EDGE Math 8

Graphs of Linear Inequalities in Two Variables

Recall that the graph of a linear inequality in one variable is a shaded portion of the number line that either includes an endpoint (represented by a solid dot) or not (represented by an open dot). On the other hand, the graph of a linear inequality in two variables is a region on the coordinate plane.

Below are the steps in graphing a linear inequality in two variables.

Given a linear inequality $A x + B y < C$ (or $A x + B y \leq C,$ $A x + B y > C,$ or $A x + B y \geq C) :$

Step 1.

Graph the equation obtained by replacing the inequality symbol with the equal sign. This equation represents the boundary line. It divides the plane into two regions called half-planes.

a.	If the inequality symbol is < or >, the points on the boundary line are not included in the solution set of the inequality. In this case, make the boundary line dashed.

b.	If the inequality symbol is $\leq$ or $\geq$ the points on the boundary line are included in the solution set. In this case, make the boundary line solid.

Step 2.

Choose any point not on the boundary line. This is called a test point. Substitute the coordinates of the test point for x and y in the given inequality.

a.	If the coordinates of the test point satisfy the inequality, shade the region that contains the test point.

b.	If the coordinates of the test point do not satisfy the inequality, shade the region that does not contain the test point.

The shaded region, together with the points on the boundary line, if applicable, constitute the graph of the given inequality and represent its solution set.