James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

, 0.2, use the Triangle Inequality to esti­
Example 9 If | x 2 4 | , 0.1 and | y 2 7 |
mate | sx 1 yd 2 11 | .
appendix A Numbers, Inequalities, and Absolute Values A9
SOLUTIOn In order to use the given information, we use the Triangle Inequality with
a − x 2 4 and b − y 2 7:
| sx 1 yd 2 11 | − | sx 2 4d 1 sy 2 7d |
< | x 2 4 | 1 | y 2 7 |
, 0.1 1 0.2 − 0.3
Thus | sx 1 yd 2 11 | , 0.3 n
1–12 Rewrite the expression without using the absolute-value
symbol.
1. | 5 2 23 | 2. | 5 | 2 | 223 |
3. | 2 | 4. | 2 2 |
5. | s5 2 5 | 6. || 22 | 2 | 23 ||
7. | x 2 2 | if x , 2 8. | x 2 2 | if x . 2
9. | x 1 1 | 10. | 2x 2 1 |
11. | x 2 1 1 | 12. | 1 2 2x 2 |
13–38 Solve the inequality in terms of intervals and illustrate the
solution set on the real number line.
13. 2x 1 7 . 3 14. 3x 2 11 , 4
15. 1 2 x < 2 16. 4 2 3x > 6
17. 2x 1 1 , 5x 2 8 18. 1 1 5x . 5 2 3x
19. 21 , 2x 2 5 , 7 20. 1 , 3x 1 4 < 16
21. 0 < 1 2 x , 1 22. 25 < 3 2 2x < 9
23. 4x , 2x 1 1 < 3x 1 2 24. 2x 2 3 , x 1 4 , 3x 2 2
25. sx 2 1dsx 2 2d . 0 26. s2x 1 3dsx 2 1d > 0
27. 2x 2 1 x < 1 28. x 2 , 2x 1 8
29. x 2 1 x 1 1 . 0 30. x 2 1 x . 1
31. x 2 , 3 32. x 2 > 5
33. x 3 2 x 2 < 0
34. sx 1 1dsx 2 2dsx 1 3d > 0
35. x 3 . x 36. x 3 1 3x , 4x 2
37.
1
x , 4 38. 23 , 1 x < 1
ature
39. The relationship between the Celsius and Fahrenheit tem per a­
ture scales is given by C − 5 9 sF 2 32d, where C is the temper-
in degrees Celsius and F is the temperature in degrees
Fahrenheit. What interval on the Celsius scale corresponds to
the temperature range 50 < F < 95?
40. Use the relationship between C and F given in Exercise 39 to
find the interval on the Fahrenheit scale corresponding to the
temperature range 20 < C < 30.
41. As dry air moves upward, it expands and in so doing cools at
a rate of about 1°C for each 100-m rise, up to about 12 km.
(a) If the ground temperature is 20°C, write a formula for the
temperature at height h.
(b) What range of temperature can be expected if a plane takes
off and reaches a maximum height of 5 km?
42. If a ball is thrown upward from the top of a building 128 ft
high with an initial velocity of 16 ftys, then the height h above
the ground t seconds later will be
h − 128 1 16t 2 16t 2
During what time interval will the ball be at least 32 ft above
the ground?
43–46 Solve the equation for x.
43. | 2x | − 3 44. | 3x 1 5 | − 1
45. | x 1 3 | − | 2x 1 1 | 46. Z
2x 2 1
x 1 1
Z − 3
47–56 Solve the inequality.
47. | x | , 3 48. | x | > 3
49. | x 2 4 | , 1 50. | x 2 6 | , 0.1
51. | x 1 5 | > 2 52. | x 1 1 | > 3
53. | 2x 2 3 | < 0.4 54. | 5x 2 2 | , 6
55. 1 < | x | < 4 56. 0 , | x 2 5 | , 1 2
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