# MATA 32&33 Solution Manual.pdf

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Published on 24 Apr 2014
School
UTSC
Department
Mathematics
Course
MATA32H3
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
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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) = yx.
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
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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(xy) = (–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
−− ⋅
=−= ==− =
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