Inequalities

Introduction and Summary

While the last chapter focused on equations and systems of equations, this chapter deals with inequalities and systems of inequalities. Inequalities in one variable are introduced in algebra I; in algebra II, we turn our attention to inequalities in two variables.

The first section explains how to graph an inequality on the xy -plane. Graphing an inequality in 2 dimensional space (a graph with 2 variables) is similar to graphing an inequality on the number line. Both involve treating the inequality as an equation, solving the equation, and testing points. In the 2 variable case, however, the solution to the equation is a line, not a single point. It is this line that divides the xy -graph into two regions: one that satisfies the inequality, and one that does not.

The second section deals with systems of inequalities. Unlike systems of equations, systems of inequalities generally do not have a single solution; rather, systems of inequalities describe an entire region. Thus, it makes sense to find this region by graphing the inequalities. This section explains how to solve systems of inequalities by graphing.

The third section provides an application of inequalities--linear programming. Linear programming is a process by which constraints are turned into inequalities and graphed, and a value is maximized or minimized. This is especially useful in economics, in which linear programming is used to maximize revenue, minimize cost, and maximize profit.

Inequalities have other applications in addition to linear programming. They are used to describe the relationship between any two quantities when one quantity "limits" the other. These relationships appear frequently in physics and chemistry, as well as in everyday life. Inequalities are also used to find viable values of variables against several constraints.