Prealgebra - 2e, 2020a

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1, 2, 3, 4, 5, …

1
1, 2, 3, 4, 5…
0 0
0, 1, 2, 3, 4, 5…
5
a b
0, 1 , 3, 5.2, 15, 105
4
a 1, 0 3, 15, and 105
b 0. 0, 3, 15, and 105
1 4
5.2
a b
0, 2 , 2, 9, 11.8, 241, 376
3

a b
0, 5 , 7, 8.8, 13, 201
3
537 735.
3 7 5.
$100 $10 $1
$374.
10 10
100
138 base-10

138.
1 1 100 100
3 3 10 30
8 8 1 +8
Sum =138
base-10
2 200.
1 10.
5 5.
2 2 100 200
1 1 10 10
5 5 1 +5
215

ase-10 215.
base-10
base-10
base-10
base-10
5,278,194
5 5,000,000.

2 200,000.
7 70,000.
8 8,000.
1 100.
9 90.
4 4.
63,407,218;
a 7 b 0 c 1 d 6 e 3
a 7
b 0
c 1
d 6
e 3
27,493,615
a 2 b 1 c 4 d 7 e 5
519,711,641,328
a 9 b 4 c 2 d 6 e 7

37,519,248
8,165,432,098,710
8,165,432,098,710
9,258,137,904,061
17,864,325,619,004
2014 327,577,529.

316,128,839.
31,536,000
a
b
a
53,401,742.
b

9,246,073,189.
b
$77
$77,000,000,000.
34
204
2013, 19,651,127
20 20
20
19,700,000
19,650,000

76 76 70 80
76 80 70. 76 80.
72. 72
72 70, 72 70.
75 75
75 70 80.
80. 75 80.
5,
5,
76
6. 6 5,
7 8. 8 76
80.
72 10.
2. 2 5,
72 70.

5.
1
843
157.
884.
a 23,658 b 3,978

a
b
17,852.
4,951.
a 147,032 b 29,504

a
b
b 1 9 10 1
0 1 3 0
63,921.
156,437.

a b
0, 2 3 , 5, 8.1, 125 0, 7 , 3, 20.5, 300 0, 4 , 3.9, 50, 221
10 9
0, 3 , 10, 303, 422.6
5
base-10

579,601
398,127
56,804,379
a b c d
e
a b c d
e
a b c d
e
78,320,465
a b c d
e
1,078 5,902 364,510
146,023 5,846,103 1,458,398
37,889,005 62,008,465
14,410
12,276
613,200
525,600
2,617,176.
2,718,782.
23,867,000
20,665,415
1,377,583,156
2016.
1,267,401,849
1, 2014.

a 386 b 2,931
a 792 b 5,647
a 13,748 b 391,794
a 28,166 b 481,628
a 63,994 b 63,949
a 1,492 b 1,497
a 163,584 b 163,246
a 2,391 b 2,795
$24,493.
$24,493.
a b
c d
1,355,692,544 2014.
a b
c
$18,549.
$18,549.
a b
c d
149,597,888
a
b c
a
b

ase-10
3 4
3 4.
3 4
3+4
3 4
+ 3+4 3 4
a 7+1 b 12 + 14
a

a 8 +4 b 18 + 11
a 21 + 16 b 100 + 200
base-10 1
10. 3+4.
10,
7 (=)
3+4=7.
2+6.
2+6 2 6

8 2+6=8.
3 + 6.
5 + 1.
10 10
5+8.
5+8 5 8.
5 + 7.
6 + 8.
17 + 26.
17 + 26

40 + 3 = 43
17 + 26 = 43
15 + 27.
16 + 29.
a 0

a +0=a
0+a = a
a 0+11 b 42 + 0
a 0 + 11 = 11
b 42 + 0 = 42
a 0 + 19 b 39 + 0
a 0 +24 b 57 + 0
2+3=5 3+2=5
4+7=11 7+4=11
8+9=17 9+8=17
a b
a + b = b + a
a 8+7 b 7+8
a
8+7
15

7+8
15
b b
9 + 7 7 + 9.
8 + 6 6+8.
28 + 61.
28
____ +61
8+1=9
2+6=8
28
____ +61
89
32 + 54.
25 + 74.
10.
10 base-10 17 26
7+6, 13 10 10
1 4 3 1
9,
10 1 10 1

9,
43 + 69.
43
____ +69
3+9=12
2
1
1+4+6=11
4 1 3
____ +69
2
4 1 3
____ +69
112
35 + 98.
72 + 89.
324 + 586.

4+6=10
0 1
1+2+8=11
1 1
1+3+5=9
9
456 +376.
269 + 578.
1,683 + 479.
1,683
______ + 479
3 + 9 = 12.
2 1
1+7+8=16
6 1
1+6+4=11
1 1
1+1=2
2
1,68 1 3
______ + 479
2
1,6 1 8 1 3
______ + 479
62
1,6 1 8 1 3
______ + 479
162
1,6 1 8 1 3
______ + 479
2,162

4,597 + 685.
5,837 + 695.
21,357 + 861 + 8,596.
21,357
861
_______ + 8,596
7+1+6=14
4 1
1+5+6+9=21
1 2
2+3+8+5=18
8 1
1+1+8=10
0 1
1+2=3
3
21,35 1 7
861
_______ + 8,596
4
21,3 2 5 1 7
861
_______ + 8,596
14
1 21 21, 357
861
_______ + 8,596
814
2 1 1 21 1, 357
861
_______ + 8,596
0814
2 1 1 21 1, 357
861
_______ + 8,596
30,814
46,195 + 397 + 6,281.

53,762 + 196 + 7,458.
1 2
3 4
5 6
8 7
9 5
6 4
1+2
3+4
5+6
7+8
9+5
4+6
19 23.
19 23
19 23
19 + 23
42
19 23 42.
17 26.
28 14.
28 31.
28 31.
28 + 31
59
28 31 59.

29 76.
37 69.
87, 93, 68, 95, and 89
87 + 93 + 68 + 95 + 89
8 3 7
93
68
95
____ +89
432
432
18 15 26
49 32
230, 165, and 325.

4+6+2+3+2+9
26
26
26

5+2 6+3 13 + 18
15 + 16 214 + 642 438 + 113
2+4 5+3 8+4
5+9 14 + 75 15 + 63
16 + 25 14 + 27

a 0+13 b 13 + 0
a 0 + 5,280
b 5,280 + 0
a 8+3 b 3+8
a 7+5 b 5+7
45 + 33 37 + 22
71 + 28 43 + 53 26 + 59
38 + 17 64 + 78 92 + 39
168 + 325 247 + 149 584 + 277
175 + 648 832 + 199 775 + 369
6,358 + 492 9,184 + 578 3,740 + 18,593
6,118 + 15,990 485,012 + 619,848 368,911 + 857,289
24,731 + 592 + 3,868 28,925 + 817 + 4,593 8,015 + 76,946 + 16,570
6,291 + 54,107 + 28,635
13 18 12 19 90 65
70 38 33 49 68 25
250 599 115 286 628 77
593 79 1,482 915 2,719 682
$1,100,
$250,
$525.
$299,
$35,
$68.
14 19
12
25
68

19
12 23
29
44
$82,572.
$79,316,
$75,298.
7
238, 120, 156, 196, 100, 132,
225.
7
$292,540, $505,875,
$423,699.
175, 192, 148, 169, 205, 181,
225
16-ounce
320
170
16-ounce 150
12-oz
420
230
12-oz 580

400
82, 91, 75, 88, and 70.
1150
210, 145, 183, 230, 159, and 164
a
b

3 + 4
324 + 586.
3 7
7−3
7−3
− 7−3 7 3
a 8−1 b 26 − 14
a
b
a 12 − 4 b 29 − 11
a 11 − 2 b 29 − 12

ase-10
7−3.
7−3=4
8−2.
8−2
8−2=6
9 − 6.
6 − 1.
13 − 8.

13 − 8 = 5
12 − 7.
14 − 8.
43 − 26.
43 − 26 43 26, 43.
26,
1 10
2 6 6 3
2 6
1 7 17.
43 − 26 = 17

42 − 27.
45 − 29.
7−3=4 4+3=7.
7 − 3 = 4 because 4 + 3 = 7
13 − 8 = 5 because 5 + 8 = 13
43 − 26 = 17 because 17 + 26 = 43
a 9−7 b 8−3.
a
2
2+7=9✓
9−7
b
5
5+3=8✓
8−3
7 − 0
6 − 2
89 − 61.

9-1=8
8-6=2
89
____ −61
89
____ −61
28
28
____ +61
89
86 − 54.
99 − 74.
26 43, 1 10
1 10
43 − 26.

13 − 6 = 7.
3−2=1.
93 − 58.
81 − 39.
207 − 64.

7−4=3.
10 − 6 = 4.
1−0=1,
439 − 52.
318 − 75.
910 − 586.

10 − 6 = 4.
10 − 8 = 2
8−5=3
832 − 376.
847 − 578.
2,162 − 479.

12 − 9 = 3
15 − 7 = 8
1−0=1.
1, 683
______ + 479
2, 162 ✓
4,585 − 697.
5,637 − 899.

5 1 5−1
9 4 9−4
7 3 7−3
5 8 8−5
1 6 6−1
a
a 13 8 b 24 43
13 − 8
b
43 − 24
a 14 9 b 21 37
a 11 6 b 18 67

73
27
73 − 27
June 1 st 77 58
2 90
73
$399. $588.
588 − 399
$499. $648.

$149. $285.

15 − 9 18 − 16 42 − 35
83 − 64 675 − 350 790 − 525
5−2 8−4 6−3
7−5 18 − 5 19 − 8
17 − 8 17 − 9 35 − 13
32 − 11 61 − 47 55 − 36
9−4 9−3 8−0
2−0 38 − 16 45 − 21
85 − 52 99 − 47 493 − 370
268 − 106 5,946 − 4,625 7,775 − 3,251
75 − 47 63 − 59 461 − 239
486 − 257 525 − 179 542 − 288
6,318 − 2,799 8,153 − 3,978 2,150 − 964
4,245 − 899 43,650 − 8,982 35,162 − 7,885
10 3 12 8 15 4
18 7 6 9 8 9
28 75 59 81 45 20
37 24 92 67 75 49
12 16 15 19 38 61

47 62
76−47 91 − 53 256 − 184
305 − 262 719 + 341 647 + 528
2,015 − 1,993 2,020 − 1,984
13 41
28 36 100
76
1,000
945
2
80
63
1
97
73
35
22
82
46
$840.
$685
$399.
$650.
$1,125
$892.
$755.
$1,600.
502 115
230
350
75, 50, 70, and 80

a
b

1,683 + 479.
605 − 321.
3
8+8+8
8
3×8
3×8
× ·
( ).

Multiplication ×
·
()
3×8
3·8
3(8)
three times eight the product of 3 and 8
a 7×6 b 12 · 14 c 6(13)
a
b
c
a 8 × 7 b 18 · 11
a (13)(7) b 5(16)
base-10
3×8.
3×8, 8
3, 3 8
24
3×8=24
8 3
4×6.

5 × 7.
0 0.
a 0·11 b (42)0
a ·0=0
0·a =0
a 0·11
0
b (42)0
0

a 0 · 19 b (39)0
a 0 · 24 b (57)0
1
1
1·a = a
a ·1=a
a (11)1 b 1·42
a (11)1
11
b 1·42
42
a (19)1 b 1 · 39
a (24)(1) b 1 × 57
8+9=17 9 + 8 = 17.
4·7=28 7·4=28
9·7=63 7·9=63
8·9=72 9·8=72
a · b = b · a

a 8·7 b 7·8
a 8·7
56
b 7·8
56
a 9 · 6 b 6 · 9
a 8 · 6 b 6 · 8
27
___ ×3
3 7.
3×7=21
1 2 2
3 2, 2 3×2=6, 6+2=8.
8
81.

15 · 4.
5 4
15
_____ ×4
4 15. 4⋅5=20.
0 2
1 2 5
_____ ×4
0
4 15. 4⋅1=4
2 4+2=6
6
1 2 5
_____ ×4
60
64 · 8.
57 · 6.
286 · 5.
5 6
286
_____ ×5
5 286. 5⋅6=30.
0 3 5
286. 5⋅8=40
3 40 + 3 = 43
3
5 286. 5 ⋅ 2 = 10.
4 10 + 4 = 14.
4 1
28 3 6
_____ ×5
0
2 4 8 3 6
_____ ×5
30
2 4 8 3 6
_____ ×5
1,430
347 · 5.

462 · 7.
9,
62(87).

7⋅2=14.
7 ⋅ 6 = 42. 42 + 1 = 43
8 ⋅ 2 = 16.
5,394.
43(78).
64(59).
a 47 · 10 b 47 · 100.

a 47 · 10
b 47 · 100
47
×10 ___
00
470 ___
470
47
_____ ×100
00
000
_____ 4700
4,700
47 10, 470. 10 47
47 100, 4,700. 100
47
47 10,000, 47
470,000.
a 54 · 10 b 54 · 100.
a 75 · 10 b 75 · 100.
(354)(438).
3 0
(265)(483).
(823)(794).
(896)201.

3 896 0.
2
10, 200, 2
12. 2·6=12
896
_____ ×201
896
__________ 17920
180,096
(718)509.
(627)804.
8⋅3⋅2
8⋅3 24 ⋅ 2
24 ⋅ 2 48
3 8
3 8
4
3 × 8, 3 · 8, (3)(8),
(3)8, or 3(8)
2·4
12 27.
12 27

12 ⋅ 27
324
13 28.
47 14.
2.
4 20
4⋅20

6
24
8 10
4
2⋅4
8 14
8⋅14
14
___ ×8
112

2 3 1 1 1
6 6
6
9⋅12
12
___ ×9
108

4×7 8×6 5·12
3·9 (10)(25) (20)(15)
42(33) 39(64)
3×6 4×5 5×9
3×9

0·15 0·41 (99)0
(77)0 1·43 1·34
(28)1 (65)1 1(240,055)
1(189,206)
a 7·6 b 6·7
a 8×9 b 9×8
(79)(5) (58)(4) 275·6
638 · 5 3,421 × 7 9,143 × 3
52(38) 37(45) 96·73
89 · 56 27 × 85 53 × 98
23 · 10 19 · 10 (100)(36)
(100)(25) 1,000(88) 1,000(46)
50 × 1,000,000 30 × 1,000,000 247 × 139
156 × 328 586(721) 472(855)
915 · 879 968 · 926 (104)(256)
(103)(497) 348(705) 485(602)
2,719 × 543 3,581 × 724
18 33 15 22
249 589

38 × 37 86 × 29 415 − 267
341 − 285 6,251 + 4,749 3,816 + 8,184
(56)(204) (77)(801) 947 · 0
947 + 0 15,382 + 1 15,382 · 1

$12
$200
a
b

27 · 3.
43 − 26.
62(87).
12
4
4 4
3
12 4
4
12 ÷ 4
12 4. 3
4 12 3
12 4.

412 12 ÷ 4, 12/4, 12 .
4
12 4
Division ÷
a
b
ba
a / b
12 ÷ 4
12
4
412
12 / 4
Twelve divided by four the quotient of 12 and 4
a 64 ÷ 8 b 42 7 c 428
a
b
c
a 84 ÷ 7 b 18 6 c 8 24
a 72 ÷ 9 b 21 3 c 6 54
24 ÷ 8.

24 ÷ 8, 8 24.
24
8
3
24÷8=3
24 ÷ 6.
42 ÷ 7.
12÷4=3 3 · 4 = 12.
24÷8=3 3 · 8 = 24.
a 42 ÷ 6 b 72 9 c 763
a
7
7·6
42 ✓
42 ÷ 6

72
9
8
8·9
72 ✓
c
9
9·7
63 ✓
763
a 54 ÷ 6 b 27 9
a 36 9 b 8 40
15 = 1 because 1 · 15 = 15
15
(except 0) 1. 1
a ÷ a =1
a ÷ 1 =a

a 11 ÷ 11 b 19 1
c 17
a
1
1·11
11 ✓
11 ÷ 11
b
19
1
19
19·1
19 ✓
c
7
7·1
7✓
17
a 14 ÷ 14 b 27 1
a 16 1 b 1 4
$0, 3
$0. 0.
$10 0. 0
10. 0 0.

0 ÷ a =0
a ÷ 0
0 10? 0
0.
a 0÷3b 10 / 0.
a
0
0·3
0✓
0÷3
b
10 / 0
a 0 ÷ 2 b 17 / 0
a 0 ÷ 6 b 13 / 0
412
78 3.

2×3 = 6
7−6
So 78 ÷ 3 = 26.
26 × 3
78.
2 1 6
___ ×3
78 ✓

2,596 ÷ 4.
25 − 24
2,596 ÷ 4 = 649
2,636 ÷ 4
2,716 ÷ 4
4,506 ÷ 6.

4,305 ÷ 5.
3,906 ÷ 6.
7,263 ÷ 9.

4,928 ÷ 7.
5,663 ÷ 7.
24
8 3 28 8?
28

3 4 4
3.
3 8 24, 4.
3
___ ×8
24
___ +4
28
1,439 ÷ 4.
1,439 ÷ 4 359 3.

3,812 ÷ 8.
4,319 ÷ 8.
1,461 ÷ 13.
13 1,461
1,462 ÷ 13
1,493 ÷ 13.
1,461 ÷ 12.
74,521 ÷ 241.

241 74,521
3×241 = 723
4×241 = 964
9×241 = 2,169
74,521 ÷ 241
78,641 ÷ 256.
76,461 ÷ 248.

12 4
12 4
4 12
12 ÷ 4
12
4
12/4
412
51 17.
the quotient of 51 and 17
Translate. 51 ÷ 17
Divide. 3
51 17
17 51 or 51
17 .
91 13.
52 13.
160-ounce 160
8-ounce
160 ÷ 8
20
9
135

4
36

54 ÷ 9 56 32
7 8
642 48 ÷ 6 63
9
763 72 ÷ 8
15 ÷ 5 10 ÷ 5 14
7
18
6
420 315
24 ÷ 6 16 ÷ 4
18 ÷ 2 14 ÷ 2 27
3
30
3
428 436
45
5
35
5
72 / 8
864 35
7
42 ÷ 7
15 15 12 12 43 ÷ 43
37 ÷ 37 23
1
29
1
19 ÷ 1 17 ÷ 1 0÷4
0÷8 5
0
9
0
26
0
32
0
12 0
16 0 72 ÷ 3 57 ÷ 3

96
8
78
6
5 465
4 528 924 ÷ 7 861 ÷ 7
5,226
6
3,776
8
4 31,324
5 46,855 7,209 ÷ 3 4,806 ÷ 3
5,406 ÷ 6 3,208 ÷ 4 4 2,816
6 3,624
91,881
9
83,256
8
2,470 ÷ 7 3,741 ÷ 7 8 55,305
9 51,492
431,174
5
297,277
4
130,016 ÷ 3 105,609 ÷ 2 15 5,735
4,933
21
56,883 ÷ 67 43,725 / 75
30,144
314
26,145 ÷ 415 273 542,195
816,243 ÷ 462
15(204) 74 · 391 256 − 184
305 − 262 719 + 341 647 + 528
25 875 1104 ÷ 23
45 15 64 16 288 24
256 32

64
2
42
3
125
5.
152
8
48
3
54
2
45
17
128
4
12
4
53
71
8
14 11
17
14
5
45
9
6
26
15
9
14
365
25
365
300 8
37 4.

a
b

+ 3+4 3 4
a 0 a +0=a 0+a = a
a b a + b = b + a
− 7−3 7 3
Multiplication ×
·
()
3×8
3·8
3(8)
three times eight the product of 3 and 8
a ⋅0=0
0⋅a =0
1⋅a = a
a ⋅1=a

a ⋅ b = b ⋅ a
Division ÷
a
b
ba
a / b
12 ÷ 4
12
4
412
12 / 4
Twelve divided by four the quotient of 12 and 4
a ÷ a =1
a ÷1=a
0÷a =0
a ÷0
0, 2, 99 0, 3, 25 0, 4, 90
0, 1, 75

ase-10
472,981
12,403,295
a 8 b 4 c 1
d 7 e 2
a 4 b 0 c 1
d 9 e 3
5,280 204,614 5,012,582
31,640,976
412 648 3,556
2,734
38,975 26,849 81,486
75,992
4+3 25 + 18 571 + 629
10,085 + 3,492
6+7 38 + 14

a 0+19 b 19 + 0 a 0 + 480 b 480 + 0 a 7+6 b 6+7
a 23 + 18 b 18 + 23 44 + 35 63 + 29
96 + 58 375 + 591 7,281 + 12,546
5,280 + 16,324 + 9,731
30 12 11 8 25 39
15 50
4
12
1 8
$24,
5
$14, $38.
14 − 5 40 − 15 351 − 249

5,724 − 2,918
18 − 4 41 − 29
8−5 12 − 7 23 − 9
46 − 21 82 − 59 110 − 87
539 − 217 415 − 296 1,020 − 640
8,355 − 3,947 10,000 − 15 54,925 − 35,647
$2,485.
86
$948
28
8×5 6·14 (10)(95)
54(72)
2×4 3×8

0·14 (256)0 1·99
(4,789)1 a 7·4 b 4·7 (25)(6)
9,261 × 3 48 · 76 64·10
1,000(22) 162 × 493 (601)(943)
3,624 × 517 10,538 · 22
15 28
575
8
6
4
150
30 24
54 ÷ 9 42 / 7 72
8

648
8÷2 312
14 ÷ 2 32 52 ÷ 4
8
26 26 97
1
0÷52
100 ÷ 0 355
5
31 1,519 7505
25
3828 ÷ 6
5,166 ÷ 42
64 16 572 52
27 3 128 4

a
b
0, 4, 87
549,362.
a 9 b 6 c 2
d 5
a
b
25,849
45 + 23 65 − 42 85 ÷ 5
1,000 × 8 90 − 58 73 + 89
(0)(12,675) 634 + 255 0
9
8 128 145 − 79 299 + 836
7 · 475 8,528 + 704 35(14)
26
0
733 − 291 4,916 − 1,538
495 ÷ 45 52 × 983
16 58 9 15 32 18
63 21 524 29 32
50 300
84
12
12
6
9
3
$3,816
$3,472.
22
24

Add: 43 + 69.
Multiply: (896)201.

Divide: 7,263 ÷ 9.
20
23, 3 12, 15. 35, 38.
3
3
g.
g +3
12 15
20 23
35 38
g g +3
x, y, a, b, and c.
a + b a plus b a b
a − b a minus b a b
a · b, (a)(b), (a)b, a(b) a times b a b
a ÷ b, a / b, a, ba b a b a b
×, 3xy
3×y y 3·x · y x times y
5 3 5 3, 5+3.

9 2 9 2, 9−2.
4 8 4 8, 4·8.
20 5 20 5, 20 ÷ 5.
a 12 + 14 b (30)(5) c 64 ÷ 8 d x − y
a
12 + 14
b
(30)(5)
c
64 ÷ 8
d
x − y
x y
x y
a 18 + 11 b (27)(9) c 84 ÷ 7 d p − q
a 47 − 19 b 72 ÷ 9 c m + n d (13)(7)

a = b is read a is equal to b
=
b a, b a “”
a < b a b
a b
a > b a b
a b
a < b and a > b
a < b is equivalent to b > a. For example, 7 < 11 is equivalent to 11 > 7.
a > b is equivalent to b < a. For example, 17 > 4 is equivalent to 4 < 17.
a ≤ b, a < b a = b. a
b. ≠,
a = b
a ≠ b
a < b
a > b
a ≤ b
a ≥ b
a b
a b
a b
a b
a b
a b
< >
< >

a
a 20 ≤ 35 b 11 ≠ 15 − 3 c 9>10÷2 d x +210÷2
d
x +2 4 ÷ 2 d x − 7 < 1
a 19 ≥ 15 b 7 = 12 − 5 c 15 ÷ 3 < 8 d y - 3 > 6
=,

a
c
b
d
e
a
b
c
d

e
symbol, =, .
a
b
symbol, =, .
a
b
()
[]
{}
8(14 − 8) 21 − 3[2 + 4(9 − 8)] 24 ÷ ⎧ ⎩13−2[1(6−5)+4] ⎨ ⎫ ⎭ ⎬
3+5 3 plus 5
n −1 n n
6·7 6 times 7
x
y x y x y

3+5=8
n −1=14
n
6·7=42
x =53
y +9=2y −3
x
y y
a 16 − 6 = 10 b 4·2+1 c x ÷25 d y +8=40
a 16 − 6 = 10
b 4·2+1
c x ÷25
d y +8=40
a 23 + 6 = 29 b 7 · 3 − 7
y ÷ 14 x − 6 = 21
4·2+1
4·2 8 1 9.
4·2+1
8+1
9
2·2·2·2·2·2·2·2·2.

2·2·2 2 3 2·2·2·2·2·2·2·2·2 2 9 . 2 3 , 2 3
means multiply three factors of 2
2 3 2·2·2
a n , a n n
a n means multiply n factors of a
a n a n th
n =2 n =3,
a 2 is read as "a squared"
a 3 is read as "a cubed"
7 2 7 7
5 3 5 5
9 4 9
12 5 12
a 16·16·16·16·16·16·16 b 9·9·9·9·9 c x · x · x · x d a · a · a · a · a · a · a · a

a 16 7
b 9 5
c x x 4
d a a 8
41 · 41 · 41 · 41 · 41
7 · 7 · 7 · 7 · 7 · 7 · 7 · 7 · 7
a 8 6 b x 5
a 8 6, 8 6 8·8·8·8·8·8
b x 5, x 5 x · x · x · x · x
a 4 8 b a 7
a 8 8 b b 6
3 4 .

3 4
3⋅3⋅3⋅3
9⋅3⋅3
27 ⋅ 3
81
a 5 3 b 1 7
a 7 2 b 0 5
4+3·7
Some students say it simplifies to 49. Some students say it simplifies to 25.
4+3·7
Since 4 + 3 gives 7. 7 · 7
And 7 · 7 is 49. 49
4+3·7
Since 3 · 7 is 21. 4 + 21
And 21 + 4 makes 25. 25

a
a 4+3·7 b (4+3) ·7
b

a 12 − 5 · 2 b (12 − 5) · 2
a 8 + 3 · 9 b (8 + 3) · 9
a
a 18 ÷ 9 · 2 b 18 · 9 ÷ 2
b
42 ÷ 7 · 3
12 · 3 ÷ 4
18 ÷ 6 + 4(5 − 2).

30 ÷ 5 + 10(3 − 2)
70 ÷ 10 + 4(6 − 2)
Simplify: 5 + 2 3 +3 ⎡ ⎣6−3(4−2) ⎤ ⎦.

9 + 5 3 − ⎡ ⎣ ⎡ 4(9 + 3) ⎤ ⎦ ⎤
7 2 − 2 ⎡ ⎣ ⎡ 4(5 + 1) ⎤ ⎦ ⎤
2 3 +3 4 ÷3−5 2 .

