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ISM_T11_C01_A

ISM_T11_C01_A - CHAPTER 1 PRELIMINARIES 1.1 REAL NUMBERS...

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CHAPTER 1 PRELIMINARIES 1.1 REAL NUMBERS AND THE REAL LINE 1. Executing long division, 0.1, 0.2, 0.3, 0.8, 0.9 " 9 9 9 9 9 2 3 8 9 œ œ œ œ œ 2. Executing long division, 0.09, 0.18, 0.27, 0.81, 0.99 " 11 11 11 11 11 2 3 9 11 œ œ œ œ œ 3. NT = necessarily true, NNT = Not necessarily true. Given: 2 < x < 6. a) NNT. 5 is a counter example. b) NT. 2 < x < 6 2 2 < x 2 < 6 2 0 < x 2 < 2. Ê Ê c) NT. 2 < x < 6 2/2 < x/2 < 6/2 1 < x < 3. Ê Ê d) NT. 2 < x < 6 1/2 > 1/x > 1/6 1/6 < 1/x < 1/2. Ê Ê e) NT. 2 < x < 6 1/2 > 1/x > 1/6 1/6 < 1/x < 1/2 6(1/6) < 6(1/x) < 6(1/2) 1 < 6/x < 3. Ê Ê Ê Ê f) NT. 2 < x < 6 x < 6 (x 4) < 2 and 2 < x < 6 x > 2 x < 2 x + 4 < 2 (x 4) < 2. Ê Ê Ê Ê Ê Ê The pair of inequalities (x 4) < 2 and (x 4) < 2 | x 4 | < 2. Ê g) NT. 2 < x < 6 2 > x > 6 6 < x < 2. But 2 < 2. So 6 < x < 2 < 2 or 6 < x < 2. Ê Ê h) NT. 2 < x < 6 1(2) > 1(x) < 1(6) 6 < x < 2 Ê Ê 4. NT = necessarily true, NNT = Not necessarily true. Given: 1 < y 5 < 1. a) NT. 1 < y 5 < 1 1 + 5 < y 5 + 5 < 1 + 5 4 < y < 6. Ê Ê b) NNT. y = 5 is a counter example. (Actually, never true given that 4 y 6) c) NT. From a), 1 < y 5 < 1, 4 < y < 6 y > 4. Ê Ê d) NT. From a), 1 < y 5 < 1, 4 < y < 6 y < 6. Ê Ê e) NT. 1 < y 5 < 1 1 + 1 < y 5 + 1 < 1 + 1 0 < y 4 < 2. Ê Ê f) NT. 1 < y 5 < 1 (1/2)( 1 + 5) < (1/2)(y 5 + 5) < (1/2)(1 + 5) 2 < y/2 < 3. Ê Ê g) NT. From a), 4 < y < 6 1/4 > 1/y > 1/6 1/6 < 1/y < 1/4. Ê Ê h) NT. 1 < y 5 < 1 y 5 > 1 y > 4 y < 4 y + 5 < 1 (y 5) < 1. Ê Ê Ê Ê Ê Also, 1 < y 5 < 1 y 5 < 1. The pair of inequalities (y 5) < 1 and (y 5) < 1 | y 5 | < 1. Ê Ê 5. 2x 4 x 2 Ê 6. 8 3x 5 3x 3 x 1 x 1   Ê   Ê Ÿ ïïïïïïïïïñqqqqqqqqp 7. 5x 3x 8x 10 x $ Ÿ ( Ê Ÿ Ê Ÿ 5 4 8. 3(2 x) 2(3 x) 6 3x 6 2x Ê 0 5x 0 x x 0 Ê Ê ïïïïïïïïïðqqqqqqqqp 9. 2x 7x 5x   Ê   " " # # 7 7 6 6 x or x Ê     " " 5 6 3 10 ˆ 10. 12 2x 12x 16 6 x 3x 4 4 2 Ê 28 14x 2 x x 2 Ê Ê qqqqqqqqqðïïïïïïïïî
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2 Chapter 1 Preliminaries 11. (x 2) (x 6) 12(x 2) 5(x 6) 4 5 3 Ê " 12x 24 5x 30 7x 6 or x Ê Ê 6 7 12. (4x 20) 24 6x Ÿ Ê Ÿ x 5 12 3x 2 4 44 10x x x 22/5 Ê Ÿ Ê Ÿ qqqqqqqqqñïïïïïïïïî 22 5 13. y 3 or y 3 œ œ 14. y 3 7 or y 3 7 y 10 or y 4 œ œ Ê œ œ 15. 2t 5 4 or 2t 4 2t 1 or 2t 9 t or t œ & œ Ê œ œ Ê œ œ " # # 9 16. 1 t 1 or 1 t 1 t or t 2 t 0 or t 2 œ œ Ê œ ! œ Ê œ œ 17. 8 3s or 8 3s 3s or 3s s or s œ œ Ê œ œ Ê œ œ 9 9 7 25 7 25 2 6 6 # # # 18. 1 1 or 1 1 2 or s 4 or s 0 s s s s # # # # œ œ Ê œ œ ! Ê œ œ 19. 2 x 2; solution interval ( 2 2) ß 20. 2 x 2; solution interval [ 2 2] x 2 2 Ÿ Ÿ ß qqqqñïïïïïïïïñqqqqp 21. 3 t 1 3 2 t 4; solution interval [ 2 4] Ÿ Ÿ Ê Ÿ Ÿ ß 22. 1 t 2 1 3 t 1; Ê solution interval ( 3 1) t 3 1 ß qqqqðïïïïïïïïðqqqqp 23. 3y 7 4 3 3y 11 1 y ; % Ê Ê 11 3 solution interval 1 ˆ ß 11 3 24. 1 2y 5 6 2y 4 3 y 2; " Ê Ê solution interval ( 3 2) y 3 2 ß qqqqðïïïïïïïïðqqqqp 25. 1 1 1 0 2 0 z 10; Ÿ Ÿ Ê Ÿ Ÿ Ê Ÿ Ÿ z z 5 5 solution interval [0 10] ß 26. 2 1 2 1 3 z 2; Ÿ Ÿ Ê Ÿ Ÿ Ê Ÿ Ÿ 3z 3z 2 3 # # solution interval 2 z 2/3 2 ß qqqqñïïïïïïïïñqqqqp 2 3 27. 3 Ê Ê " " " " " # # # # # # x x x 7 5 7 5 x ; solution interval Ê ß 2 2 2 2 7 5 7 5 ˆ 28. 3 4 3 1 1 Ê ( Ê 2 2 x x x 7 # " 2 x x 2; solution interval 2 x 2/7 2 Ê Ê ß qqqqðïïïïïïïïðqqqqp 2 2 2 7 7 7 ˆ
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Section 1.1 Real Numbers and the Real Line 3 29. 2s 4 or 2s 4 s 2 or s 2;     Ê   Ÿ solution intervals ( 2] [2 ) ß _ 30. s 3 or (s 3) s or s     Ê     " " # # # # 5 7 s or s ; Ê   Ÿ 5 7 # # solution intervals s 7/2 5/2
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