0.8e: Exercises- Linear Inequalities


A: Check a solution

Exercise $$\PageIndex{1}$$

$$\bigstar$$  Determine whether or not the given value is a solution.

 $$5 x - 1 < - 2 ; x = - 1$$ $$- 3 x + 1 > - 10 ; x = 1$$ $$2 x - 3 < - 5 ; x = 1$$ $$5 x - 7 < 0 ; x = 2$$ $$9 y - 4 \geq 5 ; y = 1$$ $$- 6 y + 1 \leq 3 ; y = - 1$$ $$12 a + 3 \leq - 2 ; a = - \dfrac { 1 } { 3 }$$ $$25 a - 2 \leq - 22 ; a = - \dfrac { 4 } { 5 }$$ $$- 10 < 2 x - 5 < - 5 ; x = - \dfrac { 1 } { 2 }$$ $$3 x + 8 < - 2 \text { or } 4 x - 2 > 5 ; x = 2$$
Answers to odd exercises:
 1. Yes 3. No 5. Yes 7. No 9. Yes

B: Solve Linear Inequalities

Exercise $$\PageIndex{2}$$

$$\bigstar$$  Graph all solutions on a number line and provide the corresponding interval notation.

 $$3 x + 5 > - 4$$ $$2 x + 1 > - 1$$ $$5 - 6 y < - 1$$ $$7 - 9 y > 43$$ $$6 - a \leq 6$$ $$- 2 a + 5 > 5$$ $$\dfrac { 5 x + 6 } { 3 } \leq 7$$ $$\dfrac { 4 x + 11 } { 6 } \leq \dfrac { 1 } { 2 }$$ $$\dfrac { 1 } { 2 } y + \dfrac { 5 } { 4 } \geq \dfrac { 1 } { 4 }$$ $$\dfrac { 1 } { 12 } y + \dfrac { 2 } { 3 } \leq \dfrac { 5 } { 6 }$$ $$2 ( 3 x + 14 ) < - 2$$ $$5 ( 2 y + 9 ) > - 15$$ $$5 - 2 ( 4 + 3 y ) \leq 45$$ $$- 12 + 5 ( 5 - 2 x ) < 83$$ $$6 ( 7 - 2 a ) + 6 a \leq 12$$ $$2 a + 10 ( 4 - a ) \geq 8$$ $$9 ( 2 t - 3 ) - 3 ( 3 t + 2 ) < 30$$ $$- 3 ( t - 3 ) - ( 4 - t ) > 1$$ $$\dfrac { 1 } { 2 } ( 5 x + 4 ) + \dfrac { 5 } { 6 } x > - \dfrac { 4 } { 3 }$$ $$\dfrac { 2 } { 5 } + \dfrac { 1 } { 6 } ( 2 x - 3 ) \geq \dfrac { 1 } { 15 }$$ $$5 x - 2 ( x - 3 ) < 3 ( 2 x - 1 )$$ $$3 ( 2 x - 1 ) - 10 > 4 ( 3 x - 2 ) - 5 x$$ $$- 3 y \geq 3 ( y + 8 ) + 6 ( y - 1 )$$ $$12 \leq 4 ( y - 1 ) + 2 ( 2 y + 1 )$$ $$- 2 ( 5 t - 3 ) - 4 > 5 ( - 2 t + 3 )$$ $$- 7 ( 3 t - 4 ) > 2 ( 3 - 10 t ) - t$$ $$\dfrac { 1 } { 2 } ( x + 5 ) - \dfrac { 1 } { 3 } ( 2 x + 3 ) > \dfrac { 7 } { 6 } x + \dfrac { 3 } { 2 }$$ $$- \dfrac { 1 } { 3 } ( 2 x - 3 ) + \dfrac { 1 } { 4 } ( x - 6 ) \geq \dfrac { 1 } { 12 } x - \dfrac { 5 } { 4 }$$ $$4 ( 3 x + 4 ) \geq 3 ( 6 x + 5 ) - 6 x$$ $$1 - 4 ( 3 x + 7 ) < - 3 ( x + 9 ) - 9 x$$ $$6 - 3 ( 2 a - 1 ) \leq 4 ( 3 - a ) + 1$$ $$12 - 5 ( 2 a + 6 ) \geq 2 ( 5 - 4 a ) - a$$
Answers to odd exercises:

11. $$( - 3 , \infty )$$;

Figure 1.8.11

13. $$( 1 , \infty )$$;

15. $$[ 0 , \infty )$$;

17. $$( - \infty , 3 ]$$;

19. $$[ - 2 , \infty )$$;

21. $$( - \infty , - 5 )$$;

23. $$[ - 8 , \infty )$$;

25. $$[ 5 , \infty )$$;

27. $$( - \infty , 7 )$$;

29. $$( - 1 , \infty )$$;

31. $$( 3 , \infty )$$;

33. $$\left( - \infty , - \dfrac { 3 } { 2 } \right]$$;

