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Graphing Inequalities in Two Variables. 11-6. Warm Up. Problem of the Day. Lesson Presentation. Pre-Algebra. Graphing Inequalities in Two Variables. 11-6. 1. y = x. 5. Pre-Algebra. Warm Up

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11-6

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

  2. Graphing Inequalities in Two Variables 11-6 1 y = x 5 Pre-Algebra 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. 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. Today’s Learning Goal Assignment Learn to graph inequalities on the coordinate plane.

  5. Vocabulary boundary line linear inequality

  6. 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. 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.

  8. ? 0 < 0 – 1 ? 0 < –1 Additional Example 1A: 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.

  9. 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).

  10. ? 0 < 0 – 4 ? 0 < –4 Try This: Example 1A 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.

  11. 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).

  12. y ≥ 2x + 1 ? 4 ≥ 0 + 1 Additional Example 1B: 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.

  13. 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).

  14. y ≥ 4x + 4 ? 3 ≥ 8 + 4 Try This: Example 1B 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.

  15. 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).

  16. 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 Additional Example 1C: 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.

  17. 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 Additional Example 1C Continued (0, 0) Choose any point not on the line.

  18. 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: Example 1C 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.

  19. 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: Example 1C Continued (0, 0) Choose any point not on the line.

  20. 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)

  21. 1 3 1.y < – x + 4

  22. 2. 4y + 2x > 12

  23. 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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