3 2 + 2 4 ÷ 2 + 4 3
6 2 − 5 3 ÷ 5 + 8 2

16 − 9 25 − 7 5·6
3·9 28 ÷ 4 45 ÷ 5
x +8 x +11 (2)(7)
(4)(8) 14 < 21 17 < 35
36 ≥ 19 42 ≥ 27 3n =24
6n =36 y −1>6 y −4>8
2≤18÷6 3≤20÷4 a ≠7·4
a ≠1·12
9·6=54 7·9=63 5·4+3
6·3+5 x +7 x +9
y −5=25 y −8=32
3·3·3·3·3·3·3 4·4·4·4·4·4 x · x · x · x · x
y · y · y · y · y · y
5 3 8 3 2 8
10 5
a 3+8·5
b (3+8) · 5
a 2+6·3
b (2+6) · 3
2 3 −12÷(9−5)

3 2 −18÷(11−5) 3·8+5·2 4·7+3·5
2+8(6+1) 4 + 6(3 + 6) 4·12/8
2·36/6 6+10/2+2 9+12/3+4
(6 + 10) ÷ (2 + 2) (9 + 12) ÷ (3 + 4) 20÷4+6·5
33 ÷ 3 + 8 · 2 20 ÷ (4 + 6) · 5 33 ÷ (3 + 8) · 2
4 2 +5 2 3 2 +7 2 (4 + 5) 2
(3 + 7) 2 3(1+9·6)−4 2 5(2 + 8 · 4) − 7 2
2[1 + 3(10 − 2)] 5[2 + 4(3 − 2)]

(=,).
83″
80″
79″
82″
80″
83″
50
14,000 feet.
14,433′
14,421′
14,420′
14,345′
74″
76″
14,336′
78″
77″
14,309′
14,294′
a
b
c
d
e
14,286′
14,270′
14,269′
a
b
c
d
e

a
b

n ÷ 5
4 5 .
1+8 ⋅ 9.
x +7
a x =3 b x =12
a 3 x
x =3, x +7 10.
b 12 x

x = 12, x +7 19.
a b
x
y + 4 when
a y = 6 b y = 15
a − 5 when
a a = 9 b a = 17
9x − 2, when
a x =5 b x =1
ab a b, 9x 9 x.
a x =5, 5 x,
b x =1, 1 x,
a 9⋅5 b 9(1).

8x − 3, when
a x = 2 b x = 1
4y − 4, when
a y = 3 b y = 5
x 2 x = 10.
10 x,
x = 10, x 2 100.
x 2 when x = 8.
x 3 when x = 6.
Evaluate 2 x when x =5.
x =5, 2 x 32.

2 x when x = 6.
3 x when x = 4.
Evaluate 3x +4y − 6 when x = 10 and y =2.
x =10 y =2, 3x +4y −6 32.
2x + 5y − 4 when x = 11 and y = 3
5x − 2y − 9 when x = 7 and y = 8
Evaluate 2x 2 +3x + 8 when x =4.
2x 2 2⋅x ⋅ x
(2x) 2 , 2x ⋅2x.
4 2
3x 2 + 4x + 1 when x = 3.

6x 2 − 4x − 7 when x = 2.
7, y, 5x 2 ,9a, and 13xy.
3x 3. x, 1,
x =1⋅x.
7 7
9a 9
y 1
5x 2 5
7 7
y
y
x +7 x, 7
2x +7y +4 2x, 7y, 4
3x 2 +4x 2 +5y +3 3x 2 ,4x 2 ,5y, 3
9b +15x 2 + a +6.
9b, 15x 2 , a, 6.
9b 9.
15x 2 15.
1. a 1.
6 6.

4x + 3b + 2
9a + 13a 2 + a 3
7 4
5x 3x x.
5x, 7,n 2 ,4,3x, 9n 2
n 2 9n 2 n 2 .
5x, 7,n 2 ,4,3x, 9n 2 ,
7 and 4 are like terms.
5x and 3x are like terms.
n 2 and 9n 2 are like terms.
a y 3 ,7x 2 , 14, 23, 4y 3 ,9x, 5x 2
b 4x 2 +2x +5x 2 +6x +40x +8xy
a y 3 ,7x 2 , 14, 23, 4y 3 ,9x, 5x 2
y 3 , x 2 , x,
y 3 4y 3 y 3 .
7x 2 5x 2 x 2 .
14 23
9x x 1.
b 4x 2 +2x +5x 2 +6x +40x +8xy
4x 2 ,2x, 5x 2 ,6x, 40x, and 8xy
4x 2 5x 2 x 2 .
2x, 6x, and 40x x.
8xy xy.
9, 2x 3 , y 2 , 8x 3 , 15, 9y, 11y 2

4x 3 + 8x 2 + 19 + 3x 2 + 24 + 6x 3
3x +6x
9x,
x 3 6
9 3 6 9
3x +6x
3x +7+4x +5.
7x + 9 + 9x + 8

5y + 2 + 8y + 4y + 5
7x 2 +8x + x 2 +4x.
8x 2 +12x
3x 2 + 9x + x 2 + 5x
11y 2 + 8y + y 2 + 7y

a b
a b
a b
b a
a b
b a
a b
a b
b a
a b
b a
a b
a b
a b
a b
a b
b a
a + b
a − b
a ⋅ b ab a(b) (a)(b)
a ÷ b a / b a ba b
a b
a b
a b
a b
a 20 4 b 10x 3
a
the difference of 20 and 4
20 minus 4
20 − 4
b
the quotient of 10x and 3
divide 10x by 3
10x ÷3
10x /3or 10x
3

a 47 41 b 5x 2
a 17 19
b 7 x
8
7
a y
b 9z
a
Eight more than y
Eight added to y
y +8
b
Seven less than 9z
Seven subtracted from 9z
9z −7
a x
b 11a
a 19 j
b 21 2x
a m n
b m n
a 5
m n.
m n
5(m + n)
b
m n.

m n
5m + n
a b
a p q
b p q
a x and 8 b x and 8
6 w
w
6
6 w
6 w
w −6
5 w
2 l
2 5 q

q
5 q
2 q
2 5q
5q −2
q
n

7x + 8 when x =2 9x + 7 when x =3 5x − 4 when x =6
8x − 6 when x =7 x 2 when x =12 x 3 when x =5
x 5 when x =2 x 4 when x =3 3 x when x =3
4 x when x =2 x 2 +3x − 7 when x =4 x 2 +5x − 8 when x =6
2x +4y − 5 when x =7,y =8
6x +3y − 9 when x =6,y =9
(x − y) 2 when x = 10, y =7
(x + y) 2 when x =6,y =9 a 2 + b 2 when a =3,b =8 r 2 − s 2 when r = 12, s =5
2l +2w when l = 15, w =12
2l +2w when l = 18, w =14
15x 2 +6x +2 11x 2 +8x +5 10y 3 + y +2
9y 3 + y +5
8a 13m 5r 2
6x 3
x 3 ,8x, 14, 8y, 5,8x 3 6z, 3w 2 ,1,6z 2 ,4z, w 2
9a, a 2 ,16ab, 16b 2 ,4ab, 9b 2
3, 25r 2 ,10s, 10r, 4r 2 ,3s
10x +3x 15x +4x 17a +9a
18z +9z 4c +2c + c 6y +4y + y
9x +3x +8 8a +5a +9 7u +2+3u +1

8d +6+2d +5 7p +6+5p +4 8x +7+4x −5
10a +7+5a −2+7a −4 7c +4+6c −3+9c −1
3x 2 +12x +11+14x 2 +8x +5
5b 2 +9b +10+2b 2 +3b −4
x 4
3 x 6 y 9 y
8x 3x 13x 3x y 3
y 8
y
y
x
y
x
$15
b
3
c
4
b
6
f
n
$5 $10
t
$750
$750
$750. $2,100,
$2,500
$2,500
$2,500.
$19,400,

x y
“4
x y” 4 x y.”
a
b

Evaluate x + 8 when x = 11.
Evaluate 5x − 3 when x = 9.
x 8.
x +2=7? 5, 5 x +2=7
5 x
x +2=7
5+2= ? 7
7=7✓
5+2=7 5
= ?
(=) (≠).

Determine whether x = 5 is a solution of 6x − 17 = 16.
x =5 6x − 17 = 16.
Is x = 3 a solution of 4x − 7 = 16?
Is x = 2 a solution of 6x − 2 =10?
Determine whether y = 2 is a solution of 6y −4=5y −2.
2 y.
y =2 2 6y −4=5y −2.
Is y = 3 a solution of 9y − 2 = 8y + 1?
Is y = 4 a solution of 5y − 3 = 3y + 5?

3 3
3 5
x,
x +3. 8 x +3 8
x +3=8.
x +3=8
5 x

x 4 x +4 x +4
5 5
x +4=5
4
1 x
x =1 1

a, b, c,
a = b
a − c = b − c
17 3 3
17, 17 − 3, 14. 3 14
3
a = b
a −3=b −3
x + 8 = 17.
x.
x =9 x +8=17 9

x + 6 = 19
x + 9 = 14
100 = y + 74.
y,
74
26 y
y =26 100 = y +74
95 = y + 67
91 = y + 45
x −5=8.
a, b c
a = b
a + c = b + c
17-year-old
27.

a = b
a +10=b +10
x −5=8.
x − 9 = 13
y − 1 = 3
27 = a − 16.
16

27 = a −16 a = 43.
19 = a − 18
27 = n − 14
6 9 15.

7 6 13.
8 6 14.
8 7 56.
6 9 54.
21 3 63.
x 3 18.
x 5 30.
y 4 16.

x 47.
x =44
x =44
x 37.
y 28.
y 14 18.
y =32
y =32
z 17 37.

x 19 45.

y +18=25
x +13=21
a x =8 b x =34
a y =7 b y =43
m −4=13
a m =9 b m =17
n −9=6
a n =3 b n =15
18d −9=27
a d =1 b d =2
7v −3=4v +36
a v =3 b v =11
3p +6=15
a p =3 b p =7
24 f −12=60
a f =2 b f =3
20h −5=15h +35
a h =6 b h =8
8q +4=20
a q =2 b q =3
8u −4=4u +40
a u =3 b u =11
18k −3=12k +33
a k =1 b k =6
a +2=18 b +5=13 p +18=23
q +14=31 r + 76 = 100 s +62=95
16 = x +9 17 = y +6 93 = p +24
116 = q +79 465 = d + 398 932 = c + 641

y −3=19 x −4=12 u −6=24
v −7=35 f − 55 = 123 g − 39 = 117
19 = n −13 18 = m −15 10 = p −38
18 = q −72 268 = y − 199 204 = z − 149
8 9
17.
7 9
16.
23 19
4.
29 12
17.
54 6
9.
3 9
27.
42 7
6.
6 8
48.
n
10 52.
14 64.
m
y
10 100.
x
4 68.
p
21.
q
40.
r 18 73.
s 13 68. d 30
52.
c 25
75.
12 u 89. 19 w 56. 325 c 799.
299 d 850.
$500
p, $1800
500 + p = 1800.
$120 $340
p,
p − 120 = 340.
$750
$5800
d,
d − 750 = 5800.
$1299. $1409,
t,
1299 + t = 1409.

x =1
8x −2=16−6x?
y −5=21
a
b

0, 4, 215
3, 5, 7.
2, 4, 6, 8, 10, 12. 12
2, 4, 6, 8, 10, 12 2. 2
2. 2
1⋅2=2
2⋅2=4
3⋅2=6
4⋅2=8
5⋅2=10
6⋅2=12
3
3. 3.
1⋅3=3
2⋅3=6
3⋅3=9
4⋅3=12
5⋅3=15
6⋅3=18
2 9

1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108
n n.
2, 5, 10, and 3
1 50. 2
2 1 50
0, 2, 4, 6, or 8. 2
2 0, 2, 4, 6, or 8,
2.
2:
a 489 b 3,714

a
b
2:
a 678 b 21,493
2:
a 979 b 17,780
5. 5 1 50.
5?
5 1 50
5 5 0. 2
5
5:
a 579 b 880

a
b
5.
a 675 b 1,578
5.
a 421 b 2,690
10 1 50. 10
10 1 50
10:
a
a 425 b 350

10:
a 179 b 3,540
10:
a 110 b 7,595
3. 3
2, 5, and 10.
3 1 50
3 3,
3.
Multiple of 3 3 6 9 12 15 18 21 24
Sum of digits 3 6 9
1+2
3
1+5
6
1+8
9
2+1
3
2+4
6
42. 4 2, 4+2=6. 6 3,
42 3.
3:
a 645 b 10,519
a 645 3?

6+4+5=15
645 ÷ 3
3 ⋅ 215 = 645
b 10,519 3?
1+0+5+1+9=16
645 ÷ 3
3,506R1
3 10,519
10,519 3,
3. 3.
10,519
3:
a 954 b 3,742
3:
a 643 b 8,379
2, 5, 10. 10
2 5. 10=2⋅5. 6=2⋅3,
6 2 3.
375 5 375 5. 375 ÷ 5 75, 375
5 ⋅ 75. 10,519 3. 10,519 3
10,519 3.
m n, m n.

2 0, 2, 4, 6, or 8
3 3
5 5 0
6 2 3
10 0
1,290 2, 3, 5, and 10.
1,290.
2 0, 2, 4, 6, or 8? 1290 ÷ 2 = 645
3
Is sum of digits divisible by 3?
1+2+9+0=12
1290 ÷ 3 = 430
5 5 0? 1290 ÷ 5 = 258
10 0? 1290 ÷ 10 = 129
1,290 2, 3, 5, and 10.
2, 3, 5, and 10.
6240
2, 3, 5, and 10.
7248
5,625 2, 3, 5, and 10.
5,625

2 0, 2, 4, 6, or 8? 5625 ÷ 2 = 2812.5
3
Is sum of digits divisible by 3?
5+6+2+5=18
5625 ÷ 3 = 1875
5 5 0? 5625 ÷ 5 = 1125
10 0? 5625 ÷ 10 = 562.5
5,625 3 5, 2, 10.
by 2, 3, 5, and 10.
4962
by 2, 3, 5, and 10.
3765
m n,
m n. 72 8 9, 72 8 72
9. 72 8 9. 8 9
72. 72=8⋅9 72.
a ⋅ b = m, a and b m, m a and b.
24
24.

1 24 1 ⋅ 24 = 24
2 12 2 ⋅ 12 = 24
3 8 3⋅8=24
4 6 4⋅6=24
6 4 6⋅4=24
8 3 8⋅3=24
12 2 12⋅2=24
24 1 24⋅1=24
24? 24
24,
1, 2, 3, 4, 6, 8, 12, 24
1.
72.
72 1.

9 8.
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72
96
80
72, 7, 1
1
1 1
2 20
2 20,
20 2, 3, 5, 7, 11, 13, 17, and 19.
1

2
a 83 b 77
a 83 2,
83?
2 83 0, 2, 4, 6, or 8.
3 8 + 3 = 11, 11 3.
5 83 5 0.
7 83 ÷ 7 = 11.857….
11 83 ÷ 11 = 7.545…
11 (7.545…)
83, 83
b 77.
77?
2 0, 2, 4, 6, or 8.
3 7 + 7 = 14, 14 3.
5 5 0.
7 77 ÷ 7 = 11
77 7,
91

137

50
2 3 4
5 6 7
8 9 10
12
2, 3, 4, 5, 6, and 10.
84 96 75
78 168 264
900 800 896
942 375 750
350 550 1430
1080 22,335 39,075
36 42 60
48 144 200
588 576
43 67 39
53 71 119
481 221 209
359 667 1771

$100
$15
0 100 100
1 100 + 15 115
2 100 + 15 ⋅ 2 130
3 100 + 15 ⋅ 3
4 100 + 15 ⋅ []
5 100 + []
6
20
x
$75
$20
0 75 75
1 75 + 20 95
2 75 + 20 ⋅ 2 115
3 75 + 20 ⋅ 3
4 75 + 20 ⋅ []
5 75 + []
6
20
x
2 3,
6?
a
b

810 2, 3, 5, 6, or 10?
127
2 ⋅ 2 ⋅ 2 ⋅ 2
1
50
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
36. 3 12. 3
12 36

3 12 3 4.
12.
3 4 2·2.
4. 2 2s.
2⋅2⋅3⋅3
2⋅2⋅3⋅3
2 2 ⋅3 2
36. 12 3,
2 18, 4 9, or 6 and 6.
48

2⋅2⋅2⋅2⋅3
2 4 ⋅3
48.
80
60

2⋅2⋅3⋅7
2 2 ⋅3⋅7
84.
126
294
36, 36 2, 36.
9 3.
3,
2⋅2⋅3⋅3
2 2 ⋅3 2

120
2⋅2⋅2⋅3⋅5
2 3 ⋅3⋅5
120.
80
60
48

2⋅2⋅2⋅2⋅3
2 4 ⋅3
126
294
10 25.
10:10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, …
25:25, 50, 75, 100, 125, …
50 100 10 25.
10 25 50.
15 20
15 20.
15: 15, 30, 45, 60, 75, 90, 105, 120
20: 20, 40, 60, 80, 100, 120, 140, 160

60, 60 15 20.
120
9 and 12
18 and 24
12 18.
12=2⋅2⋅3
18=2⋅3⋅3
12=2⋅2⋅3
18 = 2 ⋅ 3 ⋅ 3
12 18
36
15 18
LCM=2⋅3⋅3⋅5

15 and 20
15 and 35
50 100
LCM=2⋅2⋅5⋅5
55, 88
60, 72

86 78 132
455 693 420
115 225 2475
56 72 168
252 391 400
432 627 2160
2520
150 180 525
444 36 50
350 144
8, 12 4, 3 6, 15
12, 16 30, 40 20, 30
60, 75 44, 55
8, 12 12, 16 24, 30
28, 40 70, 84 84, 90
6, 21 9, 15 24, 30
32, 40

12 8.
a
b

m n m n
n n
a + b a plus b a b
a · b, (a)(b), (a)b, a(b) a times b a b
a − b a minus b a b
a ÷ b, a / b, a, ba b a b a b
a = b a b
=
a < b a b
a b
a > b a b
a b

a = b
a ≠ b
a < b
a > b
a ≤ b
a ≥ b
a b
a b
a b
a b
a b
a b
a n n n
a n n a
a n a nth
a b c

a = b
a − c = b − c
a b c
a = b
a + c = b + c
a ⋅ b = m a b m m a b

3⋅8 12 − x 24 ÷ 6
9+2a 50 ≥ 47 3y

8 4 3 6 y 5
n 4
3 4 10 6 2 7
4 3
10 + 2 ⋅ 5 (10 + 2) ⋅ 5 (30+6)÷2
30 + 6 ÷ 2 7 2 +5 2 (7 + 5) 2
4 + 3(10 − 1) (4 + 3)(10 − 1)
9x − 5 when x =7 y 3 when y =5
3a −4b when a = 10, b =1
bh when b =7,h =8
12n 2 +3n +1 4x 3 +11x +3
6y 13x 2
5x 2 ,3,5y 2 ,3x, x, 4 8, 8r 2 ,8r, 3r, r 2 ,3s
15a +9a 12y +3y + y 4x +7x +3x
6+5c +3 8n +2+4n +9 19p +5+4p −1+3p
7y 2 +2y +11+3y 2 −8 13x 2 − x +6+5x 2 +9x

x 6 10 a 3n 9
s 4 5 y 1 10
5 z
$3
c.
5
n.
y +16=40
a 24 b 56
d −6=21
a 15 b 27
4n +12=36
a 6 b 12
20q −10=70
a 3 b 4
15x −5=10x +45
a 2 b 10
22p −6=18p +86
a 4 b 23
c +8=14 v + 8 = 150 23 = x +12
376 = n + 265
y −7=16 k − 42 = 113 19 = p −15
501 = u − 399

7 33
40.
15 3
12.
4 8
32.
63 9
7.
n
3 76.
4 89.
y
x 21 a 11. q 18
35.
57.
240.
m 125
h −15=27 k −11=34 z +52=85
x + 93 = 114 27 = q +19 38 = p +19
31 = v −25 38 = u −16
50
3 2 8
10
2, by 3, by 5, by 6, and by 10.
96 250 420
625
30 70 180
378
19 51 121

219
84 165 350
572
9, 15 12, 20 25, 35
18, 40

6⋅4 15 − x
5⋅8+10 x +6=9 3 ⋅ 11 = 33
a
n ⋅ n ⋅ n ⋅ n ⋅ n ⋅ n
b 3 5
4+3⋅5 (8 + 1) ⋅ 4 1+6(3−1)
(8 + 4) ÷ 3 + 1 (1 + 4) 2 5[2 + 7(9 − 8)]
8x − 3 when x =4 y 3 when y =5 6a −2b when a =5,b =7
hw when h = 12, w =3
a 6x +8x
b 9m +10+m +3
5 x 12 y
a and b
3
r.
n −6=25 x +58=71
15 y 32. a 129
164.
4
50.
90.
1080.
24 40.

0, 1, and 3
(=, ): 2___4

0.
−1°F
1 degree 0.
−20°F, 20 degrees 0.
0 feet.
1,302 feet −1,302 feet.
0 ft
.
500 feet. −500 feet
500 ft
−500 ft
.
0

(1, 2, 3, …) +,
+2 +3,
−1
0, −2
a 3 b −3 c −2
0
a 3, 0
b −3, 0
c −2, 0

a 1 b −1 c −4
a −4 b 4 a −1
a < b a b a b
a > b a b a b
3 5
3
5 3
5, 5 3.
a
4 1 4>1.
1 4 1−3.
−3 −1 −3 < − 1.

:
a 14___6 b −1___9 c −1___−4 d 2___−20
a
14___6
14 > 6
b
−1___9
−1 < 9
c
−1___−4
−1>−4
d
−2___−20
2 > −20
< >.
a 15___7 b −2___5 c −3___−7 d 5___−17
< >.
a 8___13 b 3___−4 c −5___−2 d 9___−21
2 −2 2 −2,
−2 2 3 −3

a 7 b −10
a −7 0 7, 0. −7
7
b 10 0 −10 0. 10
−10
a 4
b −3
a 8
b −5

“−”,
10 − 4
−8
−x
− (−2)
10 − 4 4
−8
−x x
− (−2)
−2.
−a a
−a a.
−(−6).
−(−6)
−6 6. 6
−(−1)
−(−5)
0
… −3, −2, −1, 0, 1, 2, 3…
−x :
a x =8
b x =−8.

a −x x =8 8 x
−x
−8
b −x x =−8 −8 x
−x
8
−n :
a when n = 4
b when n = −4
−m :
a when m = 11
b when m = −11
5 −5 0
0. 0
5
|5|, −5 |−5|
0
n |n|.
|n| ≥ 0 for all numbers

a |3| b |−44| c |0|
a
3
|3|
b
44
|−44|
c
0
|0|
a |12| b − |−28|
a |9| b (b) − |37|
a |x| when x = −35 b |−y| when y = −20 c − |u| when u =12 d − |p| when p = −14
a |x| x = −35 :
|x|
35

| − y| y = −20 :
| − y|
|20|
20
c − |u| u =12:
− |u|
−12
d − |p| p = −14 :
− |p|
−14
a |x| when x = −17 b |−y| when y =−39 c − |m| when m = 22
d − |p| when p = −11
a |y| when y = −23 b |−y| when y = −21 c − |n| when n = 37
d − |q| when q = −49
, or =
a |−5|___ − |−5| b 8___ − |−8| c −9___ − |−9| d − |−7|___−7

a
|−5|___ − |−5|
5___−5
5>−5
b
8___ − |−8|
8___ − 8
8> −8
c
−9___ − |−9|
−9___ − 9
−9 = −9
d
− |−7|___−7
−7___ − 7
−7 = − 7
, or = for each of the follo
wing:
a |−9| ___ − |−9| b 2___ − |−2| c −8___|−8| d − |−5|___−5
, or =
a 7___ − |−7| b − |−11|___−11 c |−4|___ − |−4| d −1___|−1|

a |9−3| b 4|−2|
a
b
a |12 − 9|
b 3|−6|
a |27 − 16|
b 9|−7|
|8+7| − |5+6|.
|1 + 8| − |2 + 5|
|9−5| − |7 − 6|
24 − |19 − 3(6 − 2)|.

24 − |19 − 3(6−2)|
24 − |19 − 3(4)|
3(4) 24 − |19 − 12|
24 − |7|
24 − 7
17
19 − |11 − 4(3 − 1)|
9 − |8 − 4(7 − 5)|
−20. 20.”
a b −11 c
d
a
−14
b
− (−11)
c
−16
d
2−(−7)
a b −15 c
d
a b c
d

a 12 degrees Fahrenheit
b 3 yards.
c 1,302 feet
d $40.
a
−12ºF
b
3
c
−1,302
d
−$40
5 yards.
30 fee
t

a 2 b −2
a 5 b −5
a −8 b 8
c −5
c −2
c −6
a −7 b 7
c −1
< >.
a 9__4 b −3__6
c −8__−2 d 1__−10
a 6__2;
c −9__−1;
b −7__4;
d 9__−3
a −5__1;
c 6__10;
b −4__−9;
d 3__−8
a −7__3;
b −10__−5;
c 2__−6;
d 8__9
a 2
b −6
a 9
b −4
a −8 b 1
a −2 b 6
−(−4) −(−8) −(−15)
−(−11)
−m when
a m =3 b m =−3
− p when
a p =6 b p =−6
−c when
a c =12
b c = −12
−d when
a d =21
b d = −21

a |7| b |−25|
c |0|
a |5| b |20|
c |−19|
a |−32|
c |16|
b |−18|
a |−41|
c |22|
b |−40|
a |x| when x = −28
b | − u| when u = −15
a |y| when y = −37
b | − z| when z = −24
a − |p| when p =19
b − |q| when q = −33
a − |a| when a =60
b − |b| when b = −12
, or =
a −6__|−6|
b − |−3|__−3
a −8__|−8|
b − |−2|__−2
a |−3|__ − |−3|
b 4__ − |−4|
a |−5|__ − |−5|
b 9__ − |−9|
|8−4| |9−6| 8|−7|
5|−5| |15 − 7| − |14 − 6| |17 − 8| − |13 − 4|
18 − |2(8 − 3)| 15 − |3(8 − 5)| 8(14 − 2|−2|)
6(13 − 4|−2|)
a 8
b −6
c
d 4 3
a 11
b −4
c
d 8 2
a 20
b −5
c
d 18 7

a 15
b −9
c
d 12 5
6 degrees
14 degrees
40 feet
65 feet
12 yards
4 yards $3 $5
3
20,320 feet
282 feet
a b
$540 million.
$27 billion.
a b
58° Celsius,
90° 0° Celsius,
a
b
1,400,000
2007 2010.
110,171
2009 2010.
a b
a

x + 8 x = 6.
8 + 2(5 + 1).
3 7
2−1.
+1. −1.

5, −5 and 3 , −3.
5+3 −5+(−3) −5 + 3 5 + (−3)
5+3.
5+3 5 3
5+3=8
2 + 4
2 + 5
−5 + (−3).
−5 + (−3) −5 −3
−5+−3=−8
−2 + (−4)

−2 + (−5)
5 3
5 3 8—either 8
8
−5 + 3.
−5+3 −5 3
−5+3=−2
2 + (−4)
2 + (−5)
5+(−3).

