Introduction

1. Introduction Solving a linear inequality in two variables is similar to graphing a linear equation, with a few extra steps that will be explained on the following slides. Remember that inequalities have infinitely many solutions and all the solutions need to be represented. This will be done through the use of shading. 2.3.1: Solving Linear Inequalities in Two Variables

2. Key Concepts, continued Sometimes the line or the boundary is part of the solution; this means it’s inclusive. Inequalities that have “greater than or equal to” (≥) or “less than or equal to” (≤) symbols are inclusive. Use a solid line when graphing the solution to inclusive inequalities. Other times the line or boundary is NOT part of the solution; in other words, it’s non-inclusive. Inequalities that have “greater than” (>) or “less than” (<) symbols are non-inclusive. Use a dashed line when graphing the solution to non-inclusive inequalities. 2.3.1: Solving Linear Inequalities in Two Variables

3. Key Concepts, continued Substitute the test point into the inequality. If the test point makes the inequality true, shade the side of the line that contains the test point. If it does not make the inequality true, shade the opposite side of the line. Shading indicates that all points in that region are solutions. 2.3.1: Solving Linear Inequalities in Two Variables

4. Key Concepts, continued 2.3.1: Solving Linear Inequalities in Two Variables

5. Guided Practice Example 1 Graph the solutions to the following inequality. 2.3.1: Solving Linear Inequalities in Two Variables

6. Guided Practice: Example 1, continued Graph the inequality as a linear equation. Since the inequality is non-inclusive, use a dashed line. y = x + 3 To graph the line, plot the y-intercept first, (0, 3). Then use the slope to find a second point. The slope is 1. Count up one unit and to the right one unit and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane. 2.3.1: Solving Linear Inequalities in Two Variables

7. 2.3.1: Solving Linear Inequalities in Two Variables

8. Guided Practice: Example 1, continued Pick a test point above or below the line and substitute the point into the inequality. Choose (0, 0) because this point is easy to substitute into the inequality. y > x + 3 (0) > (0) + 3 0 > 3    This is false! 2.3.1: Solving Linear Inequalities in Two Variables

9. 2.3.1: Solving Linear Inequalities in Two Variables

10. Guided Practice: Example 1, continued Shade the appropriate area. Since the test point makes the inequality false, all points on that side of the line make the inequality false. Shade above the line instead; this is the area that does NOT contain the point. 2.3.1: Solving Linear Inequalities in Two Variables

11. ✔ 2.3.1: Solving Linear Inequalities in Two Variables

12. Guided Practice Example 2 Graph the solutions to the following inequality. 2.3.1: Solving Linear Inequalities in Two Variables

13. Guided Practice: Example 2, continued 1. Graph the inequality as a linear equation. Since the inequality is inclusive, use a solid line. To graph the line, plot the y-intercept first, (0, 2). Then use the slope to find a second point. The slope is 3. Count up three units and to the right one unit and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane. 2.3.1: Solving Linear Inequalities in Two Variables

14. 14 2.3.1: Solving Linear Inequalities in Two Variables 2.3.1: Solving Linear Inequalities in Two Variables

15. Guided Practice: Example 1, continued Pick a test point above or below the line and substitute the point into the inequality. Choose (0, 0) because this point is easy to substitute into the inequality. (0) > 3(0) + 2 0 > 0 + 2 0 > 2    This is false! 2.3.1: Solving Linear Inequalities in Two Variables

16. 16 2.3.1: Solving Linear Inequalities in Two Variables

17. Guided Practice: Example 2, continued Shade the appropriate area. Since the test point makes the inequality false, all points on that side of the line make the inequality false. Shade above the line instead; this is the area that does NOT contain the point. 2.3.1: Solving Linear Inequalities in Two Variables

18. 2.3.1: Solving Linear Inequalities in Two Variables

19. Guided Practice Example 3 Graph the solutions to the following inequality. 2.3.1: Solving Linear Inequalities in Two Variables

20. Guided Practice: Example 3, continued 1. Graph the inequality as a linear equation. Since the inequality is not inclusive, use a dashed line. To graph the line, plot the y-intercept first, (0, -1). Then use the slope to find a second point. The slope is. Count down two units and to the right three units and plot a second point. Connect the two points and extend the line to the edges of the coordinate plane. 2.3.1: Solving Linear Inequalities in Two Variables

21. 21 2.3.1: Solving Linear Inequalities in Two Variables 2.3.1: Solving Linear Inequalities in Two Variables

22. Guided Practice: Example 3, continued Pick a test point above or below the line and substitute the point into the inequality. Choose (0, 0) because this point is easy to substitute into the inequality. (0) (0) - 1 0 0 - 1 0 -1    This is false! 2.3.1: Solving Linear Inequalities in Two Variables

23. 23 2.3.1: Solving Linear Inequalities in Two Variables

24. Guided Practice: Example 3, continued Shade the appropriate area. Since the test point makes the inequality false, all points on that side of the line make the inequality false. Shade above the line instead; this is the area that does NOT contain the point. 2.3.1: Solving Linear Inequalities in Two Variables

25. 25 25 2.3.1: Solving Linear Inequalities in Two Variables

26. You Try! Graph the following inequalities: 1. 2.