9 Chapter Chapter 2 Inequalities and Absolute Value

Section 9.4 Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities

Graph a Linear Inequality in Two Variables. Objective 1 Graph a Linear Inequality in Two Variables.

Linear Inequalities in Two Variables Linear inequality in two variables Can be written in one of the forms Ax + By < C Ax + By  C Ax + By > C Ax + By  C A, B, and C are real numbers, A and B are not both 0. An ordered pair is a solution of the linear inequality if it makes the inequality a true statement. 4

Linear Inequalities in Two Variables To Graph a Linear Inequality in Two Variables 1. Graph the boundary line found by replacing the inequality sign with an equal sign. If the inequality sign is > or <, graph a dashed boundary line (indicating that the points on the line are not solutions of the inequality). If the inequality sign is  or , graph a solid boundary line (indicating that the points on the line are solutions of the inequality). 2. Choose a point, not on the boundary line, as a test point. Substitute into the original inequality. 3. If a true statement is obtained in Step 2, shade the half-plane that contains the test point. If a false statement is obtained, shade the half-plane that does not contain the test point. 5

Example Graph 7x + y > –14 Graph 7x + y = –14 as a dashed line. Pick a point not on the graph: (0,0) (0,0) Test it in the original inequality. 7(0) + 0 > –14, 0 > –14 True, so shade the side containing (0,0).

Example Graph 3x + 5y  –2. Graph 3x + 5y = –2 as a solid line. Pick a point not on the graph: (0,0), but just barely Test it in the original inequality. 3(0) + 5(0)  –2, 0  –2 False, so shade the side that does not contain (0,0).

Example Graph 3x < 15. Graph 3x = 15 as a dashed line. Pick a point not on the graph: (0,0) Test it in the original inequality. 3(0) < 15, 0 < 15 True, so shade the side containing (0,0).

Linear Inequalities in Two Variables Warning! Note that although all of our examples allowed us to select (0, 0) as our test point, that will not always be true. If the boundary line contains (0,0), you must select another point that is not contained on the line as your test point. 9

Solving a System of Linear Inequalities Objective 1 Solving a System of Linear Inequalities

Systems of Linear Inequalities Two linear inequalities make a system of linear inequalities. A solution of a system of linear inequalities is an ordered pair that satisfies each inequality in the system. Graphing the Solution of a System of Linear Inequalities Step 1: Graph each inequality in the system on the same set of axes. Step 2: The solutions (or solution region) of the system are the points common to the graphs of all the inequalities in the system. 11

Example Graph the solution of the system: Graph both inequalities on the same set of axes. Both boundary lines are dashed lines since the inequality symbols are < and >. The solution of the system is the region shown by the purple shading.

Example 3x + y < 9 Graph the solution of the system: 2x + 5y  10 The solution will be the set of all points that satisfy both of the inequalities in the system. y x 4 The boundary line is 3x + y = 9. Solution The boundary line is 2x + 5y = 10

Example y > 2x + 4 Graph the solution of the system: y < 2x – 2 1 2 3 4 1 2 3 4 y > 2x + 4 This system has no solution. y < 2x – 2

Example Graph the solution of the system: Graph both inequalities on the same set of axes.