1. Pre-Algebra 11-6 Graphing Inequalities in Two Variables 11-6 Graphing Inequalities in Two Variables Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Warm Up Find each equation of direct variation, given that y varies directly with x. 1. y is 18 when x is 3. 2. x is 60 when y is 12. 3. y is 126 when x is 18. 4. x is 4 when y is 20. y = 6x y = 7x Pre-Algebra 11-6 Graphing Inequalities in Two Variables y = 5x y = x 1 5

3. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Problem of the Day The circumference of a pizza varies directly with its diameter. If you graph that direct variation, what will the slope be? 

4. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Today’s Learning Goal Assignment Learn to graph inequalities on the coordinate plane.

5. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Vocabulary boundary line linear inequality

6. Pre-Algebra 11-6 Graphing Inequalities in Two Variables A graph of a linear equation separates the coordinate plane into three parts: the points on one side of the line, the points on the boundary line, and the points on the other side of the line.

7. Pre-Algebra 11-6 Graphing Inequalities in Two Variables

8. Pre-Algebra 11-6 Graphing Inequalities in Two Variables When the equality symbol is replaced in a linear equation by an inequality symbol, the statement is a linear inequality. Any ordered pair that makes the linear inequality true is a solution.

9. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Graph each inequality. A. y < x – 1 Additional Example 1A: Graphing Inequalities First graph the boundary line y = x – 1. Since no points that are on the line are solutions of y < x – 1, make the line dashed. Then determine on which side of the line the solutions lie. (0, 0) y < x – 1 Test a point not on the line. Substitute 0 for x and 0 for y. 0 < 0 – 1 ? 0 < –1 ?

10. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Additional Example 1A Continued Since 0 < –1 is not true, (0, 0) is not a solution of y < x – 1. Shade the side of the line that does not include (0, 0).

11. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Graph each inequality. A. y < x – 4 Try This: Example 1A First graph the boundary line y = x – 4. Since no points that are on the line are solutions of y < x – 4, make the line dashed. Then determine on which side of the line the solutions lie. (0, 0) y < x – 4 Test a point not on the line. Substitute 0 for x and 0 for y. 0 < 0 – 4 ? 0 < –4 ?

12. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Try This: Example 1A Continued Since 0 < –4 is not true, (0, 0) is not a solution of y < x – 4. Shade the side of the line that does not include (0, 0).

13. Pre-Algebra 11-6 Graphing Inequalities in Two Variables B. y  2x + 1 Additional Example 1B: Graphing Inequalities First graph the boundary line y = 2x + 1. Since points that are on the line are solutions of y  2x + 1, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y  2x + 1 lie. (0, 4) Choose any point not on the line. Substitute 0 for x and 4 for y. y ≥ 2x + 1 4 ≥ 0 + 1 ?

14. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Additional Example 1B Continued Since 4  1 is true, (0, 4) is a solution of y  2x + 1. Shade the side of the line that includes (0, 4).

15. Pre-Algebra 11-6 Graphing Inequalities in Two Variables B. y > 4x + 4 Try This: Example 1B First graph the boundary line y = 4x + 4. Since points that are on the line are solutions of y  4x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y  4x + 4 lie. (2, 3) Choose any point not on the line. Substitute 2 for x and 3 for y. y ≥ 4x + 4 3 ≥ 8 + 4 ?

16. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Try This: Example 1B Continued Since 3  12 is not true, (2, 3) is not a solution of y  4x + 4. Shade the side of the line that does not include (2, 3).

17. Pre-Algebra 11-6 Graphing Inequalities in Two Variables C. 2y + 5x < 6 Additional Example 1C: Graphing Inequalities First write the equation in slope-intercept form. 2y < –5x + 6 2y + 5x < 6 y < – x + 3 5 2 Then graph the line y = – x + 3. Since points that are on the line are not solutions of y < – x + 3, make the line dashed. Then determine on which side of the line the solutions lie. 5 2 5 2 Subtract 5x from both sides. Divide both sides by 2.

18. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Additional Example 1C Continued Since 0 < 3 is true, (0, 0) is a solution of y < – x + 3. Shade the side of the line that includes (0, 0). 5 2 (0, 0)Choose any point not on the line. y < – x + 3 5 2 0 < 0 + 3 ? 0 < 3 ?

19. Pre-Algebra 11-6 Graphing Inequalities in Two Variables C. 3y + 4x  9 Try This: Example 1C First write the equation in slope-intercept form. 3y  –4x + 9 3y + 4x  9 y  – x + 3 4 3 4 3 Then graph the line y = – x + 3. Since points that are on the line are solutions of y  – x + 3, make the line solid. Then determine on which side of the line the solutions lie. 4 3 Subtract 4x from both sides. Divide both sides by 3.

20. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Try This: Example 1C Continued Since 0  3 is not true, (0, 0) is not a solution of y  – x + 3. Shade the side of the line that does not include (0, 0). 4 3 (0, 0)Choose any point not on the line. y  – x + 3 4 3 0  0 + 3 ? 0  3 ?

21. Pre-Algebra 11-6 Graphing Inequalities in Two Variables A successful screenwriter can write no more than seven and a half pages of dialogue each day. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write a 200-page screenplay in 30 days? Additional Example 2: Career Application First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. point (0, 0) point (1, 7.5) In 1 day the writer writes no more than 7 pages. 1 2

22. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Additional Example 2 Continued With two known points, find the slope. y  7.5 x + 0 The y-intercept is 0. No more than means . Graph the boundary line y = 7.5x. Since points on the line are solutions of y  7.5x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y  7.5x lie. m = 7.5 – 0 1 – 0 7.5 1 == 7.5

23. Pre-Algebra 11-6 Graphing Inequalities in Two Variables (2, 2) Choose any point not on the line. y  7.5x Substitute 2 for x and 2 for y. Since 2  15 is true, (2, 2) is a solution of y  7.5x. Shade the side of the line that includes point (2, 2). Additional Example 2 Continued 2  7.5  2 ? 2  15 ?

24. Pre-Algebra 11-6 Graphing Inequalities in Two Variables The point (30, 200) is included in the shaded area, so the writer should be able to complete the 200 page screenplay in 30 days. Additional Example 2 Continued

25. Pre-Algebra 11-6 Graphing Inequalities in Two Variables A certain author can write no more than 20 pages every 5 days. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write 140 pages in 20 days? Try This: Example 2 First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. point (0, 0) point (5, 20) In 5 days the writer writes no more than 20 pages.

26. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Try This: Example 2 Continued 20 - 0 5 - 0 m = = 20 5 = 4 With two known points, find the slope. y  4x + 0 The y-intercept is 0. No more than means . Graph the boundary line y = 4x. Since points on the line are solutions of y  4x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y  4x lie.

27. Pre-Algebra 11-6 Graphing Inequalities in Two Variables (5, 60) Choose any point not on the line. y  4x Substitute 5 for x and 60 for y. Since 60  20 is not true, (5, 60) is not a solution of y  4x. Shade the side of the line that does not include (5, 60). Try This: Example 2 Continued 60  4  5 ? 60  20 ?

28. Pre-Algebra 11-6 Graphing Inequalities in Two Variables The point (20, 140) is not included in the shaded area, so the writer will not be able to write 140 pages in 20 days. Try This: Example 2 Continued x y 200 180 160 140 120 100 80 60 40 \ 20 Pages 5 10 15 20 25 30 35 40 45 50 Days

29. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Lesson Quiz Graph each inequality. 1. y < – x + 4 2. 4y + 2x > 12 Tell whether the given ordered pair is a solution of each inequality. 3. y < x + 15 (–2, 8) 4. y  3x – 1 (7, –1) 1 3

30. Pre-Algebra 11-6 Graphing Inequalities in Two Variables 1. y < – x + 4 1 3

31. Pre-Algebra 11-6 Graphing Inequalities in Two Variables 2. 4y + 2x > 12

32. Pre-Algebra 11-6 Graphing Inequalities in Two Variables Tell whether the given ordered pair is a solution of each inequality. 3. y < x + 15 (–2, 8) 4. y  3x – 1 (7, –1) yes no