2. Solving Inequalities

3. Objective: The student will be able to solve inequalities. Algebra 2

4. -9 -8 -7 -6 -5 -4 -3 -2 Inequalities -2x - 6 > 4 -2x > 10 -2 -2 x < -5 +6 +6 E

5. Solve and Graph the Following 6.-x - 2 > 6 7.5x - 6 ≥ 2x 8.-7 < 2x - 3 < 3

6. -12 -11 -10 -9 -8 -7 -6 -5 Inequalities -x - 2 > 6 -x > 8 -1 -1 x < -8 +2 +2 6

7. -2 -1 0 1 2 3 4 5 Inequalities 5x - 6 ≥ 2x 3x - 6 ≥ 0 3 3 x ≥ 2 -2x -2x +6 +6 3x ≥ 6 OR 7

8. -2 -1 0 1 2 3 4 5 Inequalities 5x - 6 ≥ 2x - 6 ≥ -3x -3 -3 2 ≤ x -5x -5x x ≥ 2 7

9. -3 -2 -1 0 1 2 3 4 Inequalities -7 < 2x - 3 < 3 -4 < 2x < 6 2 2 2 -2 < x < 3 +3 +3 +3 8

10. Solve and Graph the Following 9.-11 ≤ 3x - 5 ≤ -2 10.-19 < 4x - 3 ≤ 1

11. -4 -3 -2 -1 0 1 2 3 Inequalities -11 ≤ 3x - 5 ≤ -2 -6 ≤ 3x ≤ 3 3 3 3 -2 ≤ x ≤ 1 +5 +5 +5 9

12. -5 -4 -3 -2 -1 0 1 2 Inequalities -19 < 4x - 3 ≤ 1 -16 < 4x ≤ 4 4 4 4 -4 < x ≤ 1 +3 +3 +3 10

13. and versus or Inequalities E

14. -4 -3 -2 -1 0 1 2 3 x < 2 and x > -2 Inequalities { }x-2 2< < E

15. -2 -1 0 1 2 3 4 5 and x > -1 { } x > 1 Inequalities x > 1 E

16. -5 -4 -3 -2 -1 0 1 2 no solutions and x < -2 x > -1 Inequalities E

17. -3 -2 -1 0 1 2 3 4 all real numbers or all real numbers x < 1 Inequalities E

18. -4 -3 -2 -1 0 1 2 3 -1 -1 x + 1 -4 x < 2 +2 +2 and x > -2 Inequalities { }x-2 2< < 11

19. -3 -2 -1 0 1 2 3 4 +1 +1 x - 1 -2 x < 3 +1 +1 and x > -1 Inequalities { }x 3< < 12

20. -x x - 2 1 -2 < x +2 +2 3x > 3 and 3-3 x > -2 { } x > 1 Inequalities x > 1 13 -3 -2 -1 0 1 2 3 4

21. Absolute Value Equations

22. Objective: The student will be able to solve absolute value equations. Algebra 2 SOL A2.4

23. Absolute Value |6| =|6| = 6 |-6| =|-6| = 6 |7| =|7| = 7 |-7| =|-7| = 7

24. Absolute Value Equations | x | = 4 x = 4 or x = -4 | x | = 5 x = 5 or x = -5

25. Absolute Value Equations | x | = - 4 No Solutions | x+2 | = 7 x+2 = 7 or x+2 = -7

26. Absolute Value Equations | x-8 | = x x-8 = x or x-8 = -x | 2x+2 | = x 2x+2 = x or 2x+2 = -x

27. Absolute Value Equations | x-4 | = x + 2 x-4 = x+2 or x-4 = -x-2 | x+5 | = 3x-4 x+5 = 3x-4 or x+5 = -3x+4

28. | x - 1 | + 5 = 7 -5 -5 | x - 1 | = 2 Absolute Value Equations E

29. | x - 1 | = 2 +1 +1 x - 1 = 2 or x - 1 = -2 x = 3 +1 x = -1 or Answers must be checked! E

30. | x - 1 | + 5 = 7 | (3) - 1 | + 5 = 7 Absolute Value Equations E | 2 | + 5 = 7 2 + 5 = 7 7 = 7

31. | x - 1 | + 5 = 7 | (-1) - 1 | + 5 = 7 Absolute Value Equations E | -2 | + 5 = 7 2 + 5 = 7 7 = 7