5+(−3) 5 −3
5+(−3) =2
(−2) + 4
(−2) + 5
a b c d
a
4+2
6 4+2=6

−3 + 6
3 −3 + 6 = 3
c
4 + (−5)
−1 4 + (−5) = −1
d
−2 + (−3)
−5 −2 + (−3) = −5

a b c d
a b c d
37 + (−53), 37 53
37 53
53−37 = 16, 16
37 + (−53) = −16
−74 + (−27). 74 27 101
−101.
−74 + (−27) = −101
5+3 −5 + (−3)
−5+3 5+(−3)
a 19 + (−47) b −32 + 40
a 19 47.
19 + (−47)
−28
b 32 40.
−32+40
8

a 15 + (−32) b −19 + 76
a −55 + 9
b 43 + (−17)
−14 + (−36).
−14 + (−36)
−50
−31 + (−19)
−42 + (−28)
−5+3(−2+7).
−5+3(−2 + 7)
−5+3(5)
−5+15
10
−2 + 5(−4 + 7)
−4 + 2(−3 + 5)

x + 7 when
a x =−2 b x = −11.
a x +7 x =−2
b x +7 x = −11
x + 5 when
a x = −3 and
b x = −17
y + 7
a y = −5
b y = −8
n =−5,
a n +1 b −n +1.
a n +1 n =−5

−n +1 n =−5
n = −8,
a n + 2 b −n + 2
When y = −9, evaluate
a y + 8 b −y + 8.
3a + b a =12 b = −30.
a + 2b when a = −19 and b = 14.
5p + q when p = 4 and q = −7.
(x + y) 2 x = −18 y = 24.
−18 x 24 y.

(x + y) 2
(−18 + 24) 2
(6) 2
36
(x + y) 2 x = −15 y = 29.
(x + y) 3 x = −8 y = 10.
−9 5.
−9 5
−9+5
−4
−7 4
−8 −6
8 −12, 3.

8 −12 3
[8+(−12)] +3
−4+3
−1
9 −16, 4.
−8 −12, 7.
$5 −$5.
7 degrees
12 degrees.
10 degrees
14 degrees Celsius.
16 fee
t 17 fee
t.

42-yard line. 6 yards,
4 yards, 8 yards.
20-yard line. 9 yards,
7 yards, 4 yards.
25-yard line. 5 yards, 8 yards
15 yards

7+4 8+5 −6 + (−3)
−5 + (−5) −7+5 −9+6
8 + (−7)
9 + (−4)
−21 + (−59) −35 + (−47) 48 + (−16)
34 + (−19) −200 + 65 −150 + 45
2 + (−8) + 6 4 + (−9) + 7 −14 + (−12) + 4
−17 + (−18) + 6 135 + (−110) + 83 140 + (−75) + 67
−32+24+(−6)+10 −38 + 27 + (−8) + 12 19+2(−3+8)
24 + 3(−5 + 9)
x +8
a x = −26
b x = −95
y +9
a y = −29
b y = −84
y + (−14)
a y = −33 b y =30
x + (−21)
a x = −27 b x =44
a =−7,
a a +3 b −a +3
b = −11,
a b +6 b −b +6
c =−9,
a c + (−4)
b −c + (−4)
d =−8,
a d + (−9)
b −d + (−9)
m + n m = −15
n =7
p + q p =−9
q =17
r−3s r =16 s =2 2t + u t =−6
u =−5
(a + b) 2 a =−7
b =15
(c + d) 2 c =−5
d =14
(x + y) 2 x =−3
y =14
(y + z) 2 y =−3
z =15

−14 5 −22 9 8 −2
5 −1 −10 −15 −6 −20
6 −1
−12
3 −2
−8
10 −19,
4
12 −15,
1
−19°F
26°F.
−15°F
28°F.
$73
$45
$212
$105
3 pounds
2 pounds,
1 pound,
4 pounds.
5 pounds
3 pounds,
2 pounds,
1 pound.
35-yard line.
12 yards,
8 yards,
6 yards.
20-yard line.
15 yards, 3 yards
6 yards
20 minutes,
90 calories.
110 calories.
30 minutes,
168 calories.
140-calorie

Monday −504 Tuesday +142 Wednesday −449
Thursday +410 Friday +369
Monday −201 Tuesday −16 Wednesday −23
Thursday +172 Friday −34
−8 2
8 −2
a
b

12 − (8−1).
20 −15
−18 + 7.
5−3
5 3.
5−3 −5−(−3) −5−3 5−(−3)
5−3.

5−3 5 3
5−3=2
5 3 2
6 − 4
7 − 4
−5 − (−3).
−5 − (−3) −5 −3
−5 − (−3) =−2
−5 −3 −2
−6 − (−4)
−7 − (−4)

3 5 2
3 5 2
−5 − 3.
−5−3 −5 3
−5−3=−8
−5 3 −8
−6 − 4
−7 − 4
5 − (−3).

5−(−3) 5 −3
5 −3 8
5−(−3) =8
6 − (−4)
7 − (−4)
a
a b c d
8−2
8 2
6
8−2=6

−5−4
−5 4
−9
−5−4=−9
c
6−(−6)
6 −6
12
6−(−6) =12

d
−8 − (−3)
−8 −3
−5
−8 − (−3) =−5
a b c d
a b c d
a 2−8
b −3 − (−8)
a
2−8=−6

−3 − (−8) =5
a 7 − 9
b −5 − (−9)
a 4 − 7
b −7 − (−10)
−23−7.
23
7
7 7 7
23 7 30
−23 − 7 = −30
7, 7
30 − (−12).
30
12
12 30 12
30 12 42
30 − (−12) = 42
−12, 12.

a − b = a +(−b)
6−4 6 + (−4).
6−4 6−4
6 + (−4)
a 13 − 8 and 13 + (−8)
b −17 − 9 and −17 + (−9)
a
13 − 8 13 + (−8)
13 − 8 = 5
13 + (−8) =5
b
−17−9 −17 + (−9)
−17 − 9 = −26
−17 + (−9) = −26
a 21 − 13 and 21 + (−13)
b −11 − 7 and −11 + (−7)

a 15 − 7 and 15 + (−7)
b −14 − 8 and −14 + (−8)
8 − (−5) 8+5.
a
a 9 − (−15) and 9 + 15 b −7 − (−4) and −7 + 4
9−(−15) 9+15
9−(−15) =24
9 + 15 = 24
b
−7 − (−4) −7 + 4
−7 − (−4) =−3
−7+4=−3
a 6 − (−13) and 6 + 13 b −5 − (−1) and −5 + 1
a 4 − (−19) and 4 + 19 a −4 − (−7) and −4 + 7

5–3 –5 – (–3)
2 –2
–5–3 5–(–3)
–8 8
−74 − (−58).
−74 − (−58)
−16
−67 − (−38)
−83 − (−57)
7−(−4−3)−9.

8 − (−3 − 1) − 9
12 − (−9 − 6) − 14
3·7−4·7−5·8.
6 · 2 − 9 · 1 − 8 · 9.
2 · 5 − 3 · 7 − 4 · 9
x − 4 when
a x =3
b x =−6.
a x −4 x =3 3 x
b x −4 x =−6, −6 x

y − 7 when
a y = 5
b y = −8
m − 3 when
a m = 1
b m = −4
20 − z when
a z =12 b z =−12
a 20 − z when z = 12, 12 z
b 20 − z when z = −12, substitute −12 for z in the expression.
17 − k when
a k = 19
b k = −19
−5 − b when
a b = 14
b b = −14
a − b

a b
a 13 −21
b 24 −19
a
b 24 −19.
a 14 −23
b 21 −17
a 11 −19
b 18 −11

11
−9
11 −9
11 − (−9)
20
20
15 degrees Fahrenheit.
30 degrees
−6
−15 degree Fahrenheit.
14,497 282

14,497 − (−282)
14,779
14,779
10,023 foo
t
80 fee
t
340 fee
t 573 fee
t
$25 $8.
a
b $20.
c $10
a
$25 $8
$17

$17 $20
$3
c
$10 −$3
$7
$75 $27.
a
b $50.
c $20
−$78.
a $24
b $49.

8−2 9−3 −5 − (−1)
−6 − (−4) −5−4 −7 − 2
8 − (−4)
7 − (−3)
a 15 − 6
b 15 + (−6)
a 12 − 9
b 12 + (−9)
a 44 − 28
b 44 + (−28)
a 35 − 16
b 35 + (−16)
a 8 − (−9) b 8+9
a 4 − (−4) b 4+4
a 27 − (−18)
b 27 + 18
a 46 − (−37)
b 46 + 37
15 − (−12) 14 − (−11) 10 − (−19)
11 − (−18) 48 − 87 45 − 69
31 − 79 39 − 81 −31 − 11
−32−18 −17−42 −19 − 46
−103 − (−52) −105 − (−68) −45 − (−54)
−58 − (−67) 8−3−7 9−6−5
−5−4+7 −3−8+4 −14 − (−27) + 9
−15 − (−28) + 5 71 + (−10) − 8 64 + (−17) − 9
−16−(−4+1)−7 −15−(−6+4)−3 (2−7)−(3−8)
(1−8)−(2−9) −(6−8)−(2−4) −(4−5)−(7−8)
25 − [10 − (3 − 12)] 32 − [5 − (15 − 20)] 6.3 − 4.3 − 7.2

5.7 − 8.2 − 4.9 5 2 −6 2 6 2 −7 2
x − 6 when
a x =3 b x =−3
x − 4 when
a x =5 b x =−5
5−y when
a y =2 b y =−2
8−y when
a y =3 b y =−3
4x 2 −15x + 1 when x =3 5x 2 −14x + 7 when x =2
−12−5x 2 when x =6 −19−4x 2 when x =5
a 3
−10
b −20 45
a 8
−12
b −13 50
a −6
9
b −12 −16
a −8
9
b −15 −19
a 21 6
b 31 −19
a 8 −17
b −24 37
a 34 7
b 29 −50
a 5 −14
b −13 42
28° Fahrenheit. 22° Fahrenheit.
38° Fahrenheit. 35° Fahrenheit.
84° Fahrenheit.
−12° Fahrenheit.
89°,
−31°.
30-yard line.
2 yards, 7 yards,
4 yards.
20-yard line.
8 yards, 5 yards,
6 yards.

$148
$83.
$426
$152.
$210
$250.
$94
$110.
−$14
$40
−$23
$80
−7°.
−20°”.
−100 feet.
−45 feet
9 −6 15. 3 − (−4)
3+4?
a
b

20 13
−5 + (−5) + (−5).
Evaluate n + 4 when n = −7.
a · b a, b
5 −3. 5, 3
5, 3

15 5, 3 −15.
(−5)(−3), −5, 3 15.
5·3=15 −5(3)=−15
5(−3) = −15 (−5)(−3) = 15
a −9 · 3 b −2(−5) c 4(−8) d 7·6

a
–9 ⋅ 3
–27
b
–2(–5)
10
c
4(–8)
–32
d
7⋅6
42
a −6 · 8 b −4(−7) c 9(−7) d 5 · 12
a −8 · 7
b −6(−9)
c 7(−4)
d 3 · 13
1, −1?
−1
−1 · 4
−1(−3)
−4 3
−4 is the opposite of 4 3is the opposite of −3
−1,

−1
−1
−1a =−a
a −1 · 7
b −1(−11)
a
−1 ⋅ 7
−7
b
−1(−11)
11
a −1 · 9
b −1 · (−17)
a −1 · 8
b −1 · (−16)
15÷3=5 5·3=15
15 3 5 15.
5·3=15so15÷3=5
−5(3)=−15so−15÷3=−5
(−5)(−3) = 15 so 15 ÷ (−3) = −5 5(−3) = −15 so −15 ÷ −3 = 5

a −27 ÷ 3
b −100 ÷ (−4)
a
–27 ÷ 3
–9
b
–100 ÷ (–4)
25
a −42 ÷ 6
b −117 ÷ (−3)
a −63 ÷ 7
b −115 ÷ (−5)
1,
−1? −1
8÷(−1)
−9 ÷ (−1)
−8 9
−8 is the opposite of 8 9 is the opposite of −9
−1
−1
−1
a ÷ (−1) = −a

a 16 ÷ (−1)
b −20 ÷ (−1)
a
16 ÷ (–1)
–16
b
–20 ÷ (–1)
20
a 6 ÷ (−1)
b −36 ÷ (−1)
a 28 ÷ (−1)
b −52 ÷ (−1)
Simplify: 7(−2) + 4(−7) − 6.
7(−2) +4(−7)−6
−14 + (−28)−6
−42 − 6
−48

8(−3) + 5(−7)−4
9(−3) + 7(−8) − 1
a (−2) 4 b −2 4
a 4 −2. −2
(−2) 4
(−2)(−2)(−2)(−2)
4(−2)(−2)
−8(−2)
16
b 4 2. 2
−2 4
−(2⋅2⋅2⋅2)
−(4 ⋅ 2 ⋅ 2)
−(8 ⋅ 2)
−16
a (−3) 4 b −3 4
a (−7) 2 b −7 2
Simplify: 12 − 3(9 − 12).

12 − 3(9 − 12)
12 − 3(−3)
12 − (−9)
21
17 − 4(8 − 11)
16 − 6(7 − 13)
8(−9) ÷ (−2) 3 .
8(−9) ÷ (−2) 3
8(−9) ÷ (−8)
−72 ÷ (−8)
9
12(−9) ÷ (−3) 3
18(−4) ÷ (−2) 3
Simplify: −30 ÷ 2 + (−3)(−7).

−30 ÷ 2 + (−3)(−7)
−15 + (−3)(−7)
−15+21
6
−27 ÷ 3 + (−5)(−6)
−32 ÷ 4 + (−2)(−7)
Evaluate 2x 2 −3x + 8 when x =−4.
−4 x,
2·−4 2 −3·−4+8.
3x 2 − 2x + 6 when x = −3
4x 2 − x − 5 when x = −2
Evaluate 3x +4y − 6 when x = −1 and y =2.

x =−1 y = 2
7x + 6y − 12 when x = −2 and y = 3
8x − 6y + 13 when x = −3 and y = −5
−2 14.
−2 14
(−2)(14)
−28
the product of −5 and 12
the product of 8 and −13
−56 −7.

−56 ÷ (−7)
8
the quotient of −63 and −9
the quotient of −72 and −9

−4 · 8 −3 · 9 −5(7)
−8(6) −18(−2) −10(−6)
9(−7) 13(−5) −1 · 6
−1 · 3 −1(−14) −1(−19)
−24 ÷ 6 −28 ÷ 7 56 ÷ (−7)
35 ÷ (−7) −52 ÷ (−4) −84 ÷ (−6)
−180 ÷ 15 −192 ÷ 12 49 ÷ (−1)
62 ÷ (−1)
5(−6) + 7(−2)−3 8(−4) + 5(−4)−6 −8(−2)−3(−9)
−7(−4)−5(−3) (−5) 3 (−4) 3
(−2) 6 (−3) 5 −4 2
−6 2 −3(−5)(6) −4(−6)(3)
−4 · 2 · 11 −5·3·10 (8 − 11)(9 − 12)
(6 − 11)(8 − 13) 26−3(2−7) 23−2(4−6)
−10(−4) ÷ (−8) −8(−6) ÷ (−4) 65 ÷ (−5) + (−28) ÷ (−7)
52 ÷ (−4) + (−32) ÷ (−8) 9 − 2[3 − 8(−2)] 11 − 3[7 − 4(−2)]
(−3) 2 −24 ÷ (8 − 2)
(−4) 2 −32÷(12−4)

−5y + 14 when
−2x + 17 when
a x =8 b x =−8
a y =9 b y =−9
10 − 3m when
a m =5 b m =−5
18 − 4n when
a n =3 b n =−3
2w 2 −3w +7
w =−2
p 2 −5p + 5 when p =−1 q 2 −2q +9 q =−2
3u 2 −4u +5 u =−3 6x −5y +15 x =3
y =−1
3p −2q +9 p =8
q =−2
9a −2b −8 a =−6
b =−3
7m −4n −2 m =−4
n =−9
−3 −4 16 −60
−20
−20
−40
−6
a b
−7
m n
−10
p and q
−13
c and d
300
$12
3 pounds
−2 4 ≠ (−2) 4 ? −4 2 ≠ (−4) 2 ?

a
b

Evaluate x + 4 when x = −4.
Solve: y − 6 = 10.
5 x.
2x − 5 = −13:
a x =4 b x =−4 c x =−9.

a
x =4 4
b
x =−4 −4
c
x =−9 −9
2x − 8 = −14:
a x = −11 b x = 11 c x = −3
2y + 3 = −11:
a y = 4 b y = −4 c y = −7

For any numbers a, b, c,
if a = b then a − c = b − c.
For any numbers a, b, c,
if a = b then a + c = b + c.
y +9=5.
−4
y +9=5
−4+9= ? 5
5=5✓
y =−4 y +9=5
y + 11 = 7
y + 15 = −4
a −6=−8

−2
−2 a
a −6=−8 −2.
a =−2 a −6=−8
a − 2 = −8
n − 4 = −8
x + a = b x − a = b.
2 6
2 3 6÷2=3).
x
6 2x =6.

2
3 counters. 2·3=6,
3 12 counters.
12 counters 3 groups. 12 ÷ 3 = 4, 4 counters
3x = 12. 3.
3 · 4 = 12.

4 envelopes, 4 8 counters
x.
2 counters
For any numbers a, b, c, and c ≠0,
If a = b then a c = b c .

Solve: 7x = −49.
x,
7x = −49
7(−7) = ? −49
−49 = −49 ✓
−7
8a = 56
11n = 121
−3y = 63.
y,
−3y =63
−3(−21) = ? 63
63 = 63 ✓

y = −21
−8p =96
−12m = 108
x −3.
x −3
5
x +5=−3
x +5−5=−3−5
x =−8
x +5=−3
−8+5= ? −3
−3 = −3 ✓
x −2
Eleven more than y is equal to 2.
n 6 −10.

n 6 −10
6
n − 6 = −10
n −6+6=−10+6
n =−4
n − 6 = −10
−4−6= ? −10
−10 = −10 ✓
p 2 −4
q 7 −3
108 −9 y.
108 −9 y
108 = −9y
−9 108
−9 = −9y
−9
−12 = y
108 = −9y
108 = ? −9(−12)
108 = 108 ✓
132 −12 y
117 −13 z

4y − 10 = −14
4x −2=6
a x =−2 b x =−1
c x =2
a y =−6 b y =−1
c y =1
9a + 27 = −63
a a =6 b a =−6
c a = −10
7c + 42 = −56
a c =2 b c =−2
c c = −14
n +12=5 m +16=2 p +9=−8
q +5=−6 u −3=−7 v −7=−8
h −10=−4 k −9=−5 x + (−2) = −18
y + (−3) = −10 r −(−5)=−9 s − (−2) = −11

5x =45 4p =64 −7c =56
−9x =54 −14p = −42 −8m = −40
−120 = 10q −75=15y 24x = 480
18n = 540 −3z =0 4u =0
n
m
−3
p
q −7
a
−14
b 5
−2
x
y
f
g
c
d
m
n −10
a x +2=10
b 2x =10
a y +6=12
b 6y =12
a −3p =27
b p −3=27
a −2q =34
a −4=16 b −1=11
b q −2=34
−8m = −56 −6n = −48 −39 = u +13
−100 = v +25 11r = −99 15s = −300
100 = 20d 250 = 25n −49 = x −7
64 = y −4
51 cookies 3
c, 3c = 51.

24 children. 4
g, 4g = 24.
3x =15?
−3x =36
3
x +4=12
4x = 12.
4y =40
4
a
b

−a a
−a a.
n |n|
|n| ≥0
5+3 −5 + (−3)
−5 + 3
5+(−3)

5–3 –5 – (–3)
2 –2
–5–3 5–(–3)
–8 8
a − b = a +(−b)
a − (−b) = a + b

−1
−1 −1a =−a
−1
−1 a ÷ (−1) = −a
For any numbers a, b, c,
if a = b then a − c = b − c.
For any numbers a, b, c,
if a = b then a + c = b + c.
a, b, c, c ≠0
a = b a c = b c
5 −5 −3
3 −8 −7
< >.
4__8 −6__3 −5__−10
−9__−4 2__−7 −3__1

6 −2 −4
3
a −(8)
b −(−8)
a −(9)
b −(−9)
−x, when
−n, when
a x =32
b x = −32
a n =20
a n = −20
|−21| |−42| |36|
−|15| |0| −|−75|
|x| when x = −14 −|r| when r =27 −|−y| when y =33
|−n| when n =−4
, or =
−|−4|__4 −2__|−2| −|−6|__−6
−|−9|__|−9|
−(−55) and − |−55| −(−48) and − |−48| |12 − 5|
|9+7| 6|−9| |14−8| − |−2|
|9−3| − |5−12| 5+4|15 − 3|
16 −8 3
19 12 10
85 feet

3+7 −2+6 5 + (−4)
−3 + (−6)
14 + 82 −33 + (−67) −75 + 25
54 + (−28) 11 + (−15) + 3 −19 + (−42) + 12
−3+6(−1+5) 10+4(−3+7)
x + (−9) when
n + 4 when
a n =−1
b n = −20
a x =3
b x =−3
(x + y) 3 when x =−4,y =1
(u + v) 2 when u =−4,v =11
the sum of −8 and 2 4 more than −12
10 more than the sum of −5 and −6
the sum of 3 and −5, increased by 18
−4 degrees.
20 degrees
$75
$21
6−1 −4 − (−3) 2 − (−5)
−1 − 4

24 − 16 19 − (−9) −31 − 7
−40 − (−11) −52 − (−17) − 23 25−(−3−9)
(1 − 7) − (3 − 8) 3 2 −7 2
x − 7 when
a x =5
b x =−4
10 − y when
a y =15
b y = −16
2n 2 − n + 5 when n =−4
−15−3u 2 when u =−5
−12 and 5 23 −50
18 degrees. 78 degrees,
20 degrees.
−28 degrees.
−9⋅4 5(−7) (−11)(−11)
−1⋅6
56 ÷ (−8) −120 ÷ (−6) −96÷12
96 ÷ (−16) 45 ÷ (−1) −162 ÷ (−1)
5(−9) − 3(−12) (−2) 5 −3 4

(−3)(4)(−5)(−6) 42 − 4(6 − 9) (8 − 15)(9 − 3)
−2(−18) ÷ 9 45 ÷ (−3) − 12
7x − 3 when x =−9 16 − 2n when n =−8
5a +8b when a =−2,b =−6
x 2 +5x + 4 when x =−3
−12 6 3
−7 s
5x − 10 = −35
a x =−9
b x =−5 c x =5
8u + 24 = −32
a u =−7
b u =−1 c u =7
a +14=2 b − 9 = −15 c + (−10) = −17
d − (−6) = −26
8p =72 −12q =48 −16r = −64
−5s = −100

The product of −6 and y is −42
m −48.
The difference of z and −13 is −18.
The product of −21 and n is 63.

0, 2, −4,
−1
< or > or =.
a −6__3
b −1__−4
a −5__|−5|
b −|−2|__−2
a −7 b 8
−(−22) |4−9| −8 + 6
−15 + (−12) −7 − (−3) 10 − (5 − 6)
−3 ⋅ 8 −6(−9) 70 ÷ (−7)
(−2) 3 −4 2 16−3(5−7)
|21 − 6| − |−8|
35 − a when a =−4 (−2r) 2 when r =3
3m −2n when m =6, n =−8
−|−y| when y =17
25
m n.
−8°F. 12°.
$128
$65.
n +6=5 p −11=−4 −9r = −54
Eight less than y is −32.
The product of 15 and x is 75.

5 · 2 + 1.

:−2__−5
1 2
1 4
12
1
12
a b ,
a b b ≠0. a b
b
a b,

1
3
2 3 2 3
a
How many equal parts are there?
How many are shaded?
There are eight equal parts.
Five parts are shaded.
5 8 .
b
How many equal parts are there?
How many are shaded?
There are nine equal parts.
Two parts are shaded.

2 9 .
3 4
4, a
3, b
3
4
6 8

2 5
1 2
2 2 =1.
1 3
1 4
1 6
3
3 =1.
4
4 =1.
6 6 =1.
24
24
1
24 24
24 =1.
24 24
24 =1.

a =1 (a ≠0)
a 4 b 5 c 6
3
8
1
a 3 b 8 c 7
a 3 1 1
b 8 1 3
c 7 2 1

5
5
1,
5
8 5 . 8 5 1, 3, 5
1 3 ,
5
1 3 5
a b c
c ≠0.
a b c
c ≠0
5 4 , 3 2 , 5 5 ,
7 3
1 2 , 3 , 11
7 18
a b
a < b a ≥ b.

1 3
4 3 . 11 3 . 4
3
=1 1 3 .
11.
8
8.
11
11
8
=1 3 8 .
7
6
.
6
5
.
11 6
11 ⎛ ⎝ 11 6
⎞
⎠ .

5 6
11
6
=1 5 6 .
6
6 =1
⎛
⎝ 11 sixths minus 6 sixths is 5 sixths⎞ ⎠ .
9 7 .
7 4 .
1 4 5
1 4 5
5,
5 .
5
1 4 5 = 9 5 .
1 3 8 .
1 5 6 .
11 6
15 6
11
11
6
11 6 =15 6 .

11 611 6 11.
6
11 6
11
6
11 6
11 ÷ 6
quotient remainder 1
divisor
5 6
11
6
=1 5 6
13 7 .
14 9 .
remainder
divisor
33 8

33
8
33
8
833
remainder
divisor
4 1 8
33
8
=4 1 8
23 7 .
48
11 .
1 4 5
5
5 + 4 5 = 9 5
4 2 3

4 2 3
14
3
3 5 7 .
2 7 8 .
10 2 7

10 2 7
72
7
4 6 11 .
11 1 3 .
1 2 2 4
1 2 = 2 ?
4
1 2 1 4 1 2
1 4 1 2 2 4 1 2 , 2 4 = 1 2 .
1 6 1 2

1 6 1 2 3 6 1 2 .
3 = 1 .
6 2
a
b
c
a 1 8 1 2 4 8 = 1 2 .
b 1 10 1 2 5 10 = 1 2 .
c 1 12 1 2 6 12 = 1 2 .
1 20 . 1
2
?
10
20 = 1 2 .
1 2 , 2 4 , 3 6 , 4 8 , 5
10
, 6
12
, 10
20
1 2 . 2
4
, 3 6 , 4 8

1 2 . 1 8 1
2
4 8 = 1 .
2
1 2
4 8
1 2 4 8 .
1 2 4 ?
8
a, b, c b ≠0 c ≠0,
a
b = a b · · c
c
1 2 , 2 4 , 3 , 10
6 20
2 5 .

2 5 ,
2, 3, 5.
4
10
, 6
15
, 10
25 2 5 .
3 5 .
4 5 .
21 2 7 .
21.
7 3 21,
3.
21 6 7 .
100 3 10 .
3
1 5 , 4 5 ,3,31 3 , 7 4 , 9 2 ,5, 8 3
3 5
1 5 4 5 . 1 5 4 5
0 1. 5,

0 1
1 5 , 2 5 , 3 5 , 4 5 .
1 5 4 5 .
3 1 3 . 31 ?
3
3 1 3 >3. 3, 31 3
3 4. 3 4
3 1 3
7 4 , 9 2 , 8 .
3
7
4 =13 4 , 9 =4 1 2 2 , 8 =2 2 3 3
3
4
, 4 3 , 5 3 ,41 5 , 7 2 .
3 . 0 1. 0 1
4
3 4 .
4 1 . 4 5 4
5
5 4 1 5
4 5
4 3 5 3
1 2
4
3 =11 3 , 5
3
=1 2 3

7 2 . 7 =3 1 3 4.
2 2
1
3
, 5 4 , 7 4 , 23 5 , 9 2 .
2
3
, 5 2 , 9 4 , 11 4 , 32 5 .
7 −7
−7 7.
3 4 −3 . 0
4
0.
− 15 8 15 8
1 7 8 1 2 −15 8 ,
−1
−2
1
4
, − 1 4 ,11 3 ,−11 3 , 5 2 , −5 2 .

