may close into a polygon with corners found exactly.
Applications. Writing and interpreting constraints turns the limits in a situation into equations and inequalities and decides which solutions make sense in context. Linear programming finds the largest or smallest value of an amount such as P = 25x + 40y over that region; when a best value exists and the region has corners, it is reached at a corner.
A path through it
Graph one inequality, then the overlap of several. Both applications follow the overlap, and linear programming builds on writing constraints, so constraints come first.
Finish here and you can describe every plan that keeps within a set of limits, and pick the best one: a first taste of optimization.