1. 30S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Graphing Inequalities in Two Variables Learning Outcome B-1 LP-L2 Objectives: Graph a linear inequality in two variables on a Cartesian Plane

2. 30S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables When you place a line on a coordinate plane, the line creates three regions: the region that is the line itself the region to the left of the line the region to the right of the line Consider the graph of the equation y = 2x + 1 shown. It separates the coordinate plane into two half-planes. One half-plane, Region 1, is on the left of the line. The point A with coordinates (-4, 3) is in Region 1. Theory – Inequalities in Two Dimensions

3. 30S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables If the x-coordinate of point A, -4, is substituted into the equation of the line to find the corresponding y-coordinate on the line, the result is: y = 2x + 1 y = 2(-4) + 1, or y = -7. In other words, the y-value for the point A is greater than the y- value for the line. Use another point in Region 1 to verify the relationship. The y- values in Region 1 for a given x-value are greater than the y-values on the line. This region or half-plane can be represented by the inequality y > 2x + 1. Theory – Inequalities in Two Dimensions

4. 30S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables What inequality do you think represents the area in Region 2? The other half-plane (Region 2) is on the right of the line. The point B with coordinates (2, 2) is in Region 2. If the x-coordinate of point B, 2, is substituted into the equation of the line to find the corresponding y-coordinate on the line, the result is: y = 2x + 1 y = 2(2) + 1, or y = 5. In other words, the y-value for the point B is less than the y- value for the line. The y- values in Region 2 for a given x- value are less than the y-values on the line. This region or half-plane can be represented by the inequality y < 2x + 1. Theory – Inequalities in Two Dimensions

5. 30S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Summary A line divides the coordinate plane into two half-planes, which can each be represented by an inequality statement. In our example, the line y = 2x + 1 divides the coordinate plane into two halves. Given the line’s positive slope, the left half can be represented by the equation y > 2x + 1, and the right half can be represented by the equation y < 2x + 1. Theory – Inequalities in Two Dimensions

6. 30S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Example Graph the region 2x + y < 4. Steps: 1.Solve for y. 2.Determine dashed or solid line. 3.Select a point and test in inequality. 4.Shade accordingly. (If test was true, shade region that includes point. If false, shade other side of boundary.) Example – Inequalities in Two Dimensions

7. 30S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Example Graph the region 2x + y < 4. Solution Example – Inequalities in Two Dimensions

8. 30S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Graph the region: Test Yourself

9. 30S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Graph the region: Test Yourself

10. 30S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Graph the region: Test Yourself

11. 30S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Graph the region: Test Yourself

12. 30S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Graph the region: Test Yourself

13. 30S Applied Math Mr. Knight – Killarney School Slide 13 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Special Case: Graph the region: Test Yourself

14. 30S Applied Math Mr. Knight – Killarney School Slide 14 Unit: Linear Programming Lesson: Graphing Inequalities in Two Variables Special Case: Graph the region: Test Yourself