0
1 , 0 1
4
1 4
− 1 4 , 0 −1 −1 4
0.
1 1 3 1 2, 1 2 11 3
1. −1 1 3 11 3
−1 −2. −1
−2 −1 1 3
−1.
5 2 −5 2 , 21 2 −21 2 .
2 1 2
2 3,
2 3 5 2
−2 1 2
−2 −3,
−2 −3 −5 2
2
3 , − 2 3 , 21 4 , −21 4 , 3 2 , − 3 2
3
4 , − 3 4 , 11 2 , −11 2 , 7 3 , − 7 3
a > b a b
< >:
a − 2 3 ____−1 b −31 2 ____−3 c −3 7 ____ − 3 8
d −2____ −16
9

a − 2 3 >−1
b −3 1 2 :
a −3__ − 17 5
b −2 1 4 __−2 c −3 5 __ − 4 5
d −4__ − 10 3

1
2
1
3
3
4
2
5
5
6
7
8
5
8
7
10
3 8 7
4 7 7

3
3
4
4
7
4
5
3
11
6
13
8
10
3
9
4
3
2
5
3
11
4
13
5
25
6
28
9
42
13
47
15
1 2 3
1 2 5
2 1 4
2 5 6
2 7 9
2 5 7
3 4 7
3 5 9

1
4
1
3
3
8
5
6
2
7
5
9
2
3
, 5 4 , 12 5
1
3
, 7 4 , 13 5
1
4
, 9 5 , 11 3
7
10
, 5 2 , 13 8 ,3 21 3 ,−21 3
1 3 4 ,−13 5
3
4
, − 3 4 ,12 3 ,−12 3 , 5 2 , − 5 2
2
5
, − 2 5 ,13 4 ,−13 4 , 8 3 , − 8 3
< >.
−1__ − 1 4
−1__ − 1 3
−2 1 2 __ − 3
−1 3 4 __ − 2 − 5 12 __ − 7 12
− 9 10 __ − 3 10
−3__ − 13 5
−4__ − 23 6
1 1 1
2
4
4
1
3
a
1
5 b 25
1
2
a
b

21 4
0 10
a
b

48.
3 4 .
5 6 .
1,
2
3
2 3.
10 5 10 15.
15
a, b, c b ≠0,c ≠0,
a
b = a b · · c
c
and
a · c
b · c = a b .
c

10
15 .
10
15
2
3
8
12
12
16
− 18
24 .
− 18
24
− 3 4

− 21
28 .
− 16
24 .
− 56
32 .
− 56
32
− 7 4
− 54
42 .
− 81
45 .
210
385 .

210
385
210
385 = 2⋅3⋅5⋅7
5⋅7⋅11
2⋅3
11
6
11
69
120 .
120
192 .
5xy
15x .
5xy
15x
5·x · y
3·5·x
5 · x · y
3·5· x
y
3
7x
7y .

9a
9b .
1
2 · 34 . 1 2
3
4 , 1 2 3 4 .
1 4
6 8 3 4 . 1 8 1 8 3 8 .
1
2 · 34 = 3 8
1 · 34 . 2
3 4
1 2 3 4 , 1 2
3 8 3 8
1
2
3 4 3 8 , 1 2 · 34 = 3 8 .
1
2 · 35 .

1
2 · 56 .
1 2 · 34 = 3 .
8
1
2 · 34
1
2 · 34
3
8
a, b, c, and d b ≠ 0 and d ≠0,
a · = ac
b
cd bd
3
4 · 15 .
3
4 · 15
3·1
4·5
3
20
1
3 · 25 .
3
5 · 78 .
− 5 ⎛
8⎝ −2 ⎞
3⎠ .

− 5 ⎛
8⎝ −2 ⎞
3⎠
5⋅2
8⋅3
10
24
5
12
− 5 ⎛
8⎝ −2 ⎞
3⎠
5⋅2
8⋅3
5
12
− 4 ⎛
7⎝
− 5⎞ 8⎠ .
−
12
7 ⎛
⎝ − 8⎞ 9⎠ .
− 14
15 · 20
21 .

− 14
15 · 20
21
− 14
15 · 20
21
− 2·4
3·3
− 8 9
− 10
28 · 815 .
− 9 20 · 512 .
a,
a 1 . 3=3 1 ,
a 1 7 ·56
b 12 5 (−20x)
a
1
7 ·56
1
7 · 56 1
56
7
8

12
5
(−20x)
12⎛−20x⎞
5 ⎝ 1 ⎠
−12 · 20 · x
5·1
a 1 8 · 72
b 11 3 (−9a)
a 3 · 64 b 16x · 11
8 12
2 3 3 2 −10 7
− 10 7 .
1.
2
3 · 32 = 1 and − 10 ⎛
7 ⎝ − 10
7 ⎞
⎠ =1
a b b a , a ≠0 b ≠0,
1.
a ·
b
ba =1
1. r 0·r =1? 0

1.
a 4 9
b − 1 6
c − 14 5
d 7
a
4 9
4 9 9 4
4
9 ⋅ 9 4
36
36
1✓
b
- 1 6 -6 1
-6
- 1 6 ⋅ ⎛ ⎝ -6⎞ ⎠
1✓
c
- 14 5 - 5 14
- 14 5 ⋅ ⎛ ⎝ - 14
5 ⎞
⎠
70
70
1✓

d
7
7
7
1
7 1 1
7
7⋅ ⎛ ⎝ 1 ⎞
7⎠
1✓
a 5 7
b − 1 8
c − 11 4
d 14
a 3 7
b − 1
12
c − 14
9
d 21
− 3 8
1
2
9
5
−5

− 3 8
3
8
3
8
− 8 3
1
2
− 1 2
1
2
2
9
5
− 9 5
9
5
5
9
−5 5 5 − 1 5
− 5 8
1
4
8
3
−8
− 4 7
1
8
9
4
−1
12 ÷ 3 = 4? 3
12
4 3 3s 12. 12÷3=4.
1 2 ÷ 1 6 . 1 6 s

1 .
2
1 6 1 2 , 1 2 ÷ 1 6 =3.
1 4 ÷ 1 8 .
1 8 s 1 4 . 1 4 1 8 1 4
1 8 s 1 4 .
1
4
÷ 1 8 =2.
1 3 ÷ 1 6 .
1 2 ÷ 1 4 .
2÷ 1 4 .
1 s 2.
4
1 4 s 2, 2 ÷ 1 4 =8.
2 ÷ 1 3

3 ÷ 1 2
2÷ 1 4 1 4
2÷ 1 4
4 2 8 2÷ 1 4 =8.
1
2
÷ 1 6 =3. 1
2 · 61 =3 1 6 6 1
a, b, c, and d b ≠0,c ≠ 0, and d ≠0,
a
b ÷ d c = a b · dc
b ≠0,c ≠ 0 and d ≠0
2 ÷ ⎛ 5 ⎝ −3 ⎞
7⎠ .
2
5 ÷ ⎛ ⎝ −3 ⎞
7⎠
2⎛
5⎝ − 7 ⎞
3⎠
− 14
15
3
7
÷ ⎛ ⎝ ⎛ − 2
3
⎞
⎠
⎞ .
2
3
÷ ⎛ ⎝ ⎛ − 7
5
⎞
⎠
⎞ .
2
3
÷ n 5 .

2
3 ÷ n 5
2
3 · 5n
10
3n
3
5
÷ p 7 .
5
8
÷ q 3 .
− 3 4 ÷ ⎛ ⎝ −7 ⎞
8⎠ .
− 3 4 ÷ ⎛ ⎝ − 7 ⎞
8⎠
− 3 4 · ⎛⎝ − 8 ⎞
7⎠
3·8
4·7
3·4·2
4 ·7
6
7
− 2 3 ÷ ⎛ ⎝ ⎛ − 5
6
⎞
⎠
⎞ .
− 5 6 ÷ ⎛ ⎝ ⎛ − 2
3
⎞
⎠
⎞ .
7
18
÷ 14
27 .

7
18 ÷ 14
27
7
18 · 27
14
7·27
18 · 14
3
2·2
3
4
7
27
÷ 35
36 .
5
14
÷ 15
28 .

7 8 15
21 24 20
12
18
− 108
63
182
294
11x
11y
− 40
88
− 104
48
− 168
192
15a
15b
− 63
99
120
252
− 140
224
− 3x
12y
− 4x
32y 14x 2
21y
24a
32b 2
1
2 · 23
1
2 · 58
1
3 · 56
1
3 · 25
2
5 · 13
1
2 · 38
3
4 · 910
4
5 · 27
− 2 ⎛
3⎝ −3 ⎞
8⎠ − 3 ⎛
4⎝ −4 ⎞
9⎠
− 5 9 · 310
− 3 8 · 415
7 ⎛
12⎝ −
21
8 ⎞
⎠
5 ⎛
12⎝ −
15
8 ⎞
⎠
⎛ ⎝ −63 ⎞ ⎛
84⎠⎝ −44 ⎞
90⎠
⎛ ⎝ −14 ⎞
15⎠ ⎛ ⎝20
9 ⎞
⎠
⎛ ⎝ −33 ⎞
60⎠ ⎛ ⎝ −40 ⎞
88⎠
⎛ ⎝ − 10
9 ⎞
⎠ ⎛ ⎝ 25 ⎞
33⎠
4· 511
5·83
3
7 ·21n 5
6 ·30m
−28p ⎛ ⎝ −1 ⎞
4⎠ −51q ⎛ ⎝ −1 ⎞
3⎠
−8 ⎛ ⎝ 17 4
⎞
⎠

14(−15) −1 ⎛ 5 ⎝ −3 ⎞
8⎠
(−1) ⎛ ⎝ −6 ⎞
7⎠
⎛ ⎝ 2 3
⎞
3⎠
⎛ ⎝ 4 2
⎞
5⎠
⎛ ⎝ 6 4
⎞
5⎠
⎛ ⎝ 4 4
⎞
7⎠
3
4
2
3
− 5 17
− 6
19
11
8
−13
−19 −1 1
− 7 11
− 3 13
4
5
9
14
10
7
−8
15
7
−9
1
2
÷ 1 4
1
2
÷ 1 8
2÷ 1 5
3÷ 1 4
1
2
÷ 1 4
1
2
÷ 1 8
3
4
÷ 2 3
4
5
÷ 3 4
− 4 5 ÷ 4 7
− 3 4 ÷ 3 5
− 7 9 ÷ ⎛ ⎝ −7 ⎞
9⎠ − 5 6 ÷ ⎛ ⎝ −5 ⎞
6⎠
3
4
÷ x
11

2
5
÷ y 9
5
8
÷ a 10
5
6
÷ c
15
5 ÷ ⎛ 18 ⎝ −15 ⎞
24⎠ 7 ÷ ⎛ 18 ⎝ −14 ⎞
27⎠
7p
12 ÷ 21p
8
5q
12 ÷ 15q
8
8u
15 ÷ 12v
25
12r
25 ÷ 18s
35
−5 ÷ 1 2
−3 ÷ 1 4
3
4
÷ (−12)
2 ÷ (−10) −18 ÷ ⎛ 5 ⎝ −9 ⎞
2⎠ −15 ÷ ⎛ ⎝ −5 ⎞
3⎠
1 ÷ ⎛ 2 ⎝ −3 ⎞
4⎠ ÷ 7 8
11
2
÷ 7 8 · 211
3
4
a
b
1
8 , 1 4 , 1 3 , 1 , and 1
2
4
2
3
a
b
1
8 , 1 4 , 1 3 , 1 , and 1
2
5
1
4
3
4
6
6 8 3
6 4 8
3
6
1 2 · 23 is 1 3 .

a
b

(4 + 5) ÷ (10 − 7).
1
8 · 23
2 3 5
3 1 3 · 58
3 1 3 · 58
3 1 3 10
3 · 58
10 · 5
3·8
2̸ ·5·5
3·2̸ ·4
5·5
3·4
25
12

25,
12
2
12 1 2· . 112
5 2 3 · 617 .
3
7 · 5 1 4 .
2 4 ⎛
5 ⎝ −17 ⎞
8⎠ .
2 4 ⎛
5 ⎝ −17 ⎞
8⎠
14
5
⎛
⎝ − 15 8
⎞
⎠
− 14·15
5·8
−
− 7·3
4
− 21
4
2̸ ·7·5̸ ·3
5̸ ·2̸ ·4
5 5 7
⎛
⎝ − 2 5⎞ 8⎠ .
−3 2 5 · 41 6 .

3 4 7 ÷5.
3 4 7 ÷5
25
7
÷ 5 1
25
7 · 15
25 · 1
7·5
5̸ ·5·1
7·5̸
5·1
7
5
7
4 3 8 ÷ 7.
2 5 8 ÷ 3.
2 1 2 ÷11 4 .
2 1 2 ÷11 4
5
2 ÷ 5 4
5
2 · 45
5·4
2·5
5̸ ·2̸ ·2
2̸ ·1·5̸
2
1
2

2 2 3 ÷ 11 3 .
3 3 4 ÷ 11 2 .
a and b a by b , a .
b
3x 8.”
The quotient of 3x and 8.
3x 8. 3x
8
9s 14.
5y 6.
m n, p.
m p. m
n p.
m − n
p
a b, cd.
p q, r.
6
7
3
3
4
5
8
x
2
5
6

3 4 ÷ 5 8 .
3
4
5
8
3
4 .
5
8
3
4
5
8
3
4 ÷ 5 8
3
4 · 85
3·8
4·5
3·4̸ ·2
4̸ ·5
6
5
2
3 .
5
6
3
7 .
6
11
− 6 7
3 .

− 6 7 ÷3
− 6 7
3
− 6 7 · 13
− 6·1
7·3
3̸ ·2·1
−
7·3̸
− 2 7
− 8 7
4 .
− 3 .
9
10
x
2
xy .
6
x
2
xy
6
x
2 ÷ xy
6
x
2 · 6xy
x ·6
2·xy
x̸ ·3·2̸
2̸ ·x̸ · y
3 y

a
8 .
ab
6
p
2
pq .
8
2 3 4 .
1
8
2 3 4
1
8
2 3 4 ÷ 1 8
11
4
÷ 1 8
11
4 · 81
11 · 8
4·1
11 · 4̸ ·2
4̸ ·1
22
5
7
1 2 .
5
8
5
3 1 .
5
− 1 −1
3 3 , −3 1 ,

−1
−3 = 1 3
negative
negative = positive
a and b,
−a
b
= a
−b =−a b
7 −8 ?
−7
−8 , −7
8 , 7 8 , − 7 8
−8 7 −7 and − 7 8 8
−3 5 ?
−3
−5 , 3, − 3 5 5 , 3
−5
− 2 7 ?
−2
−7 , −2
7 , 2 7 , 2
−7
4+8 (4 + 8) ÷ (5 − 3).
5−3

4+8
5−3 .
4+8
5−3
12
5−3
12
2
4 + 6
11 − 2 .
3 + 5
18 − 2 .
4 − 2(3)
2 2 +2 .
4 − 2(3)
2 2 +2
4−6
4+2
−2
6
- 1 3
6 − 3(5)
3 2 + 3 .
4 − 4(6)
3 3 + 3 .

(8 − 4)2
8 2 −4 2 .
(8 − 4) 2
8 2 −4 2
(4) 2
64 − 16
16
48
1
3
(11 − 7)2
11 2 − 7 2 .
(6 + 2)2
6 2 + 2 2 .
4(−3) + 6(−2)
.
−3(2)−2
4(−3) + 6(−2)
−3(2)−2
−12 + (−12)
−6 − 2
−24
−8
3
8(−2) + 4(−3)
−5(2) + 3 .
7(−1) + 9(−3)
.
−5(3)−2

4 3 8 · 710
2 4 9 · 67
15
22 ·33 5
25
36 ·6 3 10
4 2 3 (−11 8 ) 22 5 (−22 9 )
−4 4 9 ·513 16
−1 7 20 ·211 12
5 1 3 ÷4 131 2 ÷9 −12 ÷ 3 3 11
−7 ÷ 5 1 4
6 3 8 ÷21 8
2 1 5 ÷11 10
−9 3 5 ÷(−13 5 ) −183 4 ÷(−33 4 )
5u 11 7v 13 p q
a b r
s 10
A
3 B
2
3
8
9
4
5
8
15
− 8 21
12
35
− 9 16
33
40
− 4 5
2
− 9 10
3
2
5
8
5
3
10
m
3
n
2
r
5 s
3
− x 6
− 8 9
− 3 8
− y
12
2 4 5
1
10
4 2 3
1
6
7
9
−2 4 5

3
8
−6 3 4
5
−11 ?
−5
−11 , −5
11 , 5 , − 5 11 11
−4
9 ?
−4
−9 , −4
9 , 4 9 , − 4 9
− 11 3 ?
−11
3 , 11 3 , −11
−3 , 11
−3
− 13 6 ?
13, 13
6 −6 , −13
−6 , −13
6
4+11
8
19 − 4
6
−6 + 6
8+4
15 + 9
18 + 12
4⋅3
6⋅6
7 2 +1
60
15 ⋅ 5 − 5 2
2⋅10
9+3
7
48
24 − 15
−6+3
17 − 8
5⋅8
−10
6⋅6
9⋅2
8⋅3+2⋅9
14 + 3
12 ⋅ 9 − 3 2
3⋅18
22 + 3
10
46
4+4
22 − 14
19 − 13
3⋅4
−24
4 2 −1
25
9⋅6−4⋅7
22 + 3
5⋅6−3⋅4
4⋅5−2⋅3
8⋅9−7⋅6
5⋅6−9⋅2 5 2 −3 2
3−5
6 2 −4 2
4−6
2 + 4(3)
−3 − 2 2
7 + 3(5)
−2−3 2
7⋅4−2(8−5)
9⋅3−3⋅5
9 ⋅ 7 − 3(12 − 8)
8⋅7−6⋅6
9(8 − 2)−3(15 − 7)
6(7 − 1)−3(17 − 9)
8(9 − 2)−4(14 − 9)
7(8 − 3)−3(16 − 9)

2 1 4
a
b
1
8 , 1 4 , 1 3 , 1 , and 1
2
2 2
3
a
b
10
10
3 1 2 ·51 4 151 .
8
− 1 2 , −1
2 , 1 −2
a
b

2x + 9 + 3x − 4.
3 4 .
3 + 2
6 .
2 3
1
2
3
4
4
4
one quarter + two quarters = three quarters
1
4
+ 2 4 .

1 4
1 4
3 4
1
4 + 2 4 = 3 4
3 8 + 2 8 .

1 8
1 8
1 8
1 8 3 8 + 2 8 = 5 8 .
1
8 + 4 8
1
6 + 4 6
a, b, and c c ≠0,
a c + b c = a + c
b
3
5
+ 1 5 .

3
5 + 1 5
3+1
5
4
5
3
6
+ 2 6 .
3
10
+ 7 10 .
x
3
+ 2 3 .
x
3 + 2 3
x +2
3
x and 2
x
4
+ 3 4 .
y
8 + 5 8 .
− 9 d + 3 d .
− a b = −a
b

− 9 d + 3 d
−9
d
+ 3 d
−9 + 3
d
−6
d
− 6 d
− 7 d + 8 d .
− 6 m + 9 m .
2n
11 + 5n
11 .
2n
11 + 5n
11
2n +5n
11
7n
11
3p
8 + 6p
8 .
2q
5 + 7q 5 .
−
12 3 + ⎛ ⎝ − 12
5 ⎞
⎠ .

−
12 3 + ⎛ ⎝ − 12
5 ⎞
⎠
−3 + (−5)
12
−8
12
− 2 3
− 4 15 + ⎛ ⎝ ⎛ − 6
15
⎞
⎠
⎞ .
− 5 21 + ⎛ ⎝ ⎛ − 9
21
⎞
⎠
⎞ .
12
7
12
2 2 5
12
5
12
7
12 − 2 12 = 5 12
7
12
− 2 12 .
1 12 1 12
5
12
.

4 5 − 1 5 .
1 5 1 5 1 5
7
8 − 4 8
5
6 − 4 6
a, b, and c c ≠0,
a c − b c = a − c
b
23
24 − 14
24 .
23
24 − 14
24
23 − 14
24
9
24
3
8
19
28 − 7 28 .

27
32 − 11
32 .
y
6 − 1 6 .
y
6 − 1 6
y −1
6
x
7
− 2 7 .
y
14 − 13
14 .
− 10 x − 4 x .
− 10 x
−10
x .
− 10 x − 4 x
−10−4
x
−14
x
− 14 x
− 9 x − 7 x .
− 17 a − 5 a .
3 8 + ⎛ ⎝ −5 ⎞
8⎠ − 1 8 .

3
8 + ⎛ ⎝ −5 ⎞
8⎠ − 1 8
3+(−5) −1
8
−2−1
8
−3
8
− 3 8
2 5 + ⎛ ⎝ ⎛ − 4
5
⎞
⎠
⎞ − 3
5
.
5
9
+ ⎛ ⎝ ⎛ − 4
9
⎞
⎠
⎞ − 7
9
.

2
5
+ 1 5
3
10
+ 4 10
1
6
+ 3 6
3
8
+ 3 8
4
9
+ 1 9
2
9
+ 5 9
6
13
+ 7 13
9
15
+ 7 15
x
4
+ 3 4
y
3 + 2 3
7 p + 9 p 8 q + 6 q 8b
9
+ 3b 9
5a
7
+ 4a 7
−12y
8
+ 3y
8
−11x
5
+ 7x
5
− 1 8 + ⎛ ⎝ −3 ⎞
8⎠ − 1 8 + ⎛ ⎝ −5 ⎞
8⎠ −
16 3 + ⎛ ⎝ − 16
7 ⎞
⎠
−
16 5 + ⎛ ⎝ − 16
9 ⎞
⎠
− 8 17 + 15
17
− 9 19 + 17
19
6 + ⎛ 13 ⎝ −10 ⎞
13⎠ + ⎛ ⎝ −12 ⎞
13⎠
5 + ⎛ 12 ⎝ − 12
7 ⎞
⎠ + ⎛ ⎝ −11 ⎞
12⎠
5
8
− 2 8
5
6
− 2 6
4 − 1 4 − 5 5
3 5 5
11
15 − 7 15
9
13
− 4 13
4
21
− 19
21
11
12 − 5 12
− 8 9 − 16 9
7
12
− 5 12
y
17 − 9 17
x
19
− 8 19
5y
8 − 7 8
11z
13 − 8 13

− 8 d − 3 d
− 7 c − 7 c
− 23 u − 15 u
− 29 v − 26 v
−4r
13 − 5r
13
6c − 5c
7 7
−7s
3 − 7s
3
12d
11 − 9d
11
− 3 5 − ⎛ ⎝ −4 ⎞
5⎠
− 3 7 − ⎛ ⎝ −5 ⎞
7⎠ − 7 9 − ⎛ ⎝ −5 ⎞
9⎠
−
11 8 − ⎛ ⎝ − 11
5 ⎞
⎠
− 5 18 · 910
− 3 14 · 712
n
5
− 4 5
6
11
− s
11
− 7 24 + 2 24
− 5 18 + 1 18
8
15
÷ 12 5
7
12
÷ 9 28
6
10
3
10
5
8
3
8
1 1 5 3 1 9 5
16 8 16 8 2 16 8
5 12 4
12
12 4
1
7
16
3
16
1
4

a
b

5 6 .
1 + 5 · 3
2 2 + 4 .
25 10 35
35
35
100
100. 100
25 25
100
10 10 . 25
100 100 + 100 10 35 , 35
100
1 2 1 3 .
1 2 1 3 1 2 1 3
1 4 2 1 2 1 3

1 5 1 2 1 3
1 6 3 1 2 2 1 3
1 12
1 24 1 48 , 1 2 1 3
1 2 1 6.
3
1 2 1 3
2 3, 1 2 1 . 6,
3
6 1 2 1 3 .
7 12 5 18 .

LCM = 36
7
12 5 18 7
12
5 18
7
12
11
15 .
13
15 17 5 .
15 8
15 24.
11
24 .

15 24 120. 8
15
11
24
120.
13
24
17
32 .
9
28
21
32 .
1 4 and 1 6
12. 1
12 1 4
1 12 1 6 , 1
4
= 3 12 and1 6 = 2 12 .
1 4 and 3 12 1 6 and 2 12
a, b, c b ≠0,c ≠ 0, then
a
b = a b · · c
c
and
a · c
b · c = a b
1 4 and 1 6
12
1 4 and 1 6
12,

1 4 1 6
3
4 5 , LCD =12
6
−
12 7 11, LCD = 60
15
8 11 120,
15 24

13
24
17, 96
32
9
28
27, 224
32
1 2 + 1 3 .

1
2 + 1 3
3
6 + 2 6
5
6
5 6 5 6
1 4 + 1 3 .
1 2 + 1 5 .
1 2 − ⎛ ⎝ −1 ⎞
4⎠ .
1
2 − ⎛ ⎝ − 1 ⎞
4⎠
2
4 − ⎛ ⎝ − 1 ⎞
4⎠
2−(−1)
4
3
4

1 2 − ⎛ ⎝ ⎛ − 1
8
⎞
⎠
⎞ .
1 3 − ⎛ ⎝ ⎛ − 1
6
⎞
⎠
⎞ .
7
12
+ 5 18 .
7
12 + 5 18
21
36 + 10
36
31
36
31 36.
7
12
+ 11
15 .
13
15 + 17
20 .
36, 2 2 2 3.
2, 3 3.
7 3 36.
12

2
5
18
2
36.
7 − 19
15 24 .
7
15 − 19
24
56
120 − 120
95
− 39
120
− 13 · 3
40 · 3
− 13
40
13
24 − 17
32 .
21
32 − 9
28 .
− 11
30 + 23
42 .

− 11
30 + 23
42
−
210 77 + 115
210
38
210
19·2
105·2
19
105
− 13
42 + 17
35 .
− 19
24 + 17
32 .
3 5 + x 8 .

