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Graphing Inequalities in Two Variables

Graphing Inequalities in Two Variables. 11-6. Pre-Algebra. 1. y = x. 5. 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.

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Graphing Inequalities in Two Variables

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

  2. 1 y = x 5 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. Learn to graph inequalities on the coordinate plane.

  4. Vocabulary boundary line linear inequality

  5. 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.

  6. 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.

  7. ? 0 < 0 – 1 ? 0 < –1 Example: Graphing Inequalities Graph each inequality. A. 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.

  8. Example 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).

  9. y ≥ 2x + 1 ? 4 ≥ 0 + 1 Example: Graphing Inequalities B. 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.

  10. Example 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).

  11. 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 Example: Graphing Inequalities C. 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.

  12. 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 Example Continued (0, 0) Choose any point not on the line.

  13. ? 0 < 0 – 4 ? 0 < –4 Try This Graph each inequality. A. 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.

  14. Try This 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).

  15. y ≥ 4x + 4 ? 3 ≥ 8 + 4 Try This B. 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.

  16. Try This 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. 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 Try This C. 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.

  18. 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). 4 4 3 3 Try This Continued (0, 0) Choose any point not on the line.

  19. 1 2 In 1 day the writer writes no more than 7 pages. Example: 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)

  20. 7.5 1 = = 7.5 7.5 – 0 m = 1 – 0 Example 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.

  21. ? 2  7.5 2 ? 2  15 Example 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).

  22. Example 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.

  23. Try This 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)

  24. 20 - 0 m = = 20 5 - 0 = 4 5 Try This Continued With two known points, find the slope. The y-intercept is 0. No more than means . 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.

  25. ? 60  4  5 ? 60  20 Try This 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).

  26. Try This 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.

  27. 1 3 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)

  28. 1 3 1.y < – x + 4

  29. 2. 4y + 2x > 12

  30. 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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