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Warm Up

Graph Linear Inequalities in Two Variables. Warm Up. Lesson Presentation. Lesson Quiz. Warm-Up. Tell whether each statement is true or false when x = –2 and y = 1. 1. 2 x – y < 5. 2. x + 3 y ≥ 0. true. true. ANSWER. ANSWER.

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Warm Up

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  1. Graph Linear Inequalities in Two Variables Warm Up Lesson Presentation Lesson Quiz

  2. Warm-Up Tell whether each statement is true or false when x = –2 and y = 1. 1. 2x – y < 5 2.x + 3y ≥ 0 true true ANSWER ANSWER 3. The equation 360x + 600y = 5640 models the weekly payroll for a small business. Give an example of a solution of the equation. Sample answer:(9, 4) ANSWER

  3. 3(6) + 4(– 3) = 6 > 8 3(0) + 4(2) = 8 > 8 3(–2) + 4(–1) = –10 > 8 EXAMPLE 1 SOLUTION Ordered pair Substitute Conclusion (6, –3) (6, –3) is not a solution (0, 2) is not a solution (0, 2) (–2, –1) is not a solution (–2, –1) (–3, 5) (–3, 5) is a solution 3(–3) + 4(5) = 11 > 8

  4. EXAMPLE 1 ANSWER The correct answer is D.

  5. (–1, –7) (0, –4) (2, 2) (–3, 8) Guided Practice Tell whether the given ordered pair is a solution of 5x – 2y ≤ 6. Ordered pair Conclusion ANSWER (0, – 4 ) is not a solution (2, 2 ) is a solution (–3, 8 ) is a solution (– 1, – 7 ) is not a solution

  6. EXAMPLE 2 Graph(a)y< –3 and(b)x < 2 in a coordinate plane. a.Graphthe boundary liney = –3. Useasolidline because the inequality symbol is<. Test the point (0,0). Because (0,0) is nota solution of the inequality, shade thehalf- plane that does not contain (0,0).

  7. EXAMPLE 2 b.Graphthe boundary linex = 2.Useadashedline because the inequality symbol is < . Test the point (0,0).Because (0,0) is a solution of the inequality, shade thehalf-plane that contains (0,0).

  8. EXAMPLE 3 Graph(a)y > –2xand (b) 5x – 2y ≤ –4in a coordinate plane. a.Graphthe boundary liney = –2x. Useadashedline because the inequality symbol is >. Test the point (1,1).Because (1,1) is a solution of the inequality, shade thehalf-plane that contains (1,1).

  9. EXAMPLE 3 b.Graphthe boundary line 5x –2y = –4.Useasolidline because the inequality symbol is<. Test the point (0,0).Because (0,0) is nota solution of the inequality, shade thehalf-plane that does not contain (0,0).

  10. y > –1 Guided Practice Graph the inequality in a coordinate plane.

  11. x > –4 Guided Practice Graph the inequality in a coordinate plane.

  12. y > –3x Guided Practice Graph the inequality in a coordinate plane.

  13. y < 2x +3 Guided Practice Graph the inequality in a coordinate plane.

  14. x + 3y < 9 Guided Practice Graph the inequality in a coordinate plane.

  15. 2x –6y > 9 Guided Practice Graph the inequality in a coordinate plane.

  16. EXAMPLE 4 Movie Recording A film class is recording a DVD of student-made short films. Each student group is allotted up to 300 megabytes (MB) of video space. The films are encoded on the DVD at two different rates: a standard rate of 0.4 MB/sec for normal scenes and a high-quality rate of 1.2 MB/sec for complex scenes.

  17. EXAMPLE 4 • Write an inequality describing the possible amounts of time available for standard and high-quality video. • Graph the inequality. • Identify three possible solutions of the inequality.

  18. EXAMPLE 4 SOLUTION STEP1 Write an inequality. First write a verbal model. An inequality is 0.4x +1.2y ≤ 300.

  19. EXAMPLE 4 STEP2 Graph the inequality. First graph the boundary line 0.4x + 1.2y = 300. Use a solid line because the inequality symbol is ≤ . Test the point (0, 0). Because (0, 0)is a solution of the inequality, shade the half-plane that contains (0, 0). Because xand ycannot be negative, shade only points in the first quadrant.

  20. (150, 200) 150 seconds of standard and 200 seconds of high quality (300, 120) 300 seconds of standard and 120 seconds of high quality (600, 25) 600 seconds of standard and 25 seconds of high quality EXAMPLE 4 STEP3 Identify solutions. Three solutions are given below and on the graph. For the first solution, 0.4(150) + 1.2(200) = 300, so all of the available space is used. For the other two solutions, not all of the space is used.

  21. Graph y> – 2 x – 3 + 4 in a coordinate plane. Graph the equation of the boundary, y =–2 x – 3 + 4. Use a dashed line because the inequality symbol is > . EXAMPLE 5 SOLUTION STEP1 STEP2 Test the point (0, 0). Because (0, 0)is a solution of the inequality, shade the portion of the coordinate plane outside the absolute value graph.

  22. Guided Practice What If?Repeat the steps of Example 4 if each student group is allotted up to 420 MB of video space. ANSWER STEP3 300 seconds of standard 200 seconds of high quality, 600 seconds of standard 150 seconds of high quality, or 100, seconds of standard 300 seconds of high quality

  23. y<x – 2 + 1 Guided Practice Graph the inequality in a coordinate plane. ANSWER

  24. y>–x + 3 – 2 Guided Practice Graph the inequality in a coordinate plane. ANSWER

  25. y<3x – 1 – 3 Guided Practice Graph the inequality in a coordinate plane. ANSWER

  26. 1. Graphy < 3 x + 1 – 2on the coordinate plane. ANSWER Lesson Quiz

  27. ANSWER x + 2y < 256 Lesson Quiz The memory card for your digital camera has 256 megabytes of memory. Each photo uses either 1 megabyte or 2 megabytes of memory depending on whether you take low resolution or high resolution photos. 2. Write an equation that models the number of photos that can be stored on the card when you take photos at both resolutions.

  28. ANSWER 30 low resolution and 113 high resolution, 85 low resolution and 85 high resolution,or 200 low resolution and 28 high resolution. Lesson Quiz 3. Identify three possible solutions of the inequality.

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