3
5 + x 8
24
40 + 40
5x
24 + 5x
40
24 5x
y
6 + 7 9 .
x
6
+ 7 15 .
a · = ac
b
cd bd
a
b ÷ d c = a b · dc
a c + b c = a + b c
a c − b c = a − b c
a − 1 4 + 1 6
b − 1 4 ÷ 1 6

a
− 1 4 + 1 6
− 3 12 + 2 12
− 1 12
b
− 1 4 ÷ 1 6
− 1 4 · 61
− 6 4
− 3 2
a − 3 4 − 1 6
b − 3 4 · 16
a 5 6 ÷ ⎛ ⎝ ⎛ − 1
4
⎞
⎠
⎞
b 5 6 − ⎛ ⎝ ⎛ − 1
4
⎞
⎠
⎞
a 5x
6 − 10
3
b 5x
6 · 310
a

5x
6
− 3 10
5x ·5
6·5 − 3·3
10 · 3
25x
30 − 9 30
25x −9
30
b
5x
6 · 310
5x ·3
6·10
5 · x ·3
2·3·2·5
x
4
a
(27a − 32)
36
b 2a 3
a
(24k + 25)
30
b 24k
5
3
4 = 3 5 4 ÷ 5 8
8

⎛
⎝ 1 2
⎞
2⎠
4+3 2.
⎛
⎝ 1 2
⎞
2⎠
4+3 2
1
4
4+3 2
1
4
4+9
1
4
13
1
4 ÷13
1
4 · 113
1
52
⎛1
⎛ ⎞ 2
⎝3
⎠
2 3 + 2
1 + 42
⎛1
⎛ ⎞ 2
⎝4
⎠
1
2 + 2 3
3
4 − 1 .
6

1
2 + 2 3
3
4 − 1 6
3
6 + 4 6
9
12 − 2 12
7
6
7
12
7
6 ÷ 7 12
7
6 · 12 7
7 ·6·2
6 ·7·1
1
3 + 1 2
3
4 − 1
3
2
3 − 1 2
1
4 + 1
3
x + 1 3
a x =− 1 3
b x =− 3 4 .
a x + 1 3 x =−1 3 , −1 3
x
x + 1 3
0

x + 1 3 x =−3 4 , −3 x
4
x + 1 3
− 3·3
4·3 + 1·4
3·4
− 9 12 + 4 12
− 5 12
x + 3 4
a x = − 7 4
b x = − 5 4
y + 1 2
a y = 2 3
b y = − 3 4
y − 5 6 y =−2 3 .
− 2 3 y y − 5 6
− 4 6 − 5 6
− 9 6
− 3 2
y − 1 2 y = − 1 4 .

x − 3 8 x = − 5 2 .
2x 2 y x = 1 4 y =−2 3 .
2x 2 y, x.
3ab 2 a = − 2 3 b = − 1 2 .
4c 3 d c = − 1 2 d = − 4 3 .
p + r q p =−4,q =−2, r =8.
p + q
r
− 6 8
− 3 4

a + b c a = −8, b = −7, and c = 6.
x + y
z x = 9, y = −18, and z = −6.

2
3
and 3 4
3
4
and 2 5
7
12
and 5 8
9
16
and 7 12
21 and 39
35 56
13 and 25
30 42
18 and 33
35 49
23
30 and 5 48
2
3
, 1 6 , and 3 4
2
3
, 1 4 , and 3 5
1 and 1 3 4 , LCD = 12 1 and 1 4 5 , LCD = 20 5 and 7 , LCD = 24
12 8
7 and 5 12 8 , LCD = 24 13 and − 11
16 12 , LCD = 48 11
16 and − 12 5 , LCD = 48
1, 5 3 6 , and 3 4 , LCD = 12 1, 3 3 4 , and 3 , LCD = 60
5
1
3
+ 1 5
1
4
+ 1 5
1
2
+ 1 7
1
3
+ 1 8
1 − ⎛ 3 ⎝ −1 ⎞
9⎠ 1 − ⎛ 4 ⎝ −1 ⎞
8⎠
1 − ⎛ 5 ⎝ − 10
1 ⎞
⎠ 1 − ⎛ 2 ⎝ −1 ⎞
6⎠
2
3
+ 3 4
3
4
+ 2 5
7
12
+ 5 8
5
12
+ 3 8
7
12
− 9 16
7
16
− 5 12
11
12 − 3 8
5
8
− 7 12
2
3
− 3 8
5
6
− 3 4
− 11
30 + 27
40
− 23
30 + 5 48
− 9 20 + 17
30
− 39
56 − 22
35
− 13
30 + 25
42
− 33
49 − 18
35

− 2 3 − ⎛ ⎝ −3 ⎞
4⎠ − 3 4 − ⎛ ⎝ −4 ⎞
5⎠ −
16 9 − ⎛ ⎝ −4 ⎞
5⎠
−
20 7 − ⎛ ⎝ −5 ⎞
8⎠
1+ 7 8
1+ 5 6
1− 5 9
1− 3 10
x
3
+ 1 4
y
2 + 2 3
y
4 − 3 5
x
5
− 1 4
a 3 4 + 1 6
b 3 4 ÷ 1 6
a 2 3 + 1 6
b 2 3 ÷ 1 6
a − 2 5 − 1 8
b − 2 5 · 18
a − 4 5 − 1 8
b − 4 5 · 18
a 5n 6 ÷ 8 15
b 5n 6 − 8 15
a 3a 8 ÷ 7 12
b 3a 8 − 7 12
a 9 10
⎞ · ⎛⎝ −11d
12 ⎠
b 9 10 + ⎛ ⎝ −11d 12
⎞
⎠
a
15 4 · ⎛⎝ −5q ⎞
9 ⎠
b
15 4 + ⎛ ⎝ −5q ⎞
9 ⎠
− 3 8 ÷ ⎛ ⎝ − 10
3 ⎞
⎠
−
12 5 ÷ ⎛ ⎝ −5 ⎞
9⎠
− 3 8 + 5 12
− 1 8 + 7 12
5
6
− 1 9
7 · ⎛⎝ − 8 ⎞
12 35⎠
5
9
− 1 6
− 7 15 − y 4
3 ⎞ · 8
⎛⎝ −10
21⎠
− 3 8 − x
11
11
12a · 9a
16
10y
13 · 8
15y
⎛
⎝ 1 2
⎞
5⎠
2+3 2
⎛
⎝ 1 2
⎞
3⎠
2 3 +4 2
5+2 2 ⎛
⎝ 2 2
⎞
3⎠
3 3 −3 2
2
⎛
⎝ 3 ⎞
4⎠
⎛
⎝ 3 2
⎞
5⎠
⎛
⎝ 3 2
⎞
7⎠
⎛
⎝ 3 2
⎞
4⎠
⎛
⎝ 5 2
⎞
8⎠
2
1
3 + 1 5
5
1
4 + 1 3
2
3 + 1 2
3
4 − 2 3

3
4 + 1 2
5
6 − 2 3
7
8 − 2 3
1
2 + 3 8
3
4 − 3 5
1
4 + 2 5
1
2
+ 2 3 · 512
1
3
+ 2 5 · 34
1− 3 5 ÷ 1 10
1− 5 6 ÷ 1 12
2
3
+ 1 6 + 3 4
2
3
+ 1 4 + 3 5
3
8
− 1 6 + 3 4
2
5
+ 5 8 − 3 4
12 ⎛ ⎝20 9 − 15
4 ⎞
⎠
8 ⎛ ⎝ 15
16 − 5 ⎞
6⎠
5
8 + 1 6
19
24
1
6 + 3 10
14
30
⎛ ⎝ 5 9 + 1 ⎞
6⎠ ÷ ⎛ ⎝ 2 3 − 1 ⎞
2⎠ ⎛ ⎝ 3 4 + 1 ⎞
6⎠ ÷ ⎛ ⎝ 5 8 − 1 ⎞
3⎠
x + 1 2
x + 2 3
x + ⎛ ⎝ −5 ⎞
6⎠
a x =− 1 8
b x =− 1 2
a x =− 1 6
b x =− 5 3
a x = 1 3
b x =− 1 6
x + ⎛ ⎝ −11 ⎞
12⎠
a x = 11 b x =
12
3 4
x − 2 5
a x = 3 5
b x =− 3 5
x − 1 3
a x = 2 3
b x =− 2 3
7
10
− w
5
12
− w
4p 2 q
a w = 1 2
b w =− 1 2
a w = 1 4
b w =− 1 4
p =− 1 2 and q = 5 9
5m 2 n when m =− 2 5 and n = 1 3
2x 2 y 3 whenx =− 2 3 andy =−1 2
8u 2 v 3 when u =− 3 4 and v =−1 2
u + v
w
when u =−4,v =−8,w =2 m + n
p when m =−6,n =−2, p =4 a + b when a =−3,b =8
a − b
r − s
r + s
when r = 10, s =−5

3 3
16 8
1 1
4
1 1
8
a
b

7
3
.
11 4
3 1 2
1 1 1 1 4
2 1 2 1 4
3 2
3 2 4 3 31 2
1 1 4
+2 1 4
________
3 2 4 =31 2
1 1 4 +21 4

1 1 4 +21 4
1 1 4 1 4
2 1 4
1
4
1
4
2 1 3 +12 3
1 3
1 3
1 3
1 3

4 2 1 3 +12 3 =4.
1 2 5 + 33 5
2 1 6 + 25 6
1 3 5 +23 5
1 5
1 5 s
1 5 s
1
5 s
3 6 5 . 6 5 11 5 ,
3 4 1 5 .
2 5 6 + 15 6
1 5 8 + 17 8

3 4 9 +22 9 .
3 4 9 +22 9
4 4 7 + 12 7 .
2 3 11 + 5 6 11 .
9 5 9 +57 9 .

9 5 9 +57 9
9 5 9
+5
_____
7 9
14 12 9
12 9 14 + 13 9
15 3 9
15 1 3
8 7 8 + 75 8 .
6 7 9 + 85 9 .
3 7 8 +43 8 .
3 7 8 +43 8
31
8
+ 35 8
31 + 35
8
66
8
8 2 8
8 1 4

5 5 9 + 37 9 .
3 7 10 + 2 9 10 .
3 2 5 +64 5
3 2 5
3 2 5 +64 5
+6 4 5
9 6 5
9+ 6 5
17
5
+ 34 5
51
5
10 1 5
9+1 1 5
10 1 5

1− 1 3 .
1 − 1 4 .
1 − 1 5 .
2− 3 4 .

2 − 1 5 .
2 − 1 3 .
2−1 2 5 .

2 − 1 1 3 .
2 − 1 1 4 .
$1
4 4 4
$1 5 3
1 1 4 − 3 4
4 4 1 4 5 4
3 4
1 2
1 1 3 − 2 3
1 1 5 − 4 5

5 3 5 −24 5 .
3 5 4 5 3 5 : ⎛ ⎝ 5 5 + 3 5 = 8 ⎞
5⎠
6 4 9 − 37 9 .
4 4 7 − 26 7 .

9 6 11 −710 11 .
9 6 11 −710 11
105
11 − 87
11
18
11
1 7 11
6 4 9 − 37 9 .
4 4 7 − 26 7 .
2 1 2 +52 3 .
6.

1 5 6 + 43 4 .
3 4 5 + 81 2 .
4 3 4 −27 8 .
8.
1
8 1 2 − 34 5 .
4 3 4 − 15 6 .
3 5 11 −43 4 .

4 3.
3 5 11 −43 4
3
11·4 5·4 −43·11
4·11
3 20
44 −433 44
152
44 − 209
44
− 57
44
−1 13
44
1 3 4 − 67 8 .
10 3 7 − 224 9 .

1 1 5 +31 5
2 1 3 +11 3
1 3 8 +17 8
1 5 6 +15 6
5 1 3 +61 3
2 4 9 +51 9
4 5 8 +93 8
7 9 10 +31 10
3 4 5 +64 5
9 2 3 +12 3
6 9 10 +83 10
8 4 9 +28 9
1 1 6 − 5 6
1 1 8 − 5 8
2 7 8 −13 2
8
12 7 −15 12
8 17
20 −49 20
19 13
15 −137 15
8 3 7 −44 7
5 2 9 −34 9
2 5 8 −17 8
2 5 12 −17 12
3 1 4 +61 3
2 1 6 +53 4
1 5 8 +41 2
7 2 3 +81 2
9 7 10 −21 3
6 4 5 −11 4
2 2 3 −31 2
2 7 8 −41 3

2 5 8 ·13 4
1 2 3 ·41 6
2
7
+ 4 7
2
9
+ 5 9
1 5 12 ÷ 1 12
2 3 10 ÷ 1 10
13 5 12 −97 12
15 5 8 −67 8
5
9
− 4 9
11
15 − 7 15
4− 3 4
6− 2 5
9
20
÷ 3 4
7
24
÷ 14 3
9 6 11 +710 11
8 5 13 +49 13
3 2 5 +53 4
2 5 6 +41 5
8
15 · 10
19
5
12 · 89
6 7 8 −21 3
6 5 9 −42 5
5 2 9 −44 5
4 3 8 −32 3
3 1 4
2 3 8
2 2 3
1 1 8
8 1
2
1 1
2
8
2 5
8
1 5 $3.75
8 +27 8 .
a $10
b
10 − 3 3 4 ?

4 5 12 +37 8
3 7 8 −45 12
a
b

x + 4 when x =−3
2y − 3 = 9.
y − 3 = −9
x − 3 10 = 1 2 .
a x =1 b x = 4 5
c x =− 4 5

a
x =1 1
b
x = 4 5 4
5
x − 3 10 = 1 2 .
c
x =− 4 5 −4 5
x − 2 3 = 1 6 :
a x = 1 b x = 5 6
c x = − 5 6

y − 1 4 = 3 8 :
a y =1 b y = − 5 8
c y = 5 8
a, b, and c,
if a = b, then a + c = b + c.
if a = b, then a − c = b − c.
if a = b, then a c = b c , c ≠0.
y + 9 16 = 5 16 .
9
16
y =− 1 4

y =− 1 4 y + 9 16 = 5 16
y + 11
12 = 5 12 .
y + 8 15 = 4 15 .
a − 5 9 =−8 9 .
5 9
a =− 1 3
a =− 1 3 a =−1 3
a − 3 5 = − 8 5 .
n − 3 7 = − 9 7 .
10q = 44.

10q =44
10q
10 = 44
10
q = 22 5
q = 22 5 10(22 5 )=? 44
10 2 ⎛22⎞
⎝ 5 ⎠ =? 44
44 = 44 ✓
22
5 .
12u = −76.
8m = 92.
4 x =3. 4 3.
4.
a, b, c,
if a = b, then ac = bc.
x 7 =−9.
x
7
=−9.

7
f
5 = −25.
h 9 = −27.
p
−8 = −40.
p −8. −8 p.
−8
p = 320
c
−7
= −35.
x
−11 = −12.
−1
−y = 15. y y,

−y = 15.
−y −1y, y.
−y −1y
−1.
−y y.” 15
15 −15. y = −15.
y
y = −15
−y = 48.
−c = −23.
1.
3
4 x =24
x 3 . x, 1.
4
1, x 3 .
4

3 4 x = 24.
x =32
32
3 4 x = 24,
3 4
x
2 5 n = 14.
5 6 y = 15.
− 3 8 w = 72.

− 3 8
w = −192
− 4 7 a = 52.
− 7 9 w = 84.
a, b, c,
a = b, a − c = b − c.
a, b, c, c ≠ 0
a, b, c,
a = b, a + c = b + c.
a, b, c
a = b,
a c = b c
a = b,
ac = bc
n 6 −24.

6
−144 6 −24
n 7 −21.
n 8 −56.
q −5 70.
−5
−350 −5 70
q −8 72.
p −9 81.
f 18.

3 2
27 18
f 16.
f 21.
m 5 6 3 4 .

m 5 6 3 4
m
5
6
= 3 4
5 6 m
⎛
5⎜ m 6 5
⎝ 6
⎞
⎟ = 5 ⎛
⎠ 6⎝ 3 ⎞
4⎠
m = 5·3
6·4
m = 5 8
5 8 5 6 3 4 5
8
5
6
= ? 3
4
5
8 ÷ 5 6 =? 3
4
5
8 · 65 =? 3
4
3
4 = 3 4 ✓
n 2 3 5 12 .
c 3 8 4 9 .
x

3 8
3 1 8
3
8 +31 8 =? 3 1 2
3 4 8 =? 3 1 2
3 1 2 =31 2 ✓
x
x

x − 2 5 = 10 1 :
y − 1 3 = 12 5 :
h + 3 4 = 2 5 :
a x =1 b x = 1 2
c x =− 1 2
a y =1 b y = 3 4
c y =− 3 4
a h =1 b h = 7 20
c h =− 7 20
k + 2 5 = 5 6 :
a k =1 b k = 13
30
c k =− 13
30
y + 1 3 = 4 3
m + 3 8 = 7 8
f + 9 10 = 2 5
h + 5 6 = 1 6
a − 5 8 =−7 8
c − 1 4 =−5 4
x − ⎛ ⎝ − 20
3 ⎞
⎠ =−11 20
p − 3 10 = 5 8
z − ⎛ ⎝ − 12
5 ⎞
⎠ =−7 12
s + ⎛ ⎝ −1 ⎞
2⎠ =−8 9
n − 1 6 = 3 4
k + ⎛ ⎝ −1 ⎞
3⎠ =−4 5
5j =17 7k =18 −4w =26
−9v =33
f
4 = −20 b
3
=−9
y
7 = −21
x
8
= −32
p
−5 = −40 q
−4 = −40
r
−12 =−6 s
−15 =−3 −x =23
−y =42 −h =− 5 12
−k =− 17
20
4
5
n =20 3
10
p =30 3
8
q = −48

5
2
m = −40 − 2 9 a =16 −3 7 b =9
− 6 11 u = −24
− 5 12 v = −15
3x =0 8y =0 4 f = 4 5
7g = 7 9
p + 2 3 = 1 12
q + 5 6 = 1 12
7
8
m = 1 10
1
4
n = 7 10
− 2 5 = x + 3 4
− 2 3 = y + 3 8
11
20 =−f 8
15
=−d
n −16. n −24. m −9 −7.
m −7 −8.
f −3
−18.
f −4
−20.
g
8.
g
14.
q 12.
q 20. p −63. p −28.
m 4
6.
h 2
43.
z 15.
a 2 3 x
3
1 4 . 2 .
x 1 8 .
y
− 1 8 .
y
− 1 6 .
$48. 2
3
2 3 p =48
3 5
18
3 5 h = 18.

−y = 15.
3x =24 16.
4
a
b

a a b
b
b
a
a b c c ≠0 ab c
c ≠0
a b
a < b a > b
a b b a
a ≠0 b ≠0
a =1 a ≠0
a b c
c ≠0
a b c c ≠0
ab a < b a ≥ b
remainder
divisor
a, b, c b ≠0 c ≠0 a b = a b ⋅ ⋅ c
c

a, b, c b ≠0 c ≠0 a b = a ⋅ c
b ⋅ c a ⋅ c
b ⋅ c = a b
a, b, c, d b ≠0 d ≠0 a b ⋅ d c = bd ac
1 a
b
⋅ b a =1
a, b, c, d b ≠0 c ≠0 d ≠0
a
b ÷ d c = a b ⋅ d c
a b -a = a
b -b =-a b
a, b, c c ≠0 a c + b c = a + c
c

a, b, c c ≠0 a c - b c = a c
- c
a, b c b ≠0 c ≠0
a
b = a b ⋅ ⋅ c
c a b ⋅ ⋅ c
c = a b
a
b ⋅ d c = bd
ac
a
b + d c = a b ⋅ d c
a c + b c = a + c
b
a c - b c = a - b c

a = b a + c = b + c
a = b a - c = b - c
a = b a c = b c c ≠0
ab c, a = b ac = bc

58
63
15
11
12 1 4
9 4 5
2 .
5
− 4 .
3
5
8
, 4 3 ,33 4 ,4 1
4
, − 1 4 ,11 3 ,−11 3 , 7 2 , − 7 2
< >.
−1___ − 2 5
−2 1 2 ___−3
− 63
84
− 90
120
− 14a
14b
− 8x
8y
2
5 · 813
− 1 3 · 12 7
2 ⎞ · 9
⎛⎝ −45
32⎠
6m · 411
− 1 4 (−32) 16
5 · 15 8
2 15 3
9 4

− 1 4
− 5 13
3
10
9
4
−12
2
3
÷ 1 6
⎛
⎝ −3x ⎞
5 ⎠ ÷ ⎛ ⎝ −2y ⎞
3 ⎠
4
5
÷3
8÷ 8 3
5 ÷ ⎛ 18 ⎝ −b ⎞
9⎠
3 1 5 ·17 8
−5 7 12 ·4 4 11
8÷2 2 3
8 2 3 ÷11 12
8 y V
h 6
5
8
4
5
8
9
−4
n
4
3
8
−1 5 6
− 1 12
5+16
5 8·4−5 2
3·12
8·7+5(8−10)
9·3−6·4
3
8
+ 2 8
4
5
+ 1 5
2
5
+ 1 5

15
32 + 9 32
x
10
+ 7 10
8
11
− 6 11
11
12 − 5 12
4
5
− y 5
− 31
30 − 7 30
3 − ⎛ 2 ⎝ 3 ⎞
2⎠ 11
15 − 15 5 − ⎛ ⎝ − 15
2 ⎞
⎠
1
3
and 1 12
1
3
and 4 5
8
15
and 11
20
3
4
, 1 6 , and 5 10
1 and 1 3 5 , LCD = 15 3 and 5 8 6 , LCD = 24 − 16 9 and 12 5 , LCD = 48
1, 3 3 4 and 4 , LCD = 60
5
1 + 2 11
5 3
12 − 2 3
− 9 10 − 3 4
− 11
36 − 11
20
− 22
25 + 9 40
y
10 − 1 3
2 + ⎛ 5 ⎝ −5 ⎞
9⎠
4
11
÷ 2
7d
2 + ⎛ 5 ⎝ −3n ⎞ ⎛
8 ⎠⎝ − 9n
2 ⎞
⎠
⎛
⎝ 2 2
⎞
3⎠
⎛
⎝ 5 2
⎞
8⎠
⎛
⎝ 11
12 + 3 ⎞
8⎠ ÷ ⎛ ⎝ 5 6 − 10
1 ⎞
⎠
y − 4 5
a y =− 4 5
6mn 2
m = 3 4 and n =−1 3
b y = 1 4
4 1 3 +91 3
6 2 5 +73 5
5 8 11 +24 11

3 5 8 +37 8
2 11
12 −17 12
9 13
20 −411 20
8 6 11 −29 11
2 3 10 −19 10
x − 1 2 = 1 6 :
y + 3 5 = 5 9 :
a x =1 b x = 2 3
a y = 1 2
b y = 52
45
c x =− 1 3
c y =− 2 45
n + 9 11 = 4 11
x − 1 6 = 7 6
h − ⎛ ⎝ −7 ⎞
8⎠ =−2 5
x
5
= −10 −z =23
n q
− 3 5 . 1 2 .
p −4
−8.
y 24.

19
5
3 2 7
1
2
,1 2 3 ,−23 4 , and 9 4
5 18r
20 27s
3 ·15 −36u ⎛ 5 ⎝ −4 ⎞
9⎠
1
3 · 34
−5 7 12 ·4 4 11
− 5 6 ÷ 5 12
7 ÷ ⎛ 11 ⎝ − 11
7 ⎞
⎠
9a
10
÷ 15a
8
−6 2 5 ÷4 ⎛
⎝ −15 5 ⎞
6⎠ ÷ ⎛ ⎝ −31 ⎞
6⎠
−6
6
11
p
2
q
5
− 4 15
−2 2 3
9 2 −4 2
9−4
2
d
+ 9 d
−
13 3 + ⎛ ⎝ − 13
4 ⎞
⎠
− 22
25 + 9 40
2 + ⎛ 5 ⎝ −7 ⎞
5⎠
−
10 3 + ⎛ ⎝ −5 ⎞
8⎠
− 3 4 ÷ x 3
2 3 −2 2
2
⎛
⎝ 3 ⎞
4⎠
5
14 + 1 8
9
56
x + 1 3
a x = 2 3
b x =− 5 6
y + 3 5 = 7 5
a − 3 10 =−9 10
f + ⎛ ⎝ −2 ⎞
3⎠ = 12
5

m
−2
= −16 − 2 3 c =18
p −4 −8.
p.

4,926,015
748

3 10
$3.45 $1.25
$0.33
$5.03. $5 3 $5 3
$5.03.
100 pennies = $1, 1 $0.01,
100
0.01 =
100 1
1
10 = 10
10 · 10 = 100
10 · 10 · 10 = 1000
10 · 10 · 10 · 10 = 10,000
0.1
0.01
0.001
0.0001
1
10
1
100
1
1,000
1
10,000

10,000
$5.03 $5.03
5.03
15.68

a 4.3 b 2.45 c 0.009 d −15.571.
a
b
c

d
−15.571
a 6.7 b 19.58 c 0.018 d −2.053
a 5.8 b 3.57 c 0.005 d −13.461

$1 $1.00.
$1 (or $1.00)
5 = 5.0 −2 = −2.0
5 = 5.00 −2 = −2.00
5 = 5.000 −2 = −2.000
and so on…
$5.03 5 3 100
3 3 0.03 = 3
100 100 .
0.03, 3
100 0.03.
0.03 =
100
3
$5.03 5.03
5.03 = 5
100
3

a 4.09 b 3.7 c −0.286
a

c
a 5.3 b 6.07 c −0.234
a 8.7 b 1.03 c −0.024
0.4

0.4
10 4 , 0.4 0 1.
0 1 10
0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. 0 0.0 1 1.0,
0.4
0.6
0.9
−0.74
−0.74 −
100 74 , 0 −1.
-0.10 0 −1 −0.10 −0.20 −0.74 −0.70
−0.80, −0.70
−0.63
−0.25
0.04 0.40?
$0.40 $0.04
0.40 > 0.04
a < b ‘a is less than b’ when a is to the left of b on the number line
a > b ‘a is greater than b’ when a is to the right of b on the number line
0.04 0.40
0.40 0.04. 0.40 > 0.04.
0.31 0.308?
0.31 0.308

0.31 0.308
31
100
310
1000
308
1000
308
1000
308
1000
310 > 308, 310
1000 > 1000 308 . 0.31 > 0.308.
0.31
100 31
1000 310 . 310 0.310. 0.31 0.310.
1000
31
100 = 1000 310 and 0.31 = 0.310
0.31 0.310
0.31 = 0.310
:
a 0.64 __0.6
b 0.83 __0.803

a
0.64 __0.6
0.64 __0.60
64 > 60
0.64 > 0.60
0.64 > 0.6
b
0.83 __0.803
0.830 __0.803
830 > 803
0.830 > 0.803
0.83 > 0.803
< or >:
a 0.42__0.4 b 0.76__0.706
< or >:
a 0.1__0.18 b 0.305__0.35
−2 −3
−2>−3. −9 −6
−9 < −6.
0 −1, −0.2 > −0.3 and −0.9 < −0.6.
−0.1__−0.8.

−0.1 __−0.8
−0.1
−0.8
−1 > −8, −1 −8 −0.1 > −0.8
< or >:
−0.3___−0.5
< or >:
−0.6___−0.7
$3.279
$3.28
$2.72 $2 $3? $2.72
$2.70 $2.80?
a 2.72 3 2. 2.72 3.
b 2.72 2.70 2.80. 2.72 2.7.
5?
1

18.379
1.047.
9.173.
18.379 a b

a
b
6.582 a b c
15.2175 a b c

5.5 7.8 5.01
14.02 8.71 2.64
0.002 0.005 0.381
0.479 −17.9 −31.4
1.99 5.83 15.7
18.1 0.239 0.373
0.13 0.19 0.011
0.049 −0.00007 −0.00003
6.4 5.2 7.05
9.04 4.006 2.008
10.25 12.75 1.324
2.482 14.125 20.375
0.8 0.3 −0.2

−0.9 3.1 2.7
−2.5 −1.6
.
0.9__0.6 0.7__0.8 0.37__0.63
0.86__0.69 0.6__0.59 0.27__0.3
0.91__0.901 0.415__0.41 −0.5__−0.3
−0.1_−0.4 −0.62_−0.619 −7.31_−7.3
0.67 0.49 2.84
4.63
0.845 0.761 5.7932
3.6284 0.299 0.697
4.098 7.096
a b c
5.781 1.638 63.479
84.281
$58,965.95
a
b
c
$23,795.95.
a
b
c
$1624.99
$142.186625. a
b
$1,038.99
$67.53435.
a b

100-meter 12.32 seconds.
12.3 seconds.
“10 for 0.99¢.”
a
b

100 70 .
3 10 · 910 .
−36 ÷ (−9).
3.7 + 12.4.