32. Absolute Value Equations | x - 1 | = 2 +1 +1 x - 1 = 2 or x - 1 = -2 x = 3 +1 x = -1 or {3, -1} E

33. Absolute Value Equations | x - 2 | = 4 +2 +2 x - 2 = 4 or x - 2 = -4 x = 6 +2 x = -2 or Answers must be checked! 14

34. | x - 2 | = 4 | (6) - 2 | = 4 | 4 | = 4 4 = 4 Absolute Value Equations 14

35. | x - 2 | = 4 | (-2) - 2 | = 4 | -4 | = 4 4 = 4 Absolute Value Equations 14

36. Absolute Value Equations | x - 2 | = 4 +2 +2 x - 2 = 4 or x - 2 = -4 x = 6 +2 x = -2 or {6, -2} 14

37. Absolute Value Equations E | 2x - 6 | = 4x -2x 2x - 6 = 4x or 2x - 6 = -4x -6 = 2x 2 2 -3 = x

38. Absolute Value Equations E | 2x - 6 | = 4x | 2(-3) - 6 | = 4(-3) | -6 - 6 | = -12 12 = -12  | -12 | = -12

39. Absolute Value Equations E | 2x - 6 | = 4x -2x 2x - 6 = 4x or 2x - 6 = -4x -6 = 2x -2x -6 = -6x or 2 2 -3 = x 1 = x -6

40. Absolute Value Equations E | 2x - 6 | = 4x | 2(1) - 6 | = 4(1) | 2 - 6 | = 4 4 = 4 | -4 | = 4

41. Absolute Value Equations E | 2x - 6 | = 4x -2x 2x - 6 = 4x or 2x - 6 = -4x -6 = 2x -2x -6 = -6x or 2 2 -3 = x 1 = x -6 {1}

42. | x - 2 | = 2x -x x - 2 = 2x or x - 2 = -2x -2 = x Absolute Value Equations 15

43. Absolute Value Equations 15 | (-2) - 2 | = 2(-2) | -4 | = -4 4 = -4  | x - 2 | = 2x

44. -x x - 2 = 2x or x - 2 = -2x -2 = x -x -2 = -3x or -3 = x 2 3 Absolute Value Equations 15 | x - 2 | = 2x

45. Absolute Value Equations 15 | ( ) - 2 | = 2( ) | - | = = 2323 2323 4343 4343 4343 4343 | x - 2 | = 2x

46. -x x - 2 = 2x or x - 2 = -2x -2 = x -x -2 = -3x or -3 = x 2 3 Absolute Value Equations 15 { } 2323 | x - 2 | = 2x

47. | 4x - 6 | = 2x -4x 4x - 6 = 2x or 4x - 6 = -2x -6 = -2x -2 3 = x Absolute Value Equations 16

48. | 4x - 6 | = 2x | 4(3) - 6 | = 2(3) | 12 - 6 | = 6 | 6 | = 6 6 = 6 Absolute Value Equations 16

49. | 4x - 6 | = 2x -4x 4x - 6 = 2x or 4x - 6 = -2x -6 = -2x -4x -6 = -6x or -2 3 = x 1 = x -6 Absolute Value Equations 16

50. | 4x - 6 | = 2x | 4(1) - 6 | = 2(1) | 4 - 6 | = 2 | -2 | = 2 2 = 2 Absolute Value Equations 16

51. | 4x - 6 | = 2x -4x 4x - 6 = 2x or 4x - 6 = -2x -6 = -2x -4x -6 = -6x or -2 3 = x 1 = x -6 {1, 3} Absolute Value Equations 16

52. Absolute Value Inequalities

53. Objective: The student will be able to solve absolute value inequalities. Algebra 2

54. Absolute Value Inequalities | x | ≥ 2 -4 -3 -2 -1 0 1 2 3

55. Absolute Value Inequalities | x | ≥ 2 -4 -3 -2 -1 0 1 2 3 x ≥ 2orx ≤ -2

56. | x | ≥ 1 x ≥ 1 or x ≤ -1 -4 -3 -2 -1 0 1 2 3 Absolute Value Inequalities

57. | x | ≤ 2 -4 -3 -2 -1 0 1 2 3 x ≤ 2 and x ≥ -2

58. | x | ≤ 2 x ≤ 2 and x ≥ -2 -4 -3 -2 -1 0 1 2 3 Absolute Value Inequalities { }x-22≤≤

