The linear inequalities can be represented graphically by drawing straight lines corresponding to equations which replace the inequalities, that is, the inequality symbol is replaced by an equals sign.
Once the lines have been drawn, the inequalities can be represented by shading the area to the right or left of each line, depending on the inequality.
For example, ax+by≥c, which represents the whole region (which will be infinite) to the right of the line ax+by=c, including points on the line itself (an infinite number of points). ax+by>c represents the same region, but excluding points on the line itself. ax+by≤c and ax+by<c are treated similarly but it's the area on the left instead of right.
When the shading exercise is carried out for a system of lines, it will usually be found that the lines intersect one another and enclose a finite area formed when all the shadings overlap. Geometrically, then, we often get a polygon formed by segments of the graphed lines. The area of this figure represents all the values of the two variables satisfying the system of inequalities. This method can be used to define a feasibility region in linear programming. The coordinates of the vertices of the polygon can be used for determine the maximum or minimum value of a function of the two variables. The technique provides a way of solving business problems, which specify a number of constraints (the inequalities) based on business costs (materials, labour, etc.) and availability of resources (materials, labour, etc.), and which aim to minimise costs and/or maximise profit.