Graphing Linear Inequalities in Two Variables Section 6-6 Graphing Linear Inequalities in Two Variables

Linear Inequalities x - 3y ≤ 6 0 ≤ 6 (0) – 3(0) ≤ 6 An example of a linear inequality in two variables is x - 3y ≤ 6. The solution of an inequality in two variables, x and y, is an ordered pair (x, y) that produces a true statement when substituted into the inequality. Which ordered pair is NOT a solution of x - 3y ≤ 6? A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2) Substitute each point into the inequality. If the statement is true then it is a solution. x - 3y ≤ 6 0 ≤ 6 (0) – 3(0) ≤ 6 True, therefore (0,0) is a solution.

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) ≤ 8 7 ≤ 8. (Yes, this is true.) Solution of Linear Inequalities

Graph an Inequality in Two Variables The graph of an inequality in two variables is the set of points that represent all solutions of the inequality. The BOUNDARY LINE of a linear inequality divides the coordinate plane into two HALF-PLANES. Only one half-plane contains the points that represent the solutions to the inequality.

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. Some solutions in the solution set are (0,0), (2, -2), and (-4, 2). (6, 4) is not in the solution set! Feasible Set

3x – y = 2 x y If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions. 3x – y < 2 3x – y > 2 Example: The boundary line of the solution set of 3x – y ≥ 2 is solid. Boundary lines

If the inequality is < or >, the boundary line is dotted; its points are not solutions. x y Example: The boundary line of the solution set of x + y < 2 is dotted. 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. Pick any point that is not on your line. Example: For 2x – 3y ≤ 18 graph the boundary line. x y (0, 0) Use (0, 0) as a test point. 2(0) – 3(0) ≤ 18 0 ≤ 18 Yes/True Shade towards your test point/Include your test point in the shading! The solution set/feasible set, is the set of all solutions in the shaded region. 2 -2 Test Point

Graphing an Inequality To graph the solution set/feasible set for a linear inequality: Step 1. Graph the boundary line. (Remember to check if your line is a dotted line or a solid line.) Step 2. Select a test point, not on the boundary line, and determine if it is a solution. Step 3. Shade a half-plane. Graphing an Inequality

Example: Graph an Inequality Example: Graph the solution set for x – y > 2. Step 1.) Graph the boundary line x – y = 2 as a dotted line. Step 2.) Select a test point not on the line, say (0, 0). x y x – y > 2 (0) – (0) > 2 0 > 2 No/False! (0, 0) (2, 0) (0, -2) Step 3.) Since this is a not a solution, shade in the half-plane not containing (0, 0). Or, shade away from your test point. The solution set/feasible set, is the set of all solutions in the shaded region. Example: Graph an Inequality

Step 1.) Graph the solution set for x < - 2. Solution sets for inequalities with only one variable can be graphed in the same way. Example: Graph the solution set for x < - 2. Step 1.) Graph the solution set for x < - 2. x y 4 - 4 Step 2.) Select a test point not on the line, say (0, 0). x < - 2 0 < - 2 No/False! Step 3.) Since this is a not a solution, shade in the half-plane not containing (0, 0). Or, shade away from your test point. The solution set/feasible set, is the set of all solutions in the shaded region. Inequalities in One Variable

Inequalities in One Variable Example: Graph the solution set for x ≥ 4. Step 1.) Graph the solution set for x ≥ 4. x y 4 - 4 Step 2.) Select a test point not on the line, say (0, 0). x ≥ 4 (0) ≥ 4 No/False! Since this is a not a solution, shade in the half-plane not containing (0, 0). Or, shade away from your test point. The solution set/feasible set, is the set of all solutions in the shaded region. Inequalities in One Variable

Graph the inequality y > 4x - 3. Graph an Inequality Graph the inequality y > 4x - 3. STEP 2 STEP 3 STEP 1 Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality. Graph the equation Test (0,0) in the original inequality.

Graph the inequality x + 2y ≤ 0. Graph an Inequality Graph the inequality x + 2y ≤ 0. STEP 2 STEP 3 STEP 1 Shade the half-plane that does not contain the point (1,0), because (1,0) is not a solution to the inequality. Graph the equation Test (1,0) in the original inequality.

Graph the inequality -1 ≤ x + y. Graph an Inequality Graph the inequality -1 ≤ x + y. STEP 2 STEP 3 STEP 1 Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality. Graph the equation Test (0,0) in the original inequality.

Graph the inequality x – 3y ≤ 12. Graph an Inequality Graph the inequality x – 3y ≤ 12. STEP 2 STEP 3 STEP 1 Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality. Graph the equation Test (0,0) in the original inequality.

Graph the inequality y ≥ -3. Graph an Inequality Graph the inequality y ≥ -3. STEP 2 STEP 3 STEP 1 Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality. Graph the equation Test (0,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

Graph the inequality x ≤ -1. Graph an Inequality Graph the inequality x ≤ -1. STEP 2 STEP 3 STEP 1 Shade the half-plane that does not contain the point (0,0), because (0,0) is not a solution to the inequality. Graph the equation Test (0,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.

Writing an Inequality Write an inequality for each graph. a. b.

Writing an Inequality Write an inequality for each graph. c. d.

Assignment Study Guide 6-6 (In-Class) Skills Practice/Practice Worksheet 6-6 Chapter 6 Test on

Application d. Suppose your budget for a party allows you to spend no more than $12 on peanuts and cashews. Peanuts cost $2/lb and cashews cost $4/lb. Find three possible combinations of peanuts and cashews you can buy. x = number of pounds of peanuts y = number of pounds of cashews 2x + 4y ≤ 12

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. 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