Elementary Algebra Section 7.5
Graphing Linear Inequalities

Definition: A linear inequality in two variables is an inequality that can be written in one of the following four forms:
                         

Definition: A point (x, y) is a solution of an inequality in two variables if the coordinates satisfy the inequality. (that is, if a true statement results when the coordinates are subsstituted for the variables in the inequality.)

Definition: The graph of an inequality is the set of points which are solutions of the inequality. (That is, the graph is the set of all points whose coordinates satisfy the inequality).

Definition:  If the inequality symbol in an inequality in two variables is replaced with an equality symbol, the graph of the resulting equation is called the boundary line for the inequality.

FACT: The graph of an inequality in two variables is a half-plane.
FACT: If the inequality symbol is replaced with an equal symbol, its graph forms the boundary between the half-plane consisting of all solutions of the inequality and the half-plane consisting of all points which are not solutions of the inequality.

PROCESS:  To graph a linear inequality in two variables:
                         a)  Sketch the graph of the boundary line
                                     i) as a dashed line if the inequality symbol is either ≥ or ≤.
                                    ii) as a solid line if the inequality symbol is not > or <.
                        b) Pick a point, not on the boundary line, as a test point and substitute its coordinates into the inequality.
                        c) If the result from Step b is a TRUE statement, the half-plane containing the test point is the solution.
                        d) If the result from Step b is a FALSE statement, the half-plane whcih does not contain the test point is the solution.
                        e) Shade the half-plane which is the solution and label all important points.