35. $$\emptyset$$;

37. $$( - \infty , 0 )$$;

39. $$\mathbb { R }$$;

41. $$[ - 2 , \infty )$$;

C: Solve Compound Linear Inequalities

Exercise $$\PageIndex{3}$$

$$\bigstar$$  Graph all solutions on a number line and provide the corresponding interval notation.

 $$- 1 < 2 x + 1 < 9$$ $$- 4 < 5 x + 11 < 16$$ $$- 7 \leq 6 y - 7 \leq 17$$ $$- 7 \leq 3 y + 5 \leq 2$$ $$- 7 < \dfrac { 3 x + 1 } { 2 } \leq 8$$ $$- 1 \leq \dfrac { 2 x + 7 } { 3 } < 1$$ $$- 4 \leq 11 - 5 t < 31$$ $$15 < 12 - t \leq 16$$ $$- \dfrac { 1 } { 3 } \leq \dfrac { 1 } { 6 } a + \dfrac { 1 } { 3 } \leq \dfrac { 1 } { 2 }$$ $$- \dfrac { 1 } { 6 } < \dfrac { 1 } { 3 } a + \dfrac { 5 } { 6 } < \dfrac { 3 } { 2 }$$ $$5 x + 2 < - 3 \text { or } 7 x - 6 > 15$$ $$4 x + 15 \leq - 1 \text { or } 3 x - 8 \geq - 11$$ $$8 x - 3 \leq 1 \text { or } 6 x - 7 \geq 8$$ $$6 x + 1 < - 3 \text { or } 9 x - 20 > - 5$$ $$8 x - 7 < 1 \text { or } 4 x + 11 > 3$$ $$10 x - 21 < 9 \text { or } 7 x + 9 \geq 30$$ $$7 + 2 y < 5 \text { or } 20 - 3 y > 5$$ $$5 - y < 5 \text { or } 7 - 8 y \leq 23$$ $$15 + 2 x < - 15 \text { or } 10 - 3 x > 40$$ $$10 - \dfrac { 1 } { 3 } x \leq 5 \text { or } 5 - \dfrac { 1 } { 2 } x \leq 15$$ $$9 - 2 x \leq 15 \text { and } 5 x - 3 \leq 7$$ $$5 - 4 x > 1 \text { and } 15 + 2 x \geq 5$$ $$7 y - 18 < 17 \text { and } 2 y - 15 < 25$$ $$13 y + 20 \geq 7 \text { and } 8 + 15 y > 8$$ $$5 - 4 x \leq 9 \text { and } 3 x + 13 \leq 1$$ $$17 - 5 x \geq 7 \text { and } 4 x - 7 > 1$$ $$9 y + 20 \leq 2 \text { and } 7 y + 15 \geq 1$$ $$21 - 6 y \leq 3 \text { and } - 7 + 2 y \leq - 1$$ $$- 21 < 6 ( x - 3 ) < - 9$$ $$0 \leq 2 ( 2 x + 5 ) < 8$$ $$- 15 \leq 5 + 4 ( 2 y - 3 ) < 17$$ $$5 < 8 - 3 ( 3 - 2 y ) \leq 29$$ $$5 < 5 - 3 ( 4 + t ) < 17$$ $$- 3 \leq 3 - 2 ( 5 + 2 t ) \leq 21$$ $$- 40 < 2 ( x + 5 ) - ( 5 - x ) \leq - 10$$ $$- 60 \leq 5 ( x - 4 ) - 2 ( x + 5 ) \leq 15$$ $$- \dfrac { 1 } { 2 } < \dfrac { 1 } { 30 } ( x - 10 ) < \dfrac { 1 } { 3 }$$ $$- \dfrac { 1 } { 5 } \leq \dfrac { 1 } { 15 } ( x - 7 ) \leq \dfrac { 1 } { 3 }$$ $$- 1 \leq \dfrac { a + 2 ( a - 2 ) } { 5 } \leq 0$$ $$0 < \dfrac { 5 + 2 ( a - 1 ) } { 6 } < 2$$
Answers to odd exercises:

43. $$(- 1,4 )$$;

45. $$[0,4]$$;

47. $$(−5,5]$$;

49. $$(−4,3]$$;

51. $$[−4,1]$$;

53. $$(−∞,−1)∪(3,∞)$$;

55. $$(−∞,\frac{1}{2}]∪[\frac{5}{2},∞)$$;

57. $$ℝ$$;

59. $$(−∞,5)$$;

61. $$(−∞,−10)$$;

63. $$[−3,2]$$;

65. $$(−∞,5)$$;

67. $$Ø$$;

69. $$−2$$;

71. $$(−\frac{1}{2},\frac{3}{2})$$;

73. $$[−1,3)$$;

75. $$(−8,−4)$$;

77. $$(−15,−5]$$;

79. $$(−5,20)$$;

81. $$[−\frac{1}{3}, \frac{4}{3}]$$;

.

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