3.7 + 12.4
3.7
_____ +12.4
3 1 .7
_____ +12.4
16.1
5.7 + 11.9.
18.32 + 14.79.
23.5 + 41.38.
23.5 + 41.38
4.8 + 11.69.
5.123 + 18.47.
$20 $14.65
20 − 14.65.

20 − 14.65
10 − 9.58.
50 − 37.42.
2.51 − 7.4.
7.4 2.51, 7.4 > 2.51. 2.51
7.4.
2.51 − 7.4
2.51 − 7.4 2.51 − 7.4 = − 4.89
4.77 − 6.3.
8.12 − 11.7.

(0.3)(0.7) (0.2)(0.46)
⎛
⎝10
3 ⎞
⎠ ⎛ ⎝10
7 ⎞
⎠
21
100
⎛
⎝10
2 ⎞
⎠ ⎛ ⎝100
46 ⎞
⎠
92
1000
0.21 0.092
(0.01)(0.004)?
(3.9)(4.075).

(3.9)(4.075)
(1 + 3).
(3.9)(4.075) = 15.8925
4.5(6.107).
10.79(8.12).
(−8.2)(5.19).

(−8.2)(5.19)
5.19
_____ ×8.2
5.19
_____ ×8.2
1038
_____ 4152
42558
5.19
_____ ×8.2
1038
_____ 4152
42.558
(−8.2)(5.19) = −42.558
(4.63)(−2.9).
(−7.78)(4.9).
(0.03)(0.045).
(0.03)(0.045)
(0.03)(0.045) = 0.00135

(0.04)(0.087).
(0.09)(0.067).
10
10.
1.9436 10.
1.9436(10) = 19.436
1.9436(100) = 194.36
1.9436(1000) = 1943.6
10 1 1
100 2 2
1,000 3 3
10,000 4 4
10
45.86 100, 2
2.4 100. 2
0
10.

5.63 a 10 b 100 c 1000.
a
56.3(10)
56.3
b
5.63(100)
563
c
5.63(1000)
5,630
2.58 a 10 b 100 c 1000.
14.2 a 10 b 100 c 1000.
(0.2)(4) = 0.8
0.8 ÷ 4 = 0.2
0.8 ÷ 4 = 0.2.

0.12 ÷ 3.
0.12 ÷ 3
0.12 ÷ 3 = 0.04
0.28 ÷ 4.
0.56 ÷ 7.
24 $3.99. $3.99 24,
$3.99 ÷ 24.

$3.99 ÷ 24
$0.166 ≈ $0.17
$3.99 ÷ 24 ≈ $0.17
17
$6.99 ÷ 36.
$4.99 ÷ 12.
(0.2)(4) = 0.8
0.2 0.8?” (0.2)(4) = 0.8, 0.2 0.8
0.8 0.2 4.
0.8 ÷ 0.2 = 4
4, 8 2,
0.8
0.2
(0.8)10
(0.2)10
8
2
4
10 8 4.
10

−2.89 ÷ (3.4).
−2.89 ÷ (3.4) = −0.85
−1.989 ÷ 5.1.
−2.04 ÷ 5.1.
−25.65 ÷ (−0.06).

−25.65 ÷ (−0.06)
−25.65 ÷ (−0.06) = 427.5
−23.492 ÷ (−0.04).
−4.11 ÷ (−0.12).
4 ÷ 0.05.
4 ÷ 0.05
4 ÷ 0.05 = 80
4 ÷ 0.05 = 80, 80
$4.
6 ÷ 0.03.

7 ÷ 0.02.
$50 $31.64
50 − 31.64
$35 $18.48
$24.75 $36.90. $4.32.
8 $3.529.

8($3.529)
13 $3.175.
5 $9.75.
$31.76.
$31.76 ÷ 4
$92.82.
40 $570.
6 $0.22 4 $0.49

6($0.22) + 4($0.49)
$1.32 + $1.96
3 $0.75 6 $0.62
$9.50
$6.00

16.92 + 7.56 18.37 + 9.36 256.37 − 85.49
248.25 − 91.29 21.76 − 30.99 15.35 − 20.88
37.5 + 12.23 38.6 + 13.67 −16.53 − 24.38
−19.47 − 32.58 −38.69 + 31.47 −29.83 + 19.76
−4.2 + ( − 9.3) −8.6 + ( − 8.6) 100 − 64.2
100 − 65.83 72.5 − 100 86.2 − 100
15 + 0.73 27 + 0.87 2.51 + 40
9.38 + 60 91.75 − ( − 10.462) 94.69 − ( − 12.678)
55.01 − 3.7 59.08 − 4.6 2.51 − 7.4
3.84 − 6.1
(0.3)(0.4) (0.6)(0.7) (0.24)(0.6)
(0.81)(0.3) (5.9)(7.12) (2.3)(9.41)
(8.52)(3.14) (5.32)(4.86) (−4.3)(2.71)
( − 8.5)(1.69) (−5.18)(−65.23) ( − 9.16)( − 68.34)
(0.09)(24.78) (0.04)(36.89) (0.06)(21.75)
(0.08)(52.45) (9.24)(10) (6.531)(10)
(55.2)(1,000) (99.4)(1,000)
0.15 ÷ 5 0.27 ÷ 3 4.75 ÷ 25
12.04 ÷ 43 $8.49 ÷ 12 $16.99 ÷ 9
$117.25 ÷ 48 $109.24 ÷ 36 0.6 ÷ 0.2

0.8 ÷ 0.4 1.44 ÷ ( − 0.3) 1.25 ÷ ( − 0.5)
−1.75 ÷ ( − 0.05) −1.15 ÷ ( − 0.05) 5.2 ÷ 2.5
6.5 ÷ 3.25 12 ÷ 0.08 5 ÷ 0.04
11 ÷ 0.55 14 ÷ 0.35
6(12.4 − 9.2) 3(15.7 − 8.6) 24(0.5) + (0.3) 2
35(0.2) + (0.9) 2 1.15(26.83 + 1.61) 1.18(46.22 + 3.71)
$45 + 0.08($45) $63 + 0.18($63) 18 ÷ (0.75 + 0.15)
27 ÷ (0.55 + 0.35) (1.43 + 0.27) ÷ (0.9 − 0.05) (1.5 − 0.06) ÷ (0.12 + 0.24)
⎡ ⎣$75.42 + 0.18($75.42) ⎤ ⎦ ÷5 ⎡ ⎣$56.31 + 0.22($56.31) ⎤ ⎦ ÷4
$40
$15.11
$20
$4.82
$20
$18.49
$8.92 $1.65.
$20.45
$3.15
$4.40
190
4.3
2.8
0.7
1.9
$4.50
6
$4.29.
$0.55
1,
4
5,
1.6
8,
2.1
$9.25
32
$7.99.
$0.75
4
$3.75.
$8.75
19

30
$382.50.
25
$362.50.
$6.00.
a 2?
b 3?
c 4?
d 5?
e 6?
$15.60
a 2?
b 3?
c 4?
d 5?
e 6?
4
$3.29 2
$2.74
2
$9.99 6
$2.99
$29.95
$19.95.
4 7
$8.25 $12.95
12 3
$14.04
$8.75
8
15
a
b 23
$7.95
$20.25
12
5
a
b 17

1 30.62
1 30.65
0.12 ÷ 0.04
a
b

0.24 ÷ 8.
0.64__0.6 < >.
−0.2__−0.1 < >.
4 5
4÷5 54.
3 4
3 4
3 4
0.75.

1
4
.
3
8
.
− 7 2
− 7 2
7 2.
− 7 2 = −3.5.
− 9 4 .
− 11 2 .
4 3 4
3
1. 1.
4 3.
1,
1.333…

1.333 …
1.333 … 1. 3 – . 3
3 1.333… = 1. 3 –
1.333… = 1. 3 –
4.1666… = 4.1 6 –
4.161616… = 4.16 —
–––
0.271271271… = 0.271
3
6
16
271
43
22
43 22.
120 100 54
9,
43
22 = 1.954 —
27
11
.
51
22 .
7
8
+ 6.4.

7
8 + 6.4
7 8 0.875 + 6.4
7.275
3 8 + 4.9.
5.7 + 13
20 .
3 __0.4 < >.
8
3
8 __0.4
3 8
0.375__0.4
0.375 0.4 0.375 < 0.4
3
8 < 0.4
< >.
17
20 __0.82
< >.
3
4 __0.785

−0.5___ − 3 4
< >.
−0.5___ − 3 4
− 3 4
−0.5___−0.75
−0.5 −0.75 −0.5 > −0.75
−0.5 > − 3 4
< >:
− 5 8 __−0.58
< >:
−0.53__ − 11
20
13 , 0.61, 11
20 16
13, 0.61, 11
20 16
0.65, 0.61, 0.6875
0.61, ____, _____
0.61, 0.65, _____
0.61, 0.65, 0.6875
0.61, 13
20 , 11
16
7 8 , 4 5 , 0.82.
0.835, 13
16 , 3 4 .

a
a 7(18.3 − 21.7) b 2 (8.3 − 3.8)
3
7(18.3 − 21.7)
7(−3.4)
−23.8
b
2(8.3 − 3.8)
3
4.5
2
3 (4.5)
2
3 (4.5 1 )
9
3
3
a 8(14.6 − 37.5) b 3 (9.6 − 2.1).
5
a 25(25.69 − 56.74) b 2 (11.9 − 4.2).
7
a 6 ÷ 0.6 + (0.2)4 − (0.1) 2 b ⎛ ⎝10
1 ⎞
⎠2
+ (3.5)(0.9)

a
6 ÷ 0.6 + (0.2)4 − (0.1) 2
6 ÷ 0.6 + (0.2)4 − 0.01
10 + (0.2)4 − 0.01
10 + 0.8 − 0.01
10.8 − 0.01
10.79
b
⎛
⎝10
1 ⎞
⎠2
+ (3.5)(0.9)
1
100 + (3.5)(0.9)
1
100 + 3.15
1
100 0.01 + 3.15
3.16
9 ÷ 0.9 + (0.4)3 − (0.2) 2 .
⎛1⎞ 2
⎝ 2⎠
+ (0.3)(4.2).
2,000

π
π
π
3.14 π. ≈
r is the length of the radius.
d is the length of the diameter.
The circumference is 2πr. C =2πr
The area is πr 2 . A =πr 2
C =πd.
10
r =10 π.
A =πr 2
A = π(10 2 )
A = π · 100
π 100. A = 100π
π ≈ 3.14.

A = 100π
≈ 100 · 3.14
≈ 314 square inches
10 a b
a r = 10.
C =2πr
π r C ≈ 2(3.14)(10)
C ≈ 62.8 centimeters
b r = 10.
A =πr 2
π r A ≈ (3.14)(10) 2
A ≈ 314 square centimeters
50 a b
100 a b
42.5 a b
a r = 42.5.
C =2πr
π r C ≈ 2(3.14)(42.5)
C ≈ 266.9 centimeters

= 42.5
A =πr 2
π r A ≈ (3.14)(42.5) 2
A ≈ 5671.625 square centimeters
51.8 a b
26.4 a b
π
22 7
3.14285714. 3.14, π.
22 7 π 3.14. 22 7
π,
≈
14
15
a b
a r = 14
15 .
C =2πr
22 7
π 14
15 r C ≈2(22 7 )(14 15 )
C ≈ 88
15 meters
b r = 14
15 .
A =πr 2
22 7
π 14
15 r A ≈( 22 7 )(14 15 )2
A ≈ 616 square meters
225

5 21
a b
10
33
a b

2
5
4
5
− 3 8
− 5 8
17
20
13
20
11
4
17
4
− 310
25
− 284
25
5
9
2
9
15
11
18
11
15
111
25
111
1
2
+ 6.5 1
4
+ 10.75 2.4 + 5 8
3.9 + 9 20
9.73 + 17
20
6.29 + 21
40
< >.
1
8
___0.8 1
4
___0.4 2
5
___0.25
3
5
___0.35 0.725 ___ 3 4
0.92 ___ 7 8
0.66 ___ 2 3
0.83 ___ 5 6
−0.75___ − 4 5
−0.44___ − 9 20
− 3 4 ___−0.925
−2 3 ___−0.632
3, 9 , 0.55 3, 7 , 0.36 0.702, 13
5 16 8 20 20 , 5 8
0.15, 3
16
, 1 5
−0.3, − 1 3 , − 7 20
−0.2, − 3 20 , − 1 6
− 3 4 , − 7 9 , −0.7 −8 9 , − 4 5 , −0.9

10(25.1 − 43.8) 30(18.1 − 32.5) 62(9.75 − 4.99)
42(8.45 − 5.97) 3
4
(12.4 − 4.2) 4
5
(8.6 + 3.9)
5
12
(30.58 + 17.9) 9
16
(21.96 − 9.8) 10 ÷ 0.1 + (1.8)4 − (0.3) 2
5 ÷ 0.5 + (3.9)6 − (0.7) 2
(37.1 + 52.7) ÷ (12.5 ÷ 62.5)
(11.4 + 16.2) ÷ (18 ÷ 60)
⎛ ⎝ 1 2
⎞
5⎠
+ (1.4)(6.5) ⎛ ⎝ 1 2
⎞
2⎠
+ (2.1)(8.3)
−
10 9 · + 0.25 815
− 3 8 · 14
15 + 0.72
3 1 4 − 6.5 52 5 − 8.75 10.86 ÷ 2 3
5.79 ÷ 3 4
7
8
(103.48) + 1 1 2 (361) 5
16
(117.6) + 2 1 3 (699)
3.6 ⎛ ⎝ 9 8 − 2.72⎞ ⎠ 5.1 ⎛ ⎝ 12
5 − 3.91⎞ ⎠
a b
radius = 5 in. radius = 20 in. radius = 9 ft.
radius = 4 ft. radius = 46 cm radius = 38 cm
radius = 18.6 m radius = 57.3 m radius = 7 10 mile
radius = 7 11 mile radius = 3 8 yard radius = 5 12 yard
diameter = 5 6 m
diameter = 3 4 m
2
3
$84.99.
6

a
b

x + 2 3 when x = − 1 4 .
15 − y y = −5.
n
−7 = 42.
x − 0.7 = 1.5:
a x =1b x = −0.8 c x = 2.2

a
x =1 1
b
x = −0.8 −0.8
c
x = 2.2 2.2
x − 0.6 = 1.3 : a x = 0.7 b x = 1.9 c x = −0.7
y − 0.4 = 1.7 : a y = 2.1 b y = 1.3 c −1.3
a, b, and c,
a = b, a − c = b − c.
a, b, and c, and c ≠0
a = b,
a c = b c
a, b, and c,
a = b, a + c = b + c.
a, b, and c,
a = b, ac = bc

y + 2.3 = −4.7.
y =−7 y + 2.3 = −4.7
y + 2.7 = −5.3.
y +3.6 = −4.8.
a − 4.75 = −1.39.
a = 3.36
a − 3.93 = −2.86.
n − 3.47 = −2.64.

−4.8 = 0.8n.
n.
n =−6 −4.8 = 0.8n
−8.4 = 0.7b.
−5.6 = 0.7c.
p
−1.8 = −6.5.
p
= −6.5 p = 11.7.
−1.8
c
−2.6 = −4.5.

−1.2 = −5.4.
n 4.3 2.1.
4.3
n = 6.4
n
y 4.9 2.8.
z 5.7 3.4.
−3.1 x 5.27.
−3.1
x = −1.7
x 5.27
−3.1 −1.7 5.27

−4.3 x 12.04.
−3.1 m 26.66.
p −2.4 6.5.
−2.4
p = −15.6 :
p −2.4 6.5
−15.6 −2.4
6.5
q −3.4 4.5.
r −2.6 2.5.
n 2.9 1.7.
2.9
n = −1.2 :
n 2.9 1.7
−1.2 2.9 1.7

j 3.8 2.6.
k 4.7 0.3.

y + 0.6 = −3.4
x − 0.8 = 2.3
a x =2b x = −1.5 c x = 3.1
0.75k = −3.6
a k = −0.48 b k = −4.8 c
k = −2.7
a y =−4b y = −2.8 c
y = 2.6
h
1.5
= −4.3
a h = 6.45 b h = −6.45 c
h = −2.1
y + 2.9 = 5.7 m + 4.6 = 6.5 f + 3.45 = 2.6
h + 4.37 = 3.5 a + 6.2 = −1.7 b + 5.8 = −2.3
c + 1.15 = −3.5 d + 2.35 = −4.8 n − 2.6 = 1.8
p − 3.6 = 1.7 x − 0.4 = −3.9 y − 0.6 = −4.5
j − 1.82 = −6.5 k − 3.19 = −4.6 m − 0.25 = −1.67
q − 0.47 = −1.53 0.5x = 3.5 0.4p = 9.2
−1.7c = 8.5 −2.9x = 5.8 −1.4p = −4.2
−2.8m = −8.4 −120 = 1.5q −75 = 1.5y
0.24x = 4.8 0.18n = 5.4 −3.4z = −9.18
−2.7u = −9.72 a
0.4
= −20 b
0.3
=−9
x
0.7
= −0.4
y
0.8 = −0.7 p
−5 = −1.65
q
−4 = −5.92 r
−1.2 =−6 s
−1.5 =−3
x − 5 = −11 − 2 5 = x + 3 4
p +8=−2
p + 2 3 = 1 12
−4.2m = −33.6
q + 9.5 = −14

q + 5 6 = 1 12
8.6
15 =−d 7
8
m = 1 10
j
−6.2 =−3 − 2 3 = y + 3 8
s − 1.75 = −3.2
11
20 =−f −3.6b = 2.52 −4.2a = 3.36
−9.1n = −63.7 r − 1.25 = −2.7 1
4
n = 7 10
h
−3
=−8
y − 7.82 = −16
n 1.9
3.4.
n 1.5
0.8.
−6.2 x
−4.96.
−4.6 x
−3.22.
y −1.7
−5.
z −3.6
3.
n −7.3
2.4.
n −5.1
3.8.
$78
0.75p =78
p.
$1,350.
r,
1.08r = 1,350.
1.2y = 60,
60 60?
0.8x = 200,
200 200?
a
b

4 + 9 3
+ 2 .
4(8) + 6(3).
5 2
85, 88, and 94.
3.
89
85 + 88 + 94
3
267
3
89
n
mean =
sum of values in data set
n

mean = sum of values in data set
n
n,
8, 12, 15, 9, and 6.
mean =
sum of all the numbers
n
mean = 8+12+15+9+6 n
n =5
mean = 8+12+15+9+6
5
mean = 50 5
mean = 10
8, 9, 7, 12, 10, 5.
9, 13, 11, 7, 5.
16, 26, 53, 56, 65, 70, 93, and 97

mean =
sum of all the numbers
n
mean =
16 + 26 + 53 + 56 + 65 + 70 + 93 + 97
n
n
mean =
16 + 26 + 53 + 56 + 65 + 70 + 93 + 97
8
mean = 476
8
mean = 59.5
59.5 16 97. 59.5
25, 18, 21, and 22.
4, 9, 15, 12, 10, 12, and 8.
59.5,
$42.75, $50.12, $41.54, $48.15.
n
mean =
mean =
sum of all the numbers
n
sum of all the numbers
4
mean =
mean = 182.56
4
mean = 45.64
42.75 + 50.12 + 41.54 + 48.15
4
$45.64 $41.54 $50.12.
$45.64

$6.50, $7.25, $4.90, $5.30, and $12.00.
$34.87, $42.31, $38.04, and $43.26.
59 60 65 68 70
65″,
62
59, 60, 62, 65, 68, 70
62 and 65, 62 + 65. 63.5
2
5, 6,

n.
n
n
n
12, 13, 19, 9, 11, 15, and 18.
n n =7
n
43, 38, 51, 40, 46.
15, 35, 20, 45, 50, 25, 30.
83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.

n. n =10
n
mean =
85 + 86
2
mean = 85.5
8, 7, 5, 10, 9, 12.
21, 25, 19, 17, 22, 18, 20, 24.
2, 3, 5, 8, 8, 8, 15
8 8

18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 22, 23, 24, 24, 25, 29, 30, 40, 44
7,
20
20.
3, 6, 2, 3, 7, 5, 6, 2, 4, 2.
5, 6, 3, 1, 5, 8, 1, 5, 8, 5.
56 61 63 64 65 66 67 67
60 62 63 64 65 66 67 70
60 63 63 64 66 66 67 74
61 63 64 65 66 67 67
6,
67
67

19 20 23 23 38 21 19 21 19 21 20
43 20 23 17 21 21 20 29 18 28
6 2 5 6 3 7 5 6 5 3 4 4 5
7 6 4 5 2 1 5
3 5
Probability =
number of favorable outcomes
total number of outcomes
3 5
0.6
Probability of choosing a banana = 3 5
Probability of choosing a banana = 0.6
100
a
b

a
Probability of an event =
number of favorable outcomes
total number of outcomes
Probability Cherie wins = 1
100
b
Probability = 1
100
Probability = 0.01
a b
20 20. a
b
a
b

a
Probability of an event =
number of favorable outcomes
total number of outcomes
Probability = 3 8
b
Probability = 3 8
Probability = 0.375
5 3
a b
2 6
a b

3 8 2 2 5 6 1 9 3 4 7 65 13 48 32 19 33
34 45 29 61 41 202 241 265 274 525 532 558 574
12.45 12.99 10.50
11.25 9.99 12.72
5
$32.95 $38.50
$30.00 $17.45 $24.25
.373 .360 .321
.321 .320 .312 .311
.311
28.8 32.9 32.5 27.9
30.4 32.5 31.6 32.7
21 18 16 19 24 19
60.3, 79.7, 50.9, 28.0,
47.4, 46.1
$0
$14.95 $35.25 $25.16
$74.28
$120.95 $52.40 $10.59
$35.89 $59.24
24 19 18 29 21 48 51 46 42 50 65 56 35 34 44 39
55 52 45
121 115 135 109 136
147 127 119 110
4 8 1 5 14 3 1 12 3 9 2 6 20 3 3 10
99.2 101.9 98.6 99.5
100.8 99.8
25
24 28 32 29 31
28.8 32.9 32.5 27.9
30.4 32.5 31.6 32.7
6
38, 41, 45, 36, 40, and 42.
$6.50 $7.25
$4.90 $5.30 $12.00
8
$38.65, $32.18, $40.23,
$51.50, $43.68, $30.96,
$41.37, $44.72.

2 5 1 5 2 1 2 3 2
3 1
8 5 1 3 7 1 1 7 1
8 7
18 22 17 20 19 20
22 19 29 18 23 25 22
24 23 22 18 20 22 20
42 28 32 35 24 32
48 32 32 24 35 28 30
35 45 32 28 32 42 42
30
1 4 2 3
3 2 6 2 4 2 0 3 0.
2
0 3 0 0 8 6 5 0 1 2
3.
12
5 11 10 10 11 5 11
11 11 10 12
8 5 7 8 8 6 6 6 6 9
7 8 8 8
20
24
4-year college.
10
4
258,890,850
8 $315, $236.25, $236.25,
$236.25, $315, $315, $236.25, $393.75.
a b c
$1,845, $1,520, $1,438,
$1,682, $1,850, $2,721, $2,539. a
b c

a
b

16
24 .
2.76 ÷ 11.5.
11 2
2 3 .
4
a b a to b, a, ora:b. b
a b
a to b, a
b
, ora:b.
4 4
1
a 15 to 27 b 45 to 18.

a
15 to 27
15
27
5
9
b
45 to 18
45
18
5
2
b
a 21 to 56 b 48 to 32.
a 27 to 72 b 51 to 34.
0.8 to 0.05.
100
0.8 =
10 8 0.05 = 5 .
100
8 5
10 100 100. 0.8 100,
0.05

80
5
16
1
a 4.8 to 11.2 b 2.7 to 0.54
a 4.8 to 11.2
4.8
11.2
48
112
3
7
4.8 to 11.2 3 7 .
b
2.
2
2.7 to 0.54
2.7
0.54
270
54
5
1
2.7 to 0.54 5 1 .
a 4.6 to 11.5 b 2.3 to 0.69.
a 3.4 to 15.3 b 3.4 to 0.68.

1 1 4 to 23 8
1 1 4 to 23 8
1 1 4
2 3 8
5
4
19
8
5
4 ÷ 19 8
5
4 · 819
10
19
1 3 4 to 25 8 .
1 1 8 to 23 4 .
5 1
249 39 a
b 5 1
a

total cholesterol
HDL cholesterol
249
39
83
13
b 83 13 6.4, 83
13 ≈ 6.4
1 .
185 40
204 38
1
1
rise
run
1 inch
1 foot
1 inch
12 inches
1
12
1 12. 1 12
32 1

1 54
120 2 160 4 $5 64
525 9
525 miles in 9 hours
525 miles
9 hours
175 miles
3 hours
525 9
175 miles
3 hours .
492 8
242 6
175 miles.
3 hours
175
1
1
68
68 1 68 68
68 1.
68 68 miles/1 hour.

68 1
68 miles
1 hour
68 68 68 miles per hour
36 1
36 miles
1 gallon 36 36 36 miles per gallon
$12.50
$12.50/hour $12.50
1.
$384 32 hours.
$384 last week for 32 hours
$384
32 hours
$12
1 hour
$12 / hour
$12
$630 35
$684 36
455 14
455 miles to 14 gallons of gas
455 miles
14 gallons
32.5 miles
1 gallon
32.5 32.5

423 18
406 14.5
$3.99 24
$3.99
24 bottles
$0.16625
1 bottle
$0.17
1 bottle
$0.17 $0.17.
24-pack $6.99
24-pack $12.72
$14.99 64
$15.99 80

$14.99
64 loads
$0.234…
1 load
$0.23/load
(23 cents per load.)
$15.99
80 loads
$0.199…
1 load
$0.20/load
(20 cents per load)
$0.23
$0.20
$4.59 40 $3.99 30
$1.89 26 $0.95 10.75
a
a 427 h b x 3 c y 18
427 miles per h hours
427 miles
h hours
b
x students to 3 teachers
x students
3 teachers

c
y dollars for 18 hours
$ y
18 hours
a 689 h b y 22 c d 9
a m 9 b x 8 c y 40

20 36 20 32 42 48
45 54 49 21 56 16
84 36 6.4 0.8 0.56 2.8
1.26 4.2 1 2 3 25 6
1 3 4 25 8
4 1 6 31 3
5 3 5 33 5
$18 $63
$16 $72 $1.21 $0.44 $1.38 $0.69
28 84 32 128 12 46
15 57 246 45
304 48
45
175
55
215
27 1
28 1
140 12 180 16 8.2 3
9.5 4
$595 40
488 7
$798 40
527 9
140 12 180 16 8.2 3
9.5 4
488 7
527 9
$595 40 $798 40 576 18

435 15
43 16
57 24
46 0.5 54 0.5
96,000
12
540,000
12
8 $8.69 4 $3.39 6
$7.99
3
$8.49
12
$5.79
5 $3.69
25
$14.99
50 $4.49
$3.87 3
$4.32
12
50.7-ounce 6 18
$6.99 33.8-ounce $3.19 7.8-ounce $3.99 14
$4.79
$5.19
$3.29
10.7
$2.69 14.8
$3.69
40-ounce
$2.99 64-ounce
$4.39
15-ounce
$3.49
22-ounce
$4.99
$6.49 1 $10.99 1
$3.39 1 2 $2.89 1
4
793 p 78 r $3 0.5
j 0.5 105 x 400 m

y 5x
12x y
684
45
7.5
540 29 43
25
a b
c
d
1,600
(365 days).
16-ounce
470 18
63 15
a b
c
d
3/1 1/3?
4-ounce
115
422
$0.75
15 a
1 b
a
16-ounce
110
246
a
b

(−9) 2 .
3.846
12d d = 80.
n n 2 ,
8 2 “8 squared.”
“n squared.”
64 8 8 2 = 64. 121 11, 11 2 = 121.
n 2 = m, m n.
3 2 =9
9

1 15.
(−8) 2
= (−8)(−8)
=64
−1 −15.
10 2 = 100,
100 10. 10 100.
m m.
n 2 = m, n m.
(−10) 2 = 100 −10 100. 10 −10 100.
,
m m.”
If m = n 2 , then m = n for n ≥ 0.
0 2 =0,
15
0=0.

a 25 b 121.
a
5 2 =25 5
25
b
11 2 = 121 11
121
a 36 b 169.
a 16 b 196.
100 = 10.
− 100 = −10.
a − 9b − 144.
a
− 9
−3
b
− 144
−12
a − 4 b − 225.
a − 81 b − 64.

