Graphing Linear Inequalities in Two Variables Digital Lesson Graphing Linear Inequalities in Two Variables

Solution of Linear Inequalities 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solution of Linear Inequalities

Example: The solution set for x + 2y ≤ 8 is the shaded region. The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions. Example: The solution set for x + 2y ≤ 8 is the shaded region. x y 2 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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Feasible Set

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

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

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

Example: Graph an Inequality Example: Graph the solution set for x – y > 2. 1. Graph the boundary line x – y = 2 as a dotted line. x y (0, 0) 2. Select a test point not on the line, say (0, 0). (2, 0) (0, -2) (0) – 0 = 0 > 2 is false. 3. Since this is a not a solution, shade in the half-plane not containing (0, 0). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph an Inequality

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

Solution of a System of Linear Inequalities A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities. Example: Find a solution for the system . (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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solution of a System of Linear Inequalities

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

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

Example: Graph a System of Two Inequalities Example: Graph the solution set for the system of linear inequalities: x y -2x + 3y ≥ 6 Graph the two half-planes. 2 The two half-planes do not intersect; therefore, the solution set is the empty set. 2x – 3y ≥ 12 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph a System of Two Inequalities

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