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Solving Systems of Equations

Solving Systems of Equations. Graphing Linear Inequalities. Objectives. How do we graph an inequality Define a boundary line Graphing a boundary line Define the solution for a system of inequalities Find the solution of a system of inequalities. What is the solution of an inequality.

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Solving Systems of Equations

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  1. Solving Systems of Equations Graphing Linear Inequalities

  2. Objectives • How do we graph an inequality • Define a boundary line • Graphing a boundary line • Define the solution for a system of inequalities • Find the solution of a system of inequalities

  3. What is the solution of an inequality • Solution of an inequality are all the ordered pairs (points) that make the inequality true.

  4. Graph Graphing Inequalities Consider the inequality y ≥ x y = x REMEMBER: Solution are all the ordered pairs (points) that make the inequality true. 6 5 4 3 Boundary line 2 1 1 2 3 4 5 6

  5. Graphing Inequalities Consider the inequality y ≥ x Pick two points from each side of the graph 6 5 4 (1,3) 3 2 (4,1) 1 1 2 3 4 5 6

  6. substitute into Graphing Inequalities Consider the inequality y ≥ x Check points if they make inequality true. (1,3) y ≥ x 6 5 4 (1,3) 3 2 (4,1) 1 1 2 3 4 5 6

  7. Graphing Inequalities Consider the inequality y ≥ x Check points if they make inequality true. (1,3) y ≥ x 6 substitute into 3 ≥ 1 5 4 (1,3) 3 2 (4,1) 1 1 2 3 4 5 6

  8. Graphing Inequalities Consider the inequality y ≥ x Check points if they make inequality true. (1,3) y ≥ x 6 substitute into 3 ≥ 1  5 4  (1,3) 3 2 (4,1) 1 1 2 3 4 5 6

  9. substitute into Graphing Inequalities Consider the inequality y ≥ x Check points if they make inequality true. (4,1) y ≥ x 6 5 4  (1,3) 3 2 (4,1) 1 1 2 3 4 5 6

  10. Graphing Inequalities Consider the inequality y ≥ x Check points if they make inequality true. (4,1) y ≥ x 6 substitute into 1 ≥ 4 5 4  (1,3) 3 2 (4,1) 1 1 2 3 4 5 6

  11. Graphing Inequalities Consider the inequality y ≥ x Check points if they make inequality true. (4,1) y ≥ x 6 substitute into 1 ≥ 4 X 5 4  (1,3) 3 2 X (4,1) 1 1 2 3 4 5 6

  12. Graphing Inequalities Consider the inequality y ≥ x Shade the side where the correct point lies. 6 5 4  (1,3) 3 2 X (4,1) 1 1 2 3 4 5 6

  13. Graphing Inequalities Consider the inequality y ≥ x Shade the side where the correct point lies. 6 5 4  (1,3) 3 2 1 1 2 3 4 5 6

  14. 1 y = x - 2 2 Graphing Inequalities Consider the inequality x - 2y ≤ 4 x - 2y = 4 Graph 3 2 1 1 2 3 4 5 6 -1 -2

  15. 1 y = x - 2 2 (0,1) (6,0) Graphing Inequalities Consider the inequality x - 2y ≤ 4 x - 2y = 4 Graph ¡¡TEST POINTS !! 3 2 1 1 2 3 4 5 6 -1 -2

  16. substitute into Graphing Inequalities Consider the inequality x - 2y ≤ 4 (0,1) x - 2y ≤ 4 3 2 (0,1) 1 (6,0) 1 2 3 4 5 6 -1 -2

  17. Graphing Inequalities Consider the inequality x - 2y ≤ 4 (0,1) x - 2y ≤ 4 substitute into 0 - 2(1) ≤ 4  -2 ≤ 4 3 2  (0,1) 1 (6,0) 1 2 3 4 5 6 -1 -2

  18. Graphing Inequalities Consider the inequality x - 2y ≤ 4 (6,0) x - 2y ≤ 4 substitute into 3 2  (0,1) 1 (6,0) 1 2 3 4 5 6 -1 -2

  19. Graphing Inequalities Consider the inequality x - 2y ≤ 4 (6,0) x - 2y ≤ 4 substitute into 6 - 2(0) ≤ 4 X 6 ≤ 4 3 2  (0,1) 1 (6,0) X 1 2 3 4 5 6 -1 -2

  20. Graphing Inequalities Consider the inequality x - 2y ≤ 4 ¡¡ SHADE CORRECT REGION !! 3 2  (0,1) 1 (6,0) X 1 2 3 4 5 6 -1 -2

  21. 2 GRAPH y = x + 3 3 6 5 4 3 2 1 1 2 3 4 5 6 Examples 1. 3y - 2x ≥ 9

  22. (0,5) 6 5 4 3 (3,0) 2 1 1 2 3 4 5 6 Examples 1. 3y - 2x ≥ 9 2 GRAPH y = x + 3 3 TEST!! (0, 5)  3(5) - 2(0) ≥ 9  15 - 0 ≥ 9 X

  23. (0,5) 1 y = x + 1 3 6 Graph 5 4 3 2 1 1 2 3 4 5 6 Examples 2. x - 3y > -3 TEST!! (0, 5) X 0 - 3(5) > -3 0 - 15 > -3 X

  24. 3 2 1 -3 -2 -1 1 2 3 -1 Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2

  25. Graph 3 2 (0,0) 1 -3 -2 -1 1 2 3 -1 Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 y = - x - 1 TEST: (0,0) 0 + 0 ≥ -1  0 ≥ -1

  26. Graph 3 2 1 -3 -2 -1 1 2 3 (0,0) -1 Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 y = 2x + 2 TEST: (0,0) -2(0) + 0 < 2  0 < 2

  27. 3 3 2 2 1 1 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 -1 -1 x + y ≥ -1 -2x + y < 2

  28. 3 2 1 -3 -2 -1 1 2 3 -1 Solving a system of Inequalities Consider the system x + y ≥ -1 -2x + y < 2 SOLUTION: • Lies where the two shaded • regions intersect each • other.

  29. Graph 3 2 y = x - 2 1 (0,0) -2 -1 1 2 3 4 2 3 Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 12 X TEST: (0,0) -2(0) + 3(0) < -6 -1 0 < -6 X -2

  30. Graph 3 2 y = - x + 3 1 (0,0) -2 -1 1 2 3 4 5 4 Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 12  TEST: (0,0) 5(0) + 4(0) < 12 -1  0 < 12 -2

  31. Graph 3 2 1 (0,0) -2 -1 1 2 3 4 Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 12 SOLUTION:  • Lies where the two shaded • regions intersect each • other. -1 -2

  32. Graph 3 2 1 (0,0) -2 -1 1 2 3 4 Solving a system of Inequalities Consider the system -2x + 3y < -6 5x + 4y < 12 NOTE:  • All order pairs in dark • region are true in both • inequalities. -1 -2

  33. Graph 6 4 2 (0,0) 2 4 6 8 10 12 -2 -4 -6 Solving a system of Inequalities Consider the system x - 4y ≤ 12 4y + x ≤ 12 TEST: (0,0) (0) - 4(0) ≤ 12 0 - 0 ≤ 12  0 ≤ 12

  34. Graph 6 4 2 (0,0) 2 4 6 8 10 12 -2 -4 -6 Solving a system of Inequalities Consider the system x - 4y ≤ 12 4y + x ≤ 12 TEST: (0,0) 4(0) + (0) ≤ 12  0 ≤ 12

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