Refresher - Inequalities

System of Linear Inequalities

To graph a linear inequality – with two variables – we begin by graphing the linear equality, which will define the boundary. When graphed, a linear inequality represents a region, or half plane, which contains solutions that will make the statement true.  Since the solution set contains an infinite number of solutions, it is customary to draw a graph Links to an external site. rather than write the set.

A system of inequalities would be two or more inequalities whose solutions must satisfy all the inequalities in the system. On a graph, the intersection of the regions will be shaded.

Example:  You want to spend no more than $9 on some pencils ${\displaystyle \left(x\right)}$ and pens ${\displaystyle \left(y\right)}$.   Pens cost more than twice the cost of a pencil.  ${\displaystyle [y>2x]}$  A graph of the intersection (where the two shaded regions intersect) of these two inequalities will show us the possible combination of pencils and pens that we could buy within the constraints of our budget.  Since the number of pencils and pens are positive, we will only look at those points in quadrant 1
(${\displaystyle x>0}$ and ${\displaystyle y>0}$).

A set of inequalities define the feasible regions Links to an external site. of linear programming Links to an external site..