Published on 24 Apr 2014

Department

Mathematics

Course

MATA32H3

Professor

Table of Contents

Chapter 0 1

Chapter 1 35

Chapter 2 54

Chapter 3 89

Chapter 4 132

Chapter 5 160

Chapter 6 177

Chapter 7 231

Chapter 8 295

Chapter 9 333

Chapter 10 357

Chapter 11 378

Chapter 12 423

Chapter 13 469

Chapter 14 539

Chapter 15 614

Chapter 16 658

Chapter 17 670

1

Chapter 0

Problems 0.1

1. True; –13 is a negative integer.

2. True, because −2 and 7 are integers and 7 ≠ 0.

3. False, because the natural numbers are 1, 2, 3,

and so on.

4. False, because 0

0.

1

=

5. True, because 5

5.

1

=

6. False, since a rational number cannot have

denominator of zero. In fact, 7

0 is not a number

at all because we cannot divide by 0.

7. False, because 25 5,= which is a positive

integer.

8. True; 2 is an irrational real number.

9. False; we cannot divide by 0.

10. False, because the natural numbers are 1, 2, 3,

and so on, and 3 lies between 1 and 2.

11. True

12. False, since the integer 0 is neither positive nor

negative.

Problems 0.2

1. False, because 0 does not have a reciprocal.

2. True, because

73 21 1.

37 21

⋅= =

3. False; the negative of 7 is −7 because

7 + (−7) = 0.

4. False; 2(3 · 4) = 2(12) = 24, but

(2 · 3)(2 · 4) = 6 · 8 = 48.

5. False; –x + y = y + (–x) = y – x.

6. True; (x + 2)(4) = (x)(4) + (2)(4) = 4x + 8.

7. True; 22

1.

2222

xxx+

=

+=+

8. True, because .

bab

acc

⎛⎞

=

⎜⎟

⎝⎠

9. False; the left side is 5xy, but the right side is

2

5.

x

y

10. True; by the associative and commutative

properties, x(4y) = (x ⋅ 4)y = (4 ⋅ x)y = 4xy.

11. distributive

12. commutative

13. associative

14. definition of division

15. commutative and distributive

16. associative

17. definition of subtraction

18. commutative

19. distributive

20. distributive

21. 2x(y − 7) = (2x)y − (2x)7 = 2xy − (7)(2x)

= 2xy − (7 · 2)x = 2xy − 14x

22. (a − b) + c = [a + (−b)] + c = a + (−b + c)

= a + [c + (−b)] = a + (c − b)

23. (x + y)(2) = 2(x + y) = 2x + 2y

24. 2[27 + (x + y)] = 2[27 + (y + x)] = 2[(27 + y) + x]

= 2[(y + 27) + x]

25. x[(2y + 1) + 3] = x[2y + (1 + 3)] = x[2y + 4]

= x(2y) + x(4) = (x · 2)y + 4x = (2x)y + 4x

= 2xy + 4x

26. (1 + a)(b + c) = 1(b + c) + a(b + c)

= 1(b) + 1(c) + a(b) + a(c) = b + c + ab + ac

Chapter 0: Review of Algebra ISM: Introductory Mathematical Analysis

2

27. x(y − z + w) = x[(y − z) + w] = x(y − z) + x(w)

= x[y + (−z)] + xw = x(y) + x(−z) + xw

= xy − xz + xw

28. –2 + (–4) = –6

29. –6 + 2 = –4

30. 6 + (–4) = 2

31. 7 – 2 = 5

32. 7 – (–4) = 7 + 4 = 11

33. −5 − (−13) = −5 + 13 = 8

34. −a − (−b) = −a + b

35. (–2)(9) = –(2 · 9) = –18

36. 7(–9) = –(7 · 9) = –63

37. (–2)(–12) = 2(12) = 24

38. 19(−1) = (−1)19 = −(1 · 19) = −19

39. 1

9

19

19

1

−⎛⎞

=− − =

⎜⎟

−⎝⎠

40. –(–6 + x) = –(–6) – x = 6 – x

41. –7(x) = –(7x) = –7x

42. –12(x – y) = (–12)x – (–12)(y) = –12x + 12y

(or 12y – 12x)

43. –[–6 + (–y)] = –(–6) – (–y) = 6 + y

44. 33131

315 15 15 5 3 5

−⋅

−÷ = =− =− =−

⋅

45. 99911

9(27) 27 27 9 3 3

−⋅

−÷− = = = =

−⋅

46. ()() aa

abbb

−

−÷−= =

−

47. 2(–6 + 2) = 2(–4) = –8

48. 3[–2(3) + 6(2)] = 3[–6 + 12] = 3[6] = 18

49. (–2)(–4)(–1) = 8(–1) = –8

50. (−12)(−12) = (12)(12) = 144

51. X(1) = X

52. 3(x – 4) = 3(x) – 3(4) = 3x – 12

53. 4(5 + x) = 4(5) + 4(x) = 20 + 4x

54. –(x – 2) = –x + 2

55. 0(–x) = 0

56. 1818

811 11 11

⋅

⎛⎞

=

=

⎜⎟

⎝⎠

57. 55

1

=

58. 14 2 7 2

21 3 7 3

x

xx

yyy

⋅⋅

==

⋅⋅

59. 33 3

2(2)2

x

xx

==−

−−

60. 21 21 2

333

x

xx

⋅

⋅= =

⋅

61. (3 ) 3

(3 )

aabab

b

ccc

==

62. 7

(5 ) 7

5

aa

⎛⎞

=

⎜⎟

⎝⎠

63. aby a by by

ax a x x

−−⋅

==

−−⋅

64. 71 71 7

yx yx xy

⋅

⋅= =

⋅

65. 25 25 10

x

yxyxy

⋅

⋅= =

⋅

66. 1132325

2366 6 6

+

+

=+= =

67. 535 95914277

12 4 12 12 12 12 2 6 6

+⋅

+

=+= == =

⋅

68. 37 914914 5 51 1

10 15 30 30 30 30 5 6 6

−− ⋅

−

=−= ==− =−

⋅