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12-6

Graphing Inequalities in Two Variables. 12-6. Course 3. Warm Up. Problem of the Day. Lesson Presentation. Graphing Inequalities in Two Variables. 12-6. 1. y = x. 5. Course 3. Warm Up

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12-6

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  1. Graphing Inequalities in Two Variables 12-6 Course 3 Warm Up Problem of the Day Lesson Presentation

  2. Graphing Inequalities in Two Variables 12-6 1 y = x 5 Course 3 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 y = 5x

  3. Graphing Inequalities in Two Variables 12-6 Course 3 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. Graphing Inequalities in Two Variables 12-6 Course 3 Learn to graph inequalities on the coordinate plane.

  5. Graphing Inequalities in Two Variables 12-6 Course 3 Insert Lesson Title Here Vocabulary boundary line linear inequality

  6. Graphing Inequalities in Two Variables 12-6 Course 3 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. Graphing Inequalities in Two Variables 12-6 Course 3

  8. Graphing Inequalities in Two Variables 12-6 Course 3 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. Graphing Inequalities in Two Variables 12-6 ? 0 < 0 – 1 ? 0 < –1 Course 3 Additional Example 1A: Graphing Inequalities Graph each inequality. y < x – 1 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) Test a point not on the line. y < x – 1 Substitute 0 for x and 0 for y.

  10. Graphing Inequalities in Two Variables 12-6 Helpful Hint Any point on the line y = x -1 is not a solution of y < x- 1 because the inequality symbol < means only “less than” and does not include “equal to.” Course 3

  11. Graphing Inequalities in Two Variables 12-6 Course 3 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). (0, 0)

  12. Graphing Inequalities in Two Variables 12-6 y ≥ 2x + 1 ? 4 ≥ 0 + 1 Course 3 Additional Example 1B: Graphing Inequalities y 2x + 1 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.

  13. Graphing Inequalities in Two Variables 12-6 Helpful Hint Any point on the line y = 2x + 1 is a solution of y ≥ 2x+ 1 because the inequality symbol ≥ means “greater than or equal to.” Course 3

  14. Graphing Inequalities in Two Variables 12-6 Course 3 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). (0, 4)

  15. Graphing Inequalities in Two Variables 12-6 5 2 y < – x + 3 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 Course 3 Additional Example 1C: Graphing Inequalities 2y + 5x < 6 First write the equation in slope-intercept form. 2y + 5x < 6 2y < –5x + 6 Subtract 5x from both sides. Divide both sides by 2.

  16. Graphing Inequalities in Two Variables 12-6 y < – x + 3 ? 0 < 0 + 3 ? 0 < 3 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 5 2 2 Course 3 Additional Example 1C Continued (0, 0) Choose any point not on the line. Substitute 0 for x and 0 for y. (0, 0)

  17. Graphing Inequalities in Two Variables 12-6 ? 0 < 0 – 4 ? 0 < –4 Course 3 Check It Out: Example 1A Graph each inequality. y < x – 4 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) Test a point not on the line. y < x – 4 Substitute 0 for x and 0 for y.

  18. Graphing Inequalities in Two Variables 12-6 Course 3 Check It Out: 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). (0, 0)

  19. Graphing Inequalities in Two Variables 12-6 y ≥ 4x + 4 ? 3 ≥ 8 + 4 Course 3 Check It Out: Example 1B y> 4x + 4 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.

  20. Graphing Inequalities in Two Variables 12-6 Course 3 Check It Out: 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). (2, 3)

  21. Graphing Inequalities in Two Variables 12-6 4 3 y – x + 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 4 3 Course 3 Check It Out: Example 1C 3y + 4x 9 First write the equation in slope-intercept form. 3y + 4x 9 3y –4x + 9 Subtract 4x from both sides. Divide both sides by 3.

  22. Graphing Inequalities in Two Variables 12-6 y – x + 3 ? 0  0 + 3 ? 0  3 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). (0, 0) 4 4 3 3 Course 3 Check It Out: Example 1C Continued (0, 0) Choose any point not on the line. Substitute 0 for x and 0 for y.

  23. Graphing Inequalities in Two Variables 12-6 1 2 In 1 day the writer writes no more than 7 pages. Course 3 Additional Example 2: Career Application 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? 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)

  24. Graphing Inequalities in Two Variables 12-6 Helpful Hint The phrase “no more” can be translated as less than or equal to. Course 3

  25. Graphing Inequalities in Two Variables 12-6 7.5 1 = = 7.5 7.5 – 0 m = 1 – 0 Course 3 Additional Example 2 Continued With two known points, find the slope. y 7.5 x + 0 The y-intercept is 0. 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.

  26. Graphing Inequalities in Two Variables 12-6 ? 2  7.5 2 ? 2  15  Course 3 Additional Example 2 Continued (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).

  27. Graphing Inequalities in Two Variables 12-6 Course 3 Additional Example 2 Continued 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.

  28. Graphing Inequalities in Two Variables 12-6 Course 3 Check It Out: Example 2 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? First find the equation of the line that corresponds to the inequality. In 0 days the writer writes 0 pages. point (0, 0) In 5 days the writer writes no more than 20 pages. point (5, 20)

  29. Graphing Inequalities in Two Variables 12-6 20 - 0 m = = 20 5 - 0 = 4 5 Course 3 Check It Out: Example 2 Continued With two known points, find the slope. The y-intercept is 0. y 4x + 0 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.

  30. Graphing Inequalities in Two Variables 12-6 ? 60  4  5 ? 60  20  Course 3 Check It Out: Example 2 Continued (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).

  31. Graphing Inequalities in Two Variables 12-6 Course 3 Check It Out: Example 2 Continued y 200 180 160 140 120 100 80 60 40\ 20 Pages x 5 10 15 20 25 30 35 40 45 50 Days 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.

  32. Graphing Inequalities in Two Variables 12-6 1 3 Course 3 Lesson Quiz Part I Graph each inequality. 1.y < – x + 4

  33. Graphing Inequalities in Two Variables 12-6 Course 3 Lesson Quiz Part II 2. 4y + 2x > 12

  34. Graphing Inequalities in Two Variables 12-6 Course 3 Lesson Quiz: Part III 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

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