−25? −25?
( ) 2 = −25?
−25.
−25.
a −169 b − 121.
a −169.
−169
b 121.
− 121
−11
a −196 b − 81.
a −49 b − 121.
a 25 + 144 b 25 + 144.
a
25 + 144
5+12
17

169
13
25 + 144
a 9 + 16 b 9 + 16.
a 64 + 225 b 64 + 225.
a b
a b
4 9 2 3,
2< 5

38
84
x
≈
10-digit 5 2.236067977.
5.
5 ≈ 2.236067978
5 ≈ 2.24
5
5.
17
2.236067978 2 = 5.000000002
2.24 2 = 5.0176
17
4.123105626
4.12
17 ≈ 4.12
11
13
9x 2 , x ≥0. 9x 2 ?
(?) 2 = 9x 2
(3x) 2 = 9x 2 so 9x 2 =3x

x 2 .
x 2 (?) 2 = x 2
x 2
(x) 2 = x 2
x
y 2 .
m 2 .
16x 2 .
16x 2
Since (4x) 2 =16x 2
4x
64x 2 .
169y 2 .
− 81y 2 .
− 81y 2
Since (9y) 2 =81y 2
−9y
− 121y 2 .

− 100p 2 .
36x 2 y 2 .
36x 2 y 2
Since (6xy) 2 =36x 2 y 2
6xy
100a 2 b 2 .
225m 2 n 2 .
A A
9 9 =3
144 144 =12
A
A
200

200 A
A
A
A A = 200 200
14.142135...
14.1 feet
14.1
370
2704
h
h.
4
64 64
4 .
64
4
8
4
2
2 64
400

h
4
h 4 h = 400 400
4
20
4
5
1296
196
d 24d.
190
24d
24d d = 190. 24 · 190
4,560
67.527772...
67.5

76
122

36 4 64
144 − 4 − 100
− 1 − 121 −121
−36 −9 −49
9+16 25 + 144 9 + 16
25 + 144
70 55 200
172
19 21 53
47
y 2 b 2 49x 2
100y 2 − 64a 2 − 25x 2
144x 2 y 2 196a 2 b 2
75
130
1,024

350
216
4,000
175
54
117
625
2,025
−64? 9 2 9?
a
b

n
mean = sum of values in data set
n
a b
a b a a : b b

d
2πr C =2πr
πr 2 A = πr 2
a b c
If a = b
then a − c = b − c
a b c ≠0
If a = b
then a c = b c
a b c
If a = b
then a + c = b + c
a b c
If a = b
then a ⋅ c = b ⋅ c
mean =
sum of values in data set
n
n

n
n
n
m m
m = n 2 m = n n ≥0
0.8 0.375 0.007
5.24 −12.5632 −4.09
0.43 0.825 9.7
3.64
a 0.6
b −0.9
c 2.2
d −1.3

.
0.6___0.8 0.2___0.15 0.803____0.83
−0.56____−0.562
a b c
12.529 4.8447 5.897
5.75 + 8.46 32.89 − 8.22 24 − 19.31
10.2 + 14.631 −6.4 + (−2.9) 1.83 − 4.2
(0.3)(0.7) (−6.4)(0.25) (−3.35)(−12.7)
(15.4)(1000)
0.48 ÷ 6 4.32 ÷ 24 $6.29 ÷ 12
(−0.8) ÷ (−0.2) 1.65 ÷ 0.15 9 ÷ 0.045
$40 8
$9.32
$3.528.
$16.99
16
$6.72.
$2.49.
$0.35

3 7 − 19
5 8 20
− 21 4
1
3
6
11
< >.
1
2
___0.2 3
5
___0. − 7 8 ___−0.84
−
12 5 ___−0.42 0.625___13
20
0.33___ 5 16
2, 17
3 20 , 0.65 7, 0.75, 11
9 15
4(10.3 − 5.8) 3(15.44 − 7.4) 30 ÷ (0.45 + 0.15)
4
1.6 + 3 8
52(0.5) + (0.4) 2 − 2 · + 0.14
5
910
a b
radius = 6 in. radius = 3.5 ft. radius = 7 33 m
diameter = 11 cm
y + 3.2 = −1.5
x − 0.4 = 2.1
a x = 1.7 b x = 2.5
0.45v = −40.5
a v = −18.225 b v = −90
a y = 1.7 b y = −4.7
u = −12.5 2.5
a u =−5b u = −31.25
m + 3.8 = 7.5 h + 5.91 = 2.4 a + 2.26 = −1.1

p − 4.3 = −1.65 x − 0.24 = −8.6 j − 7.42 = −3.7
0.6p = 13.2 −8.6x = 34.4 −22.32 = −2.4z
a
0.3
= −24
p
−7 = −4.2 s
−2.5 = −10
n −5.9 x
15.2 4.4.
−3.54.
y −1.8
−9.
6.8.
m (−4.03)
2, 4, 1, 0, 1, and 1 $270 $310.50 $243.75
$252.15
3, 0, 8, 1, and 8
$31.45, $48.76, and $42.60.
41 45 32 60 58 25 23 24 26 29 19
18 32
52, 63, 45, 51, 55, 75, 60, and 59.
18, 7, 12, 16, and 20.
6 4 4, 5 6, 6 4 4 4
3 5
2 0 3 2 4
1 6 5 4 1 2 3

3
200 12
28 40 56 32 3.5 0.5
1.2 1.8 1 3 4 to 15 8
2 1 3 to 51 4
64 30 28 3
180 8 90 7.5
126 4
$612.50 35
180 8 90 7.5
126 4
$612.50 35
3 $8.97 6 $2.52
$11.
12
8
$13.20.
12
$4.29 22 $7.29?
60
$6.49 100 $11.99?

535 h hours a 45 4y
x 10
19
3 n
64 144 − 25
− 81 −9 −36
64 + 225 64 + 225
28 155
15 57
q 2 64b 2 − 121a 2
225m 2 n 2 − 100q 2 49y 2
4a 2 b 2 121c 2 d 2
225
30
576
216

1.73
5 8
16.749
a b c
4
5 , −0.1, 0.804, 2 , −7.4, 0.21
9
15.4 + 3.02 20 − 5.71 (0.64)(0.3)
(−4.2)(100) 0.96 ÷ (−12) −5 ÷ 0.025
−0.6 ÷ (−0.3) (0.7) 2 24 ÷ (0.1 + 0.02)
4(10.3 − 5.8) 1.6 + 3 8
2
3
(14.65 − 4.6)
m + 3.7 = 2.5 h = 4.38 −6.5y = −57.2
0.5
1.94 = a − 2.6
$79.35.
12.
a b
[Use 3.14 for π .]
10
55 55 50 51 52 50 53
51 55 49
a b c
16
12
4
a
b
64
$10.99 48 $8.49
36 + 64 144n 2 54
225

33 5”
3 5

0.62
100 100
100. %,
100.
(2015), 57%
57 100
100 57
100 57 .
100
57
25% 25
100 ,3% 3
100 100% 100.
100
100% 100 , 100 =1, 100% 1
100 100
(2010), 44%
44%
44
100
89%
72%

2007, 21 100
4-year
21 100 21 100
21
100
21%
62
100
41 100
100,
100.
100.
a
a 36% b 125%
36%
36
100
9
25

125%
125
100
5
4
a 48% b 110%
a 64% b 150%
1. 125% 125
100 , 5 .
4
a
a 24.5% b 33 1 3 %
24.5%
24.5
100
24.5(10)
100(10)
245
1000
5·49
5 · 200
49
200

33 1 3 %
33 1 3
100
100
3
100
100
1 100
3 ÷ 100
1
100
3 ⋅ 1
100
1
3
a 64.4% b 66 2 3 %
a 42.5% b 8 3 4 %
100.
a 6% b 78%
100

a
6%
6
100
0.06
b
78%
78
100
0.78
a 9% b 87%
a 3% b 91%
a
a 135% b 12.5%
135%
135
100
1.35
b
12.5%
12.5
100
0.125

a 115% b 23.5%
a 123% b 16.8%
6% 0.06
78% 0.78
135% 1.35
12.5% 0.125
%
6
6.0, 6% 6.0%.
77%
a b
a
77%
77
100

77
100
0.77
a b
24%
a b
2013, 44% 25
25%.
a
b
a
25%
25
100
1
4

1
4
0.25
a b
30%.
a b
12.5%.
100,
100,
100.
a 0.05 b 0.83
a
0.05
5
100
5%
b
0.83
83
100
83%

a 0.01 b 0.17.
a 0.04 b 0.41
a 1.05 b 0.075
a
0.05
1 5
100
105
100
105%
1.05 > 1, 100%.
b
0.075
75
1,000
7.5
100
7.5%
a 1.75 b 0.0825
a 2.25 b 0.0925

0.05 5%
0.83 83%
1.05 105%
0.075 7.5%
%
a 3 4 b 11 8 c 21 5
a
3
4
75%

11
8
137.5%
c
2 1 5
11
5
220%
a 5 8 b 11 4 c 32 5
a 7 8 b 9 4 c 13 5
5 7
0.714
71.4%
5
7
3 7
4 7

4 3 1.3¯ .
1 3 1 3
1
3
0.333 …
33 1 3 %
33.3 _ %
1 9
1
1 6

2014,
6.0%.
2015,
29%
3.2% 2014.
2014 7.3%.
a
b
57 100
80 100
42 100
71 100
4% 8% 17%
19% 52% 78%
125% 135% 37.5%
42.5% 18.4% 46.4%
9 1 2 % 81 2 % 51 3 %
6 2 3 %
5% 9% 1%
2% 63% 71%
40% 50% 115%
125% 150% 250%
21.4% 39.3% 7.8%
6.4%

a b
2010, 1.5% 2000, 4.2%
2013
70.23%
25
2014, 37.1%
15 60%.
25%.
50
2%.
0 9.
3
10%.
0.01 0.03 0.18
0.15 1.35 1.56
3 4 0.009
0.008 0.0875 0.0625
1.5 2.2 2.254
2.317
1
4
1
5
3
8
5
8
7
4
9
8
6 4 5
5 1 4
5
12
11
12
2 2 3
1 2 3
3
7
6
7
5
9
4
9

1
4
1
5
2012, 7
20
2013, 85%
1
1
2
1
3
2
0.45
0.008
18%
1
4
2
3
3
0.65
0.004
22%
9.05%.
10%.
a 10%
b a
$95
25%
a 25%
b a
$2,400
A 4/5
B 3.5/4
C 85%

C’s
A’s
B’s
25%, 50%, 75%, and 100%
500%
30 years 500%
1
10
, 2
10
, 3
10
, 4
10
, 5
10
, 6
10
, 7
10
, 8
10
,
9
10
2008
600% 2001.
600%
a
b

3
4
x 24.
(4.5)(2.38).
3.5 = 0.7n.
$80. 20%
20% $80. $80
0.20(80), $16

20% $80 $16.
35% 90?
n =
31.5 35% 90
45% 80?
55% 60?
125% 28
a =
125% 28 35

100 1. 125% 28 35,
28.
150% 78
175% 72
36 75%
b =
17 25%
40 62.5%
6.5% $1.17?
b =
7.5% $1.95?
8.5% $3.06?

36 9?
p =
76 57?
120 96?
144 96?
p =
110 88?
126 72?

$68.50. 18%
18%
t =
$81.50. 18%
15%
$14.40,
85

2%
a =
7 29%
190 8%

480 240
p =
230 60
190 70
increase = new amount − original amount
2011, $26 $36

p =
2011, 55.5 51
1995, $1.50. 2008, $2.25.
decrease = original amount − new amount
2014 $3.71. $3.64.

p =
672,000 2010.
630,000 2020.
$42,000. $37,800.

45%
120?
65%
100?
24%
112?
36%
124?
250% 65
150% 90
800% 2,250
600% 1,740
28 25%
36 25% 81 75% 93 75%
8.2%
$2.87?
6.4%
$2.88?
11.5%
$108.10?
12.3%
$92.25?
260 78? 215 86?
1,500
540?
1,800
846?
30 20?
50 40? 840 480? 790
395?
$74.25.
16%
$90.50.
18%
12%
$2,165.
8%
$1,485.
2,070
86%
8
33%
650
27%
67
22%
380
171

190 114
$3,000
$750
$3,200
$960
$15.50
$17.55.
$24.50
$25.48.
$4,000 2000
$9,000 2014.
$12.50 $50.
$2.80 $2.73.
2000 2010,
951,000
714,000.
(7/19/2011)
22 1960
38
$8.75 $8.54.
$203,400 $192,300.
$225,400 $241,500.
$49,500
$44,055.
$1,150
$1,030
$1.65. $2.00
$249. 5%
“44 80%
44
44?
50% 200%
20%
300?”
300
300?
150%

a
b

0.0875(720)
12.96 ÷ 0.04

Sales Tax = Tax Rate · Purchase Price
Total Cost = Purchase Price + Sales Tax
6.5%
a
a b $392?
t =

c =
a b $724
6.25%
a b $250
8.2%
$499
$42.42.

=
$26,525. $2,387.25.
$7,594 $569.55
commission = rate of commission · total sales
3%
$260,000?

c =
7%
$3,900
18%
$2,190?
$87 $1,450

=
$1,140 $28,500.
$451 $8,200

amount of discount = discount rate · original price
sale price = original price − discount
$10 $39.
s =
$75 $525.
$30 $112.
$10.
35% $140. a
b
a

d =
b
s =
a b 40%
$29.
a b 65%
$395.

$13.95. $31. a
b
a
d =
b
r =

a b $375.20.
$560.
a b
$340. $400.
amount of mark-up = mark-up rate · wholesale price
list price = wholesale price + mark up
$250. 40%. a
b
a
m =

p =
$1,200. 50%. a
b
$8,500. 35%. a
b

a b
$84. 5%
$1,242. 8%
$129.
7.5%
$350.
8.5%
$250.
6.85%
$400.
5.75%
6-drawer
$1,199.
5.125%
$1,350.
4.225%
$300.
$19.50.
$400.
$38.
$3,600.
$246.60.
$2,100.
$152.25.
$225
9%
8%
$193.
12%
$8,125
9%
$32,575
17.5%
$4,910?
10.5%
$7,420?
$11,250
$375,000
$364
$5,200.
$420 $2,625

$1,393.74
$15,486
$7,052.50
$45,500
$4,487.50
$35,900
$50
$189.
$15
$96.
$165.
$40
$859,
$125
a b
60%
$44.95
40%
$49.95.
$389.
33%
25%
$245.
35%
$949.95.
45%
$514.75.
a b
$1,344.
$1,920.
$240.
$300.
$301.75.
$355.
$20.80.
$32.
$480.
$850.
$725.
$1,299.
a b
$16
45%.
$120
55%.
$0.60
33%
$0.74
85%
$115.
225%
$8,500.
150%

$20
25%
$195.
a
$20 25%
b
25% 20%
c
$35 6%
$5
20%
a
b
c
40%
$150.
$150,
60% $150.
50%
$32.90.
a
b
a
b

0.6y = 45.
n
1.45 = 4.6.
P, I.
r.
t,
P, t r,
I,
I = Prt
I = interest
P = principal
r = rate
t = time
I = Prt.
3 $500 6%.
I = ?
P = $500
r = 6%
t = 3 years

I = Prt
I = ⎛ ⎝500) ⎛ ⎝0.06)(3)
I =90
4 $800 5%.
2 $700 4%.
$178 2 4%.
I = $178
P = ?
r = 4%
t = 2 years
I = Prt
178 = P(0.04)(2)
178
0.08 = 0.08P
0.08
2,225 = P
I = Prt
178 = ? 2,225(0.04)(2)
178 = 178 ✓
$495 3 6%.

$1,246 5 7%.
$8,200 $3,772 4
I = $3,772
P = $8,200
r = ?
t = 4 years
I = Prt
3,772 = 8,200r(4)
3,772 = 32,800r
3,772
32,800 = 32,800r
32,800
0.115 = r
11.5% = r
I = Prt
3,772 = ? 8,200 ⎛ ⎝0.115)(4)
3,772 = 3,772 ✓
$5,000 $1,350 6
$9,000 $1,755 3
$12,500 4%

5
I.
I = ?
P = $12,500
r = 4%
t = 5 years
I = Prt
I = (12,500)(0.04)(5)
I = 2,500
$950 3%.
5
$36,000 6.5%.
3
$3,000 4 years $3,000 $660
r.
I = 660
P = $3,000
r = ?
t = 4 years

I = Prt
660 = (3,000)r(4)
660 = (12,000)r
660
12,000 = (12,000)r
12,000
0.055 = r
5.5% = r
I = Prt
660 = ? (3,000)(0.055)(4)
660 = 660 ✓
$5,000 3 $5,000,
$900
$7,500 5 $1,500
$7,500
7.5%,
5
$6,596.25
P.
I = 6,596.25
P = ?
r = 7.5%
t = 5 years

I = Prt
6,596.25 = P(0.075)(5)
6,596.25 = 0.375P
6,596.25
0.375 = 0.375P
0.375
17,590 = P
I = Prt
6,596.25 = ? (17,590)(0.075)(5)
6,596.25 = 6,596.25 ✓
$4,866.25 8.5% 5
5 $2,400 5%.
$900 10-month 2.1%
I.
I = ?
P = $900
r = 2.1%
t = 10 months

I = Prt
10
12 I = $900(0.021)⎛ ⎝ 10 ⎞
12⎠
I = 15.75
$4,500 8 1.9%
$2,460 20 3.5%

$1200 3% 5
$1500 2% 4
$4410 4.5% 7
$2112 3.2% 6
$577.08 $4580 2
$528.12 $3260 3
5 $600
3%.
4 $900
6%.
2 $8,950
3.24%.
3 $6,510
2.85%.
8 $15,500
11.425%.
6 $23,900
12.175%.
$656 5
4%.
$177 2
3%.
$70.95 3
2.75%.
$636.84
6
4.35%.
$15,222.57
6
10.28%.
$10,953.70
5
11.04%.

$5,400 $432
2
$2,600 $468
6
$11,000 $1,815
3
$8,500 $3,230
4
$1,450 $5,720 $31,000
4%.
2
6%.
4
5.2%.
3
$16,400
3.9%.
8
$8,000
$8,000,
$1,200
$1,200
$1,200, $96
$20,000
$20,000,
$3,000
$50,000
$50,000,
$9,375
10
5.25%
$18,375
25
4.75% $2,375
$1,244.34 3
12.4%.
$7,683.20 5
9.8%.
$8,200
18-month
2.7%
$6,100
9-month
1.8%
$3,900
15 4%
$840
5 6%

a
b
$8,000 5
a
b
$8,000 5
a
b

1
3
4 .
x
4
= 20.
24 2
a b = d c , b ≠0,d ≠0.
“a b, c d”.
1 2 = 4 8 1 2 = 4 8
8”.
“1 2 4
20 students = 60 students
1 teacher 3 teachers
a 3 7 15 35.
b 5 8 30 48
c $1.50 6 $2.25 9

a
3
7 = 15
35
b
hits
at-bats = at-bats
hits
5
8 = 30
48
c
$
ounces = $
ounces
1.50
6 = 2.25
9
a 5 9 20 36.
b 7 11 28 44
c $2.50 8 $3.75 12
a 6 7 36 42.
b 8 36 12 54
c $3.75 6 $2.50 4
1 2 = 4 8 2 3 = 6 .
9

a b = d c , b ≠0,d ≠0,
a 4 9 = 12
28
b 17.5
37.5 = 7 15
a
28 ⋅ 4 = 112 9 ⋅ 12 = 108
28 · 4 ≠ 9 · 12,
b
15 ⋅ 17.5 = 262.5 37.5 ⋅ 7 = 262.5
15 · 17.5 = 37.5 · 7,
a 7 9 = 54
72
b 24.5
45.5 = 7 13
a 8 9 = 56
73
b 28.5
52.5 = 8 15

x = 4 63 7 .
x
n
84
= 11
12 .
y
96 = 13
12 .
144 a = 9 4 .

4a.
91
b
= 7 5 .
39 c = 13 8 .
52
91 = −4 y .

84
98 = −6 x .
−7 y = 105
135 .
5 25
80

a =
5 25
60
1 15
12
120 3.5

c =
16 240
20
100 160
8
$325
$1 12.54

p =
$800 $1
0.738
$600 94.1
60% =
100 60 60
100 = 3 . 60
5 100 = 3
5
amount = percent
base 100
3
5 = 100
60
100.
amount = percent
base 100
The amount is to the base as the percent is to one hundred.

The amount out of the base is the same as the percent out of one hundred.
75% 90?
n = number
n
90 = 100
75
60% 105?
40% 85?
19 25%
n = number
19 n = 25
100
36 25%
27 36%
27 9?

p = percent 9
27
= p
100
52 39?
92 23?
45% 80?
n =
65% 40?
85% 40?
100,

125% 25
n =
125% 64
175% 84
6.5% $1.56?

n =
8.5% $3.23?
7.25% $4.64?
72 9?

n =
1 8 1 8
72 27?
92 23?

4 15 36 135. 7 9 35 45. 12 5 96 40.
15 8 75 40. 5 7
115 161
4 9
36 81
8 1
48 6
$3.92 8
$1.47 3
6 1
48 8
$18.04 11
$4.92 3
$9.36 18
$2.60 5
$12.42 27
$5.52 12
7
15
= 56
120
5
12
= 45
108
11
6
= 21
16
9
4
= 39
34
13.5
8.5 = 31.05
19.55
12
18 = 4.99
7.56
10.1
8.4 = 3.03
2.52
9
16
= 2.16
3.89
x = 7 n = 56 8
8 91 13
49
63 = z 9
56
72 = y 9
5 a = 65
117
4
b
= 64
144
98
154 = −7 p 72
156 = −6 q a = −42
−8 48
b
−7
= −30
42
2.6
3.9 = c 3
2.7
3.6 = d 4
2.7
j
= 0.9
0.2
2.8
k
= 2.1
1.5
1
2
1 = m 8
1
3
3 = 9 n

5
25
45
6
15
1
10 19
140
160
27 10
106 8
12
12
150
32
1
2
2
3
1 4
2
10
$250
$1
$1.01
$450
12.29
$600
480
$350
385
$12.00
$20
30
285
$3.09
370
18.5
$3.19
812
3 190
280 2
152

4,000
13,500
350
2000
35%
250?
75%
920?
47?
110%
64?
150%
45 30%
25 80%
90 150%
77 110%
85 17?
92 46? 260
340?
180
220?
65%
180?
55%
300?
22% 74 175% 26
18% 92
250% 61
300% 488? 500% 315? 17%
$7.65?
19%
$6.46?
$13.53 8.25%
$18.12 7.55%
56 14? 80 28? 96 12?
120 27?
3 5
12
2
15 6

45% 350”
125 25”
a
b

100
a b = c b ≠0 d ≠0
d
a b c d
P t r
I I = Prt
increase = new amount − original amount

decrease = original amount − new amount
sales tax = tax rate ⋅ purchase price
total cost = purchase price + sales tax
commission = rate of commission ⋅ original price
amount of discount = discount rate ⋅ original price
sale price = original price – discount
amount of mark-up = mark-up rate wholesale price
list price = wholesale price + mark up
P t r
I I = Prt
a b = d c b ≠0 d ≠0
a b c d
a b = d c b ≠0 a ⋅ d = b ⋅ c
amount
base
= percent
100
32%
53.3%
13 100
9 100

48% 175% 64.1%
8 1 4 %
6% 23% 128%
a b
2012, 13.5%
65
12.5%.
2012, 6.5%
5
50%.
0.04 0.15 2.82
3 0.003 1.395
3
4
11
5
3 5 8
2
9
2
5
3
4
46% 120% 55
350?
84 35%
15 8% 200%
50?
7.9%
$4.74?
120
81.6?
340
595?