59. | x | ≤ 3 x ≤ 3 and x ≥ -3 -4 -3 -2 -1 0 1 2 3 Absolute Value Inequalities { }x-33≤≤

60. | x + 4 | > 2x - 5 Write the compound inequality that is equivalent to the given. | -2x + 4 | ≤ x Absolute Value Inequalities

61. < Less Thand > Greator> Greater < Less Than Absolute Value Inequalities

62. | x + 4 | > 2x - 5 x + 4 > 2x - 5orx + 4<-2x+ 5 Write the compound inequality that is equivalent to the given. Absolute Value Inequalities

63. | -2x + 4 | ≤ x -2x + 4 ≤ x and -2x + 4 ≥-x Write the compound inequality that is equivalent to the given. Absolute Value Inequalities

64. |x - 3| < 2x -x -3 < x -x -x and -3 > -3x Absolute Value Inequalities E1 x > -3 -3 1 < x x > 1 x - 3 < 2xandx - 3>-2x

65. -3 -2 -1 0 1 2 3 4 Absolute Value Inequalities E1 and x > -3 x > 1 {x > 1}

66. |4x| < 2x - 6 -2x 2x < -6 +2x +2x 6x > 6 and 6 62 2 x < -3 x > 1 4x < 2x - 6and4x>-2x+ 6 Absolute Value Inequalities 17

67. |4x| < 2x - 6 -5 -4 -3 -2 -1 0 1 2 no solutions and x < -3 x > 1 Absolute Value Inequalities 17

68. |2x| > 2x - 4 -2x 0 > -4 +2x +2x 4x < 4 or 4 4 all real numbers x < 1 2x > 2x - 4or2x<-2x+ 4 Absolute Value Inequalities 18

69. |2x| > 2x - 4 -3 -2 -1 0 1 2 3 4 all real numbers or all real numbers x < 1 Absolute Value Inequalities 18

70. Absolute Value Inequalities | x | ≥ 2 -4 -3 -2 -1 0 1 2 3

71. Absolute Value Inequalities | x | ≥ 2 -4 -3 -2 -1 0 1 2 3 x ≥ 2orx ≤ -2

72. | x | ≥ 1 x ≥ 1 or x ≤ -1 -4 -3 -2 -1 0 1 2 3 Absolute Value Inequalities

73. | x | ≤ 2 -4 -3 -2 -1 0 1 2 3 x ≤ 2 and x ≥ -2

74. | x | ≤ 2 x ≤ 2 and x ≥ -2 -4 -3 -2 -1 0 1 2 3 Absolute Value Inequalities { }x-22≤≤

75. | x | ≤ 3 x ≤ 3 and x ≥ -3 -4 -3 -2 -1 0 1 2 3 Absolute Value Inequalities { }x-33≤≤

76. | x + 4 | > 2x - 5 x + 4 > 2x - 5orx + 4<-2x+ 5 Absolute Value Inequalities Write the compound inequality that is equivalent to the given.

77. | -2x + 4 | ≤ x -2x + 4 ≤ x and -2x + 4 ≥-x Absolute Value Inequalities Write the compound inequality that is equivalent to the given.

78. |x - 3| < 2x -x -3 < x -x -x and -3 > -3x Absolute Value Inequalities E1 x > -3 -3 1 < x x > 1 x - 3 < 2xandx - 3>-2x

79. -3 -2 -1 0 1 2 3 4 Absolute Value Inequalities E1 and x > -3 x > 1 {x > 1}

80. |4x| < 2x - 6 -2x 2x < -6 +2x +2x 6x > 6 and 6 62 2 x < -3 x > 1 4x < 2x - 6and4x>-2x+ 6 Absolute Value Inequalities 19

81. |4x| < 2x - 6 -5 -4 -3 -2 -1 0 1 2 no solutions and x < -3 x > 1 Absolute Value Inequalities 19

82. |2x| > 2x - 4 -2x 0 > -4 +2x +2x 4x < 4 or 4 4 all real numbers x < 1 2x > 2x - 4or2x<-2x+ 4 Absolute Value Inequalities 20

83. |2x| > 2x - 4 -3 -2 -1 0 1 2 3 4 all real numbers or all real numbers x < 1 Absolute Value Inequalities 20