Graphing Linear Inequalities in Two Variables Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Expressions of the type x + 2y ≤ 8 and 3x – y > 6 are called linear inequalities in two variables. A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true. Example: (1, 3) is a solution to x + 2y ≤ 8 since (1) + 2(3) = 7 ≤ 8. Solution of Linear Inequalities

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions. The solution set is a half-plane. It consists of the line x + 2y ≤ 8 and all the points below and to its left. The line is called the boundary line of the half-plane. Feasible Set Example: The solution set for x + 2y ≤ 8 is the shaded region. x y 2 2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 x y x y If the inequality is, the boundary line is dotted; its points are not solutions. If the inequality is ≤ or ≥, the boundary line is solid; its points are solutions. Example: The boundary line of the solution set of x + y < 2 is dotted. Example: The boundary line of the solution set of 3x – y ≥ 2 is solid. 3x – y < 2 Boundary lines 3x – y = 2 3x – y > 2

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 x y Example: For 2x – 3y ≤ 18 graph the boundary line. The solution set is a half-plane. A test point can be selected to determine which side of the half-plane to shade. Shade above and to the left of the line. Use (0, 0) as a test point. (0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions. (0, 0) 2 -2 Test Point

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 To graph the solution set for a linear inequality: 2. Select a test point, not on the boundary line, and determine if it is a solution. 3. Shade a half-plane. 1. Graph the boundary line. Graphing an Inequality

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 x y Example: Graph the solution set for x – y > Graph the boundary line x – y = 2 as a dotted line. 2. Select a test point not on the line, say (0, 0). (0) – 0 = 0 > 2 is false. 3. Since this is a not a solution, shade in the half-plane not containing (0, 0). (0, 0) (2, 0) (0, -2) Example: Graph an Inequality

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Solution sets for inequalities with only one variable can be graphed in the same way. Example: Graph the solution set for x < - 2. x y x y Example: Graph the solution set for x ≥ 4. Inequalities in One Variable

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities. (5, 4) is a solution of x + y > 8. (5, 4) is also a solution of 2x – y ≤ 7. Since (5, 4) is a solution of both inequalities in the system, it is a solution of the system. Example: Find a solution for the system. Solution of a System of Linear Inequalities

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 The set of all solutions of a system of linear inequalities is called its solution set. 1. Shade the half-plane of solutions for each inequality in the system. To graph the solution set for a system of linear inequalities in two variables: 2. Shade in the intersection of the half-planes. Solution Set

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 x y Graph the solution set for x + y > 8. The intersection of these two half-planes is the wedge-shaped region at the top of the diagram. Graph the solution set for 2x – y ≤ 7. Example: Graph the solution set for the system 2 2 Example: Graph a System of Two Inequalities

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Graph the solution set for the system of linear inequalities: Graph the two half-planes. The two half-planes do not intersect; therefore, the solution set is the empty set. x y 2x – 3y ≥ 12 -2x + 3y ≥ Example: Graph a System of Two Inequalities

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 x y Example: Graph the solution set for the linear system. Graph each linear inequality. The solution set is the intersection of all the half-planes. Example: Graph a System of Four Inequalities (1) (2) (3) (4)