A84 APPENDIX Review Skill Building In Problems 13–18, express the graph shown in blue using interval notation. Also express each as an inequality involving x. 13. 3 2 0 1 -1 14. 2 1 -1 0 -2 15. 3 2 0 1 -1 16. 2 1 -1 0 -2 17. 3 2 0 1 -1 18. 3 2 0 1 -1 In Problems 19–24, an inequality is given. Write the inequality obtained by: (a) Adding 3 to both sides of the given inequality. (b) Subtracting 5 from both sides of the given inequality. (c) Multiplying both sides of the given inequality by 3. (d) Multiplying both sides of the given inequality by −2. 19. 3 5 < 20. 2 1 > 21. 4 3 >− 22. 3 5 − >− 23. x2 1 2 + < 24. x 1 2 5 − > In Problems 25–32, write each inequality using interval notation, and graph each inequality on the real number line. 25. x 0 4 ≤ ≤ 26. x 1 5 − < < 27. x 4 6 ≤ < 28. x 2 0 − < < 29. x 3 ≥− 30. x 5 ≤ 31. x 4 <− 32. x 1 > Concepts and Vocabulary 3. A(n) , denoted a b , [ ] , consists of all real numbers x for which a x b. ≤ ≤ 4. The state that the sense, or direction, of an inequality remains the same if both sides are multiplied by a positive number, while the direction is reversed if both sides are multiplied by a negative number. In Problems 5–8, assume that a b < and c 0. < 5. True or False a c b c + < + 6. True or False a c b c − < − 7. True or False ac bc > 8. True or False a c b c < 9. True or False The square of any real number is always nonnegative. 10. True or False A half-closed interval must have an endpoint of either −∞ or .∞ 11. Multiple Choice Which of the following will not change the direction, or sense, of an inequality? (a) Dividing both sides by a negative number (b) Interchanging sides (c) Taking the reciprocal of both sides (d) Subtracting a positive number from both sides 12. Multiple Choice Which pair of inequalities is equivalent to x 0 3? < ≤ (a) x 0 > and x 3 ≥ (b) x 0 < and x 3 ≥ (c) x 0 > and x 3 ≤ (d) x 0 < and x 3 ≤ In Problems 41–58, fill in the blank to form a correct inequality statement. 41. If x 5, < then x 5 − 0. 42. If x 4, <− then x 4 + 0. 43. If x 4, >− then x 4 + 0. 44. If x 6, > then x 6 − 0. 45. If x 4, ≥− then 3 x 12. − 46. If x 3, ≤ then 2 x 6. 47. If x 6, > then x2− 12. − 48. If x 2, >− then x4− 8. 49. If x 5, ≥ then x4− 20. − 50. If x 4, ≤− then x3− 12. 51. If x8 40, > then x 5. 52. If x3 12, ≤ then x 4. 53. If x 1 2 3, − ≤ then x 6. − 54. If x 1 4 1, − > then x 4. − In Problems 33–40, write each interval as an inequality involving x, and graph each inequality on the real number line. 33. 2, 5 [ ] 34. 1, 2 ( ) 35. 4, 3 ( ] − 36. 0, 1 [ ) 37. 4, [ )∞ 38. , 2 ( ] −∞ 39. , 3 ( ) −∞ − 40. 8, ( ) − ∞ A.9 Assess Your Understanding ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 1. Graph the inequality: x 2 ≥− (pp. A4 – A5 ) 2. True or False 5 3 − >− (pp. A4 – A5 ) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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