$83.50.
20%
2
8%
190
54
$4,600
$253.
$19.00 $19.76.
$30,000.
$24,000.
a b
$750. 6%
$577. 8.75%
$4,600. $200.
$333.50.
$16.75.
16.5%
3%
$380,000?
$895?
$675 $80.75
$950
$4,500
$30 $145
$75.
$312.

a b
30%
$84.90.
65%
$45.
a b
$37.
$50.
$49.99.
$79.99.
a
$0.80
60%
b
$17.40
325%
4 $2,250
5%.
7 $12,000
8.5%.
$660 5
3%.
$2,898
$23,000
3
$5,000
$5,000
$375
$10,000
4.5%.
2
6
5.5% $594
$23,000
5
$23,000
$9,200
3 8 12 32. 95 3
475 15
1 18
23 414
$7.35 15
$2.94 6

5
13
= 30
78
16
7
= 48
23
12
18 = 6.99
10.99
11.6
9.2 = 37.12
29.44
x
36
= 5 9
7 a = −6
84
1.2
1.8 = d 6
1
2
2 = m 20
5
25
10
21
60
8
272
10
$1,200
0.75
494 38
100
18
62%
395?
42 70% 1,000
15?
140
210?
85%
900?
6%
$24?
$3.51 4.5%
3,100
930?

a b
24% 5% 350%
3
7 1 11
8 3 12
65 260? 27%
3,000?
60?
150%
$3,825. $918
84,000 74,000.
7.72%
$600.
$31.80
$795
5 $3000
4.2%.
$899.
$1,200.
a
b
$400
$400 $50
$20.
250%
4
144
10 360
12 a = −15
65
10
12
35

3.19
5 11

144.
1, 2, 3, 4…
0, 1, 2, 3, 4…
…−3,−2,−1,0,1,2,3,4…
q p , p q q ≠ o.
4
5 , − 7 8 , 13 4 , and − 20 3
3= 3 1
−8 = −8
1
−8 −8.0.
0= 0 1
7.3. 7.3 7 3 10 ,
73. 7.3 73 10.
10
7.3 −1.2684)

a −15 b 6.81 c −3 6 7 .
a
−15
−15
1
b
6.81
6 81
100
681
100
c
−3 6 7
− 27 7
a −24 b 3.57.
a −19 b 8.41.
a = a a.
1
Integer −2, −1, 0, 1, 2, 3
Decimal −2.0, −1.0, 0.0, 1.0, 2.0, 3.0 These decimal numbers stop.
Ratio of Integers 4, − 7 5 8 , 13
14 , −20 3
Decimal Forms 0.8, −0.875, 3.25, −6.666… These decimals either stop or repeat.
—
−6.66

4
5 , − 7 8 , 13 4 , −20
3
4
5 , −7
8 , 13 4 , −20
3
−2, −1, 0, 1, 2, 3
−2
1
, −1
1 , 0 1 , 1 1 , 2 1 , 3 1
0.8, −0.875, 3.25, −6.6 – , −2.0, −1.0, 0.0, 1.0, 2.0, 3.0
π
π = 3.141592654.......
5 = 2.236067978.....
a 0.583 – b 0.475 c 3.605551275…
a 0.583 –
3 0.583 –
b 0.475
5
c 3.605551275…
(…)
a 0.29 b 0.816 – c 2.515115111…
a 0.23 – b 0.125 c 0.418302…

a 36 b 44
a 36 6 2 = 36. 36 =6. 36
b 6 2 =36 7 2 = 49, 44
44
a 81 b 17
a 116 b 121

a b c d
e
−7, 14 ,8, 5, 5.9, − 64
5
a 0, 1, 2, 3,… 8
b 0. −7 8
64 8 − 64 =−8. −7, 8, − 64.
c −7, 8, and − 64
14 5
and 5.9
d 5
5
e
−7 ✓ ✓ ✓
14
5
✓ ✓
8 ✓ ✓ ✓ ✓
5 ✓ ✓
5.9 ✓ ✓
− 64 ✓ ✓ ✓
a b c d
e −3, − 2, 0.3 – , 9 , 4, 49.
5
a b c d
e − 25, − 3 , −1, 6,
8
121, 2.041975…

a 5 b 3.19
a 8 b −1.61
a −12 b 9.279
a −16 b 4.399
0.75 0.223 – 1.39174… 0.36 0.94729… 2.528 – —
0.45 1.919293… 3.59
0.13 – , 0.42982… 1.875
a 25 b 30
a 225 b 216
a 44 b 49
a 164 b 169
−8 0, 1.95286.... 12 36 9 −9 −3 4 5 9 − 9 0.409 — 11 7 6
− 100 −7 − 8 3 −1 0.77 31 4
5th
147 44
a
b
c
12
40
a
b
c

a
b

7y + 2 + y + 13.
2 3 · 18.
15.
5 3.
5+3=3+5
5+3 3+5
8 8
5 3.
5·3 3·5
15 15
5·3=3·5.
a b
a + b = b + a
a b
a · b = b · a

a −1 + 3 = _____
b 4 · 9 = _____
a
−1 + 3 = _____
−1+3=3+(−1)
b
4 · 9 = _____
4·9=9·4
a −4 + 7 = _____
b 6 · 12 = _____
a 14 + (−2) = _____
b 3(−5) = _____
7−3 3−7?
7−3 3−7
4 −4
4≠−4
The results are not the same. 7 − 3 ≠ 3 − 7
12 ÷ 4 4 ÷ 12
12
4
4
12
3 1
3
3≠ 1 3
The results are not the same. So 12 ÷ 4 ≠ 4 ÷ 12
7+8+2
7+8is15 15 + 2 is 17. 8 + 2 makes 10
7 + 10 makes 17.

5·13 ·3
a, b, c
(a + b)+c = a +(b + c)
a, b, c
a
a (3 + 0.6) + 0.4 = __________ b ⎛ ⎝
(a · b)·c = a ·(b · c)
(3 + 0.6) + 0.4 = __________
⎞
−4 · 25 ⎠· 15 = __________
(3 + 0.6) + 0.4 = 3 + (0.6 + 0.4)
0.6 + 0.4 1,

⎛ ⎞
⎝−4 · 25 ⎠· 15 = __________
⎛ ⎞
⎝−4 · 25 ⎠· 15 = −4 · ⎛⎝ 2 5 ·15⎞ ⎠
2 ·15 6.
5
a (1 + 0.7) + 0.3 = __________ b (−9 · 8) · = __________
34
a (4 + 0.6) + 0.4 = __________ b (−2 · 12) · = __________
56
6(3x).
6(3x)
(6 · 3)x
18x
6·3, 3·x x.
8(4x).
−9(7y).
x = 7 8 .
a x + 0.37 + ( − x) b x +(−x) + 0.37

a
7 8 x
b
7 8
a b
x + 0.37 + ( − x) =x +(−x) + 0.37. b
y = 3 8
: a y + 0.84 + ( − y) b y + ( − y) + 0.84.
f =
20 17 : a f + 0.975 + ( − f ) b f + ( − f ) + 0.975.
n = 17.
a 4 ⎛
3⎝ 3 4 n⎞ ⎠ b ⎛ ⎝ 4 3 · 34
⎞
⎠ n
a

a b
4⎛3 3⎝ n ⎞ 4 ⎠ = ⎛ ⎝ 4 3 · 34
⎞
⎠n.
p = 24: a 5 ⎛9 ⎛ ⎞ 9
⎝ p 5 ⎠
b ⎛ 5
⎝ · 9 ⎞
9 5 ⎠ p.
q = 15: a
11
7 ⎛11 ⎛
⎝ q ⎞ 7 ⎠ b ⎛ 7
⎝ 11
· 11
7
⎞
⎠ q
b
a 0. b
1.
−84n + (−73n)+84n.
−84n + (−73n)+84n
−84n +84n + (−73n)
0 + (−73n)
−73n
−27a + (−48a) + 27a.
39x + (−92x) + (−39x).
1.

7
15 · 823 · 15 7 .
7
15 · 823 · 15 7
7
15 · 15 7 · 823
1· 823
8
23
9
16 · 549 · 16 9 .
6
17 · 11
25 · 17 6 .
⎛ ⎝13 5 + 3 ⎞
4⎠ + 1 4 .
⎛
⎝13 5 + 3 ⎞
4⎠ + 1 4
5
13 + ⎛ ⎝ 3 4 + 1 ⎞
4⎠
5
13 + ⎛ ⎝ 4 ⎞
4⎠
5
13 +1
1 5 13
18
13
⎛ 7
⎝ + 5 ⎞
15 8⎠
+ 3.
8

⎛2 ⎝ + 7
⎞
9 12⎠
+ 5 . 12
(6.47q + 9.99q) + 1.01q.
(6.47q + 9.99q) + 1.01q
6.47q + (9.99q + 1.01q)
6.47q + (11.00q)
17.47q
99 1
9.99 + 1.01?
(5.58c + 8.75c) + 1.25c.
(8.79d + 3.55d) + 5.45d.
⎡ ⎣1.67(8) ⎤ ⎦(0.25).
(8)(0.25) 1.67(8)
8 $2.)
⎡
⎣1.67(8) ⎤ ⎦(0.25)
1.67 ⎡ ⎣(8)(0.25) ⎤ ⎦
1.67[2]
3.34
⎡ ⎣ ⎡ 1.17(4) ⎤ ⎦ ⎤ (2.25).
⎡ ⎣ ⎡ 3.52(8) ⎤ ⎦ ⎤ (2.5).

6(9x).
6(9x)
(6·9)x
54x
8(3y).
12(5z).
3x +7+4x +5 3x +4x +7+5
7x + 12.
18p +6q + (−15p)+5q.
18p +6q + (−15p)+5q
18p + (−15p)+6q +5q
3p +11q
23r + 14s + 9r + (−15s).
37m + 21n + 4m + (−15n).

8 + 9 = ___ 7 + 6 = ___ 8(−12) = ___
7(−13) = ___ (−19)(−14) = ___ (−12)(−18) = ___
−11 + 8 = ___ −15 + 7 = ___ x + 4 = ___
y + 1 = ___ −2a = ___ −3m = ___
(11 + 9) + 14 = ___ (21 + 14) + 9 = ___ (12 · 5) · 7 = ___
(14·6)·9=___ (−7+9)+8=___ (−2+6)+7=___
⎛ ⎞
⎝16 · 45 ⎠· 15 = ___ ⎛ ⎞
⎝13 · 23 ⎠· 18 = ___ 3(4x) = ___
4(7x) = ___ (12 + x) + 28 = ___ (17 + y)+33=___
y = 5 8 ,
a y + 0.49 + ( − y)
b y +(−y) + 0.49
z = 7 8 ,
a z + 0.97 + ( − z)
b z +(−z) + 0.97
c =− 11 4 ,
a c + 3.125 + ( − c)
b c +(−c) + 3.125
d =− 9 4 ,
a d + 2.375 + ( − d)
b d +(−d) + 2.375
j = 11,
a 5 ⎛
6⎝ 6 5 j⎞ ⎠ b ⎛ ⎝ 5 6 · 65
⎞
⎠ j
k = 21,
a
13
4 ⎛
⎝ 13 4 k⎞ ⎠
b ⎛ ⎝13 4 · 13 ⎞
4 ⎠ k
m = −25,
a − 3 ⎛
7⎝ 7 3 m⎞ ⎠
b ⎛ ⎝ −3 7 · 73
⎞
⎠ m
n =−8,
a −
21
5 ⎛
⎝ 21 5 n⎞ ⎠
b ⎛ ⎝ − 21 5 · 21 ⎞
5 ⎠ n
−45a +15+45a
9y +23+(−9y)
1 + 7 2 8 + ⎛ ⎝ −1 ⎞
2⎠

2 + 5 5 12 + ⎛ ⎝ −2 ⎞
5⎠
3
20 · 49
11 · 20 3
13
18 · 25 7 · 18
13
7
12 · 917 · 24 7
−36 · 11 · 49
3
10 · 13
23 · 50 3
⎛ ⎝ 5 6 + 15
8 ⎞
⎠ + 15
7
−24 · 7 · 38
⎛ ⎝12 1 + 4 ⎞
9⎠ + 5 9
5
13
+ 3 4 + 1 4
8
15
+ 5 7 + 2 7
(4.33p + 1.09p) + 3.91p
(5.89d + 2.75d) + 1.25d 17(0.25)(4) 36(0.2)(5)
⎡ ⎣2.48(12) ⎤ ⎦(0.5) ⎡ ⎣9.731(4) ⎤ ⎦(0.75) 7(4a)
9(8w) −15(5m) −23(2n)
12 ⎛ ⎝ 5 6 p⎞ ⎠
20 ⎛ ⎝ 3 5 q⎞ ⎠
14x +19y +25x +3y
15u +11v +27u +19v
43m + (−12n) + (−16m)+(−9n)
−22p +17q + (−35p)+(−27q)
3
8
g + 1 12 h + 7 8 g + 5 12 h 5
6
a + 3 10 b + 1 6 a + 9 10 b
6.8p + 9.14q + (−4.37p) + (−0.88q)
9.6m + 7.22n + (−2.19m) + (−0.65n)
$0.49 $0.02
$0.49 $0.02
a
b
c
d $0.49
e $0.02
$5 $10 $20
$5
$10 $20
a
b
c
d $5
e $10
f $20

a
b

3(0.25).
10 − (−2)(3).
9y + 17 + 3y − 2.
$9.25; 9 1
3 $9, $27, 3 1 75 $27.75.
a, b, c
a(b + c) =ab + ac

3(9.25)
3(9 + 0.25)
3(9) + 3(0.25)
27 + 0.75
27.75
3(x +4),
x 4,
3(x +4).
3(x +4)
3·x +3·4
3x +12
4(x +2).
6(x + 7).
6(5y +1).
9(3y + 8).
5(5w + 9).
a(b + c).

a, b, c
a(b + c) =ab + ac
a(b − c) =ab − ac
(b + c)a = ba + ca
2(x −3).
7(x − 6).
8(x − 5).
3(n + 12). 4
2 (p + 10).
5
3 (u + 21).
7
8 ⎛ ⎝ 3 8 x + 1 ⎞
4⎠ .

6 ⎛5 ⎝ y + 1 ⎞
6 2⎞
⎠ .
12 ⎛1 ⎝ n + 3⎞ 3 4⎠ .
100(0.3 + 0.25q).
100(0.7 + 0.15p).
100(0.04 + 0.35d).
m(n −4).
m · 4 as 4m.
r(s − 2).

y(z − 8).
(b + c)a = ba + ca.
(x +8)p.
(x + 2)p.
(y + 4)q.
−2(4y +1).
−3(6m + 5).
−6(8n + 11).
−11(4 − 3a).
33a − 44.

−5(2 − 3a).
−7(8 − 15y).
−a =−1·a.
−(y +5).
−(z − 11).
−(x − 4).
8−2(x +3).
9 − 3(x + 2).
7x − 5(x + 4).
4(x −8)−(x +3).

6(x − 9) − (x + 12).
8(x −1) − (x + 5).
a
b
y =10 a 6(5y +1) b 6·5y +6·1.
a
6(5y +1)
6(51)
306
b
y = 10,
6(5y +1)=6·5y +6·1.
y.
w = 3: a 5(5w + 9) b 5 · 5w + 5 · 9.

y = 2: a 9(3y + 8) b 9 · 3y + 9 · 8.
y =3, a −2(4y +1)b −2 · 4y + (−2) · 1.
a
−2(4y +1)
−2(13)
−26
b
−2 · 4y + (−2) · 1
−24 − 2
y =3,
−26
−2(4y +1)=−8y −2
n = −2: a −6(8n + 11) b −6 · 8n + (−6) · 11.
m = −1: a −3(6m + 5) b −3 · 6m + (−3) · 5.
y =35 a −(y +5) b −y −5 −(y +5)=−y −5.
a
−(y +5)
−(40)
−40

−y −5
−40
y = 35, −(y +5)=−y −5
x = 36: a −(x − 4) b −x + 4 −(x − 4) = − x + 4.
z = 55: a −(z − 10) b −z + 10 −(z − 10) = − z + 10.

4(x +8) 3(a +9) 8(4y +9)
9(3w +7) 6(c − 13) 7(y − 13)
7(3p −8) 5(7u −4) 1
2
(n +8)
1
3
(u +9) 1
4
(3q + 12) 1
5
(4m + 20)
9 ⎛ ⎝ 5 9 y − 1 ⎞
3⎠ 10 ⎛ ⎝10 3 x − 2 ⎞
5⎠
12 ⎛ ⎝ 1 4 + 2 3 r⎞ ⎠
12 ⎛ ⎝ 1 6 + 3 4 s⎞ ⎠
r(s − 18) u(v − 10)
(y +4)p (a +7)x −2(y + 13)
−3(a + 11) −7(4p +1) −9(9a +4)
−3(x −6) −4(q −7) −9(3a −7)
−6(7x −8) −(r +7) −(q + 11)
−(3x −7) −(5p −4) 5+9(n −6)
12 + 8(u −1) 16 − 3(y +8) 18 − 4(x +2)
4 − 11(3c −2) 9 − 6(7n −5) 22 − (a +3)
8−(r −7) −12 − (u + 10) −4−(c − 10)
(5m −3)−(m +7) (4y −1)−(y −2) 5(2n + 9) + 12(n −3)
9(5u +8)+2(u −6) 9(8x −3)−(−2) 4(6x −1)−(−8)
14(c −1)−8(c −6) 11(n −7)−5(n −1) 6(7y +8)−(30y − 15)
7(3n +9)−(4n − 13)

v =−2,
u =−1,
a 6(4v +7)
b 6·4v +6·7
a 8(5u + 12)
b 8·5u +8·12
n = 2 3 ,
a 3 ⎛ ⎝ n + 5 ⎞
6⎠
b 3·n +3·56
y = 3 4 ,
a 4 ⎛ ⎝ y + 3 ⎞
8⎠
b 4·y +4·38
y = 7 12 ,
a −3(4y + 15)
b −3 · 4y + (−3) · 15
p = 23
30 ,
a −6(5p + 11)
b −6 · 5p + (−6) · 11
m = 0.4,
a −10(3m − 0.9)
b −10 · 3m − (−10)(0.9)
n = 0.75,
a −100(5n + 1.5)
b −100 · 5n + (−100)(1.5)
y = −25,
a −(y − 25)
b −y +25
w = −80,
a −(w − 80)
b −w +80
p = 0.19,
a −(p + 0.72)
b − p − 0.72
q = 0.55,
a −(q + 0.48)
b −q − 0.48
$1.99
12
$23.88.
a
12
$1.99
$2 − $0.01.
b
$3.97
3-pack
$10.49.
a
3
b
3-pack
8 ⎛ ⎝ x − 1 ⎞
4⎠
4($5.97)
$5.97 6 − 0.03

a
b

−4.
5 2 .
3a
5 · 92a .
0
13 + 0 −14 + 0 0 + (−3x)
13 −14 −3x
1
43·1 −27·1 1·6y
43 −27
a,
6y
5
a +0=a 0+a = a
0 is called the additive identity
a
a ·1=a 1·a = a
1 is called the multiplicative identity
5

a 7+0=7
b −16(1) = −16
a
7+0=7
b
−16(1) = −16
a 23 + 0 = 23 b −37(1) = −37.
a 1 · 29 = 29 b 14 + 0 = 14.
5 + _____ = 0
−6 + _____ = 0
−a a.
0,
2 3
1? 1?
2
3 · ___ = 1
2 1? 1?

2 · ___ = 1
1 a a(a ≠0).
1,
a,
a +(−a) =0
−a is the additive inverse of a.
a ≠0,
a · 1a =1
1
a is the multiplicative inverse of a.
a 13 b − 5 8 c 0.6
a 13 −13. b − 5 8 5
8
.
c 0.6 −0.6.
a 18 b 7 9 c 1.2
a 47 b 7 13 c 8.4
a 9 b − 1 9 c 0.9
a 9 1.
9
b − 1 9
−9.
c 0.9, 0.9
9
10
.
10
9
.

a 5 b − 1 7 c 0.3
a 18 b − 4 5 c 0.6
0? 0
0 0.
a,
a ·0=0 0·a =0
a −8 · 0 b
12 5 ·0 c 0(2.94)
a
−8⋅0
0
b
5
12 ·0
0
c
0(2.94)
0
a −14 · 0 b 0 · c (16.5) · 0.
23
a (1.95) · 0 b 0(−17) c 0 · 54 .

0?
0 0
0÷3=0
0 ÷ 3 = 0 because 0 · 3 = 0.
a, 0, 0 a =0 0÷a =0.
a 0÷5 b 0 −2 c 0÷7 8
a
0÷5
0
b
0
0
−2
c
0÷ 7 8
0
a 0 ÷ 11 b
0 −6 c 0 ÷ 3 10
a 0 ÷ 8 3
b 0 ÷ (−10) c 0 ÷ 12.75
4 0?
0 4?
4 ÷ 0 = ___ means ___ · 0 = 4
0 0, 0 4.
4÷0,

a, a
0
, a ÷0
a 7.5 ÷ 0 b −32
0
c 4 9 ÷0
a
7.5 ÷ 0
b
−32
0
c
4
9 ÷0
a 16.4 ÷ 0 b −2
0 c 1 5 ÷ 0
a −5
0 b 96.9 ÷ 0 c 4 15 ÷ 0
a,
a ·0=0 0·a = 0 The product of any number and 0 is 0.
a, a ≠0
0
a =0
a
0

3x +15−3x.
3x +15−3x
3x −3x 0+15
15
−12z + 9 + 12z.
−25u − 18 + 25u.
4(0.25q).
4(0.25q)
⎡
⎣4(0.25) ⎤ ⎦q
1.00q
q
2(0.5p).
25(0.04r).
0 n ≠−5
n +5
0
0
n +5
0 m ≠ −7
m + 7

0
d − 4 d ≠ 4
10 − 3p
.
0
10 − 3p
0
18 − 6c.
0
15 − 4q
.
0
3 · (6x + 12).
4
43
3 · (6x + 12)
4
43
1(6x + 12)
6x +12
2 · (20y + 50).
5
52
3 · (12z + 16).
8
83

a + b = b + a a · b = b · a
(a + b)+c = a +(b + c) (a · b)·c = a ·(b · c)
0 1
a +0=a
0+a = a
a ·1=a
1·a = a
−a
a
a, a ≠0
1 / a
a
a +(−a) =0 a ·1a =1
a, b, c a(b + c) =ab + ac
a ⋅0=0
0⋅a =0
a, a ≠0
0
a =0
a
0

101 + 0 = 101 3(1) = 3 −9 · 1 = −9
5 5
0 + 64 = 64
8 14 −17
−19 7
12
8
13
− 3 10
− 5 12
0.8
0.4 −0.2 −0.5
48 · 0 0 3
6 0
22·0 0÷ 11
12
6
0
0
3
0÷ 7 15
0· 815
(−3.14)(0) 5.72 ÷ 0
1
10
0
19a +44−19a 27c +16−27c 38 + 11r −38
92 + 31s −92 10(0.1d) 100(0.01p)
5(0.6q) 40(0.05n) 0 r ≠ −20
r +20
0
s +13 s ≠ −13 0
u − 4.99 u ≠ 4.99 0 v ≠ 65.1
v − 65.1
0÷ ⎛ ⎝ x − 1 ⎞
2⎠ x ≠ 1 2
0÷ ⎛ ⎝ y − 1 ⎞
6⎠ y ≠ 1 6
32 − 5a
0
32 − 5a ≠0

28 − 9b
0
28 − 9b ≠0
2.1 + 0.4c
0
2.1 + 0.4c ≠0
1.75 + 9 f
0
1.75 + 9 f ≠0
⎛
⎝ 3 4 + 10 9 m⎞ ⎛
⎠ ÷0 ⎝16 5 n − 3 ⎞
7⎠ ÷0 9 · 10 10 9
3
4 + 10 9 m ≠0 5
16 n − 3 7 ≠0
(18p − 21)
5 · (20q − 35) 15 · (4d + 10) 18 · (15h + 24)
7
75
35
56
5
$80.
20%
a
5
b 5(0.20)(80).
c
24-pound
20
a
24 · 20
60 1
b 24 ⎛ ⎞
⎝20 · 160⎠ .
c
24 ⎛ ⎞
⎝20 · 160⎠
(24 · 20)
60 1 ?
a
b

4.29(1000).
30
54 .
7
15 · 25
28 .
1,

1 12
1 3
1 5280
1 16
1 2000
3 1
16 1
1 8
1 2
1 2
1 4
1 60
1 60
1 24
1 7
1 365
For any real number a, a ·1=a 1·a = a
1
1 12 1
1 ft
12 in .
12 in
1 ft
1.
1 ft
12 in
12 in ?
1 ft
60
60 in · 1 ft =5 ft
12 in
5
5 ft · 12 in = 60 in
1ft
1; 1
66

66 ·1
66 inches · 1 foot
12 inches
66 inches · 1 foot
12 inches
66 inches ·1 foot
12 inches
66 feet
12
5.5 feet
5.5
30
18
3.2
3.2
2000 pounds
1 .
1 ton

3.2 tons
3.2 tons · 1
3.2 tons · 2000 lbs
1 ton
3.2 tons · 2000 lbs
1ton
6400 lbs
4.3
51,000
9
1.
7days , 24 hours, 60 minutes
1 week 1day 1 hour
9 weeks
9·7·24·60min
1·1·1·1
= 90, 720 min
250,000
15
1

1 gal
1 · 4 qt
1 gal · 2 pt
1 qt · 1 2 pt C · 8 fl oz
1 C
1gal
1
·
1gal· 4qt
1qt· 2pt
1pt· 2C 8 fl oz
1C
1·4·2·2·8 fl oz
1·1·1·1·1
1
1
14 1 2 1 6

3 12 3 3 2 9
8 7 12 11
6 4
4
1 8

5 3
10.
1000
1
100 1
100
1 1000
1 100
1 10
1 1
1 0.1
1 0.01
1 0.001
1 100
1 1000
1 1000
1 100
1 10
1 1
1 0.1
1 0.01
1 0.001
1 100
1 1000
1 1000
1 100
1 10
1 1
1 0.1
1 0.01
1 0.001
1 100
1 1000
5 k 10 k
10-kilometer

5-km
2.5
3200
3.2
2800
4500
10, 100, or 1000, 1, 2, or 3
0.1, 0.01, or 0.001 1, 2, or 3
3200 1 (or 0.001).
1000

3
a 350 b 4.1
a 1 kiloliter = 1000 liters.
b 1 liter = 1000 milliliters.
a 7.25 b 6.3
a 350 b 4.1
10.

1.6 85
85 2 85
0.85
0.75
1.60 m
_______ −0.85 m
0.75 m
1.58 75
2 96
150
3 · 150 mL
450 mL
450 mL · 0.001 L
1mL
0.45 L
250
8.
400 6
2-liter

500-mg 5-K
1 2.54
1 0.305
1 0.914
1 1.61
1 0.45
1 28
1 0.95
1 30
1 3.28
1 2.2
1 1.06
500
500 mL · 1 fl oz
30 mL
500 fl oz
30
16.7 fl. oz.
2-liter
4
100

100 kilometers · 1 mile
1.61 kilometers
100 · 1mi
1.61 km
100 mi
1.61
5,895
5,586
22°C.
°F. °C.
37°C
98.6°F.

F, C,
C = 5 (F −32)
9
C, F,
F = 9 5 C +32
50°F
50°F C.
C = 5 (F − 32)
9
C = 5 9 (18)
C =10
59°F.
41°F.
20°C.
20°C F.
F = 9 5 C +32
F =36+32
F =68

15°C.
10°C.

6
2
18
45
6 4
4 10
160
30
1.5
5,183
4.6
150
35,000
220,000
2 5
6
18
1 1
2
2 1 4
14
8
7
3
6
15
20
500
2
4 1 11
4 14
1 6
2 3
8
45, 10, 8, 65, 20, and 35 minutes.

6
5, 2, 8, 12, 6, and 3.
8
18
8
6-foot-6-inch
3-foot-8-inch
24
12
6 4
2-foot-10-inch
5 8
1.55
2.45
1,500
25
750
3,072
3
2
300
10,911
91.6
3.8
1.8 1.6
89 95
345
1.2
2-kilogram
180
5
420
560
30
200
8
6
7

75
42
24
9
1,650
90,000
5K 5
1.6
20
15
14
8
86°F 77°F 104°F
14°F 72°F 4°F
0°F 120°F
5°C 25°C −10°C
−15°C 22°C 8°C
43°C 16°C
40
1
16
65° 75°
a
b
a
b
c

a
b

−a
1 a
q p
q ≠0
a, b a + b = b + a
a, b a ⋅ b = b ⋅ a
a, b, c (a + b) + c = a + (b + c)
a, b, c (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)

a, b, c
a(b + c) = ab + ac
(b + c)a = ba + ca
a(b - c) = ab - ac
a +0=a 0+a = a
a ⋅1=a 1⋅a = a
a + ( - a) =0 -a
⎛
⎝ a ≠0⎞ ⎠ a ⋅ 1 a =1 1 a
a ⋅0=0 0⋅a = 0 The product of any number and 0 is 0.
0
a =0 0+a = 0 Zero divided by any real number, except itself, is zero.
0 a a ÷0
6 −5 2.9
1.8
0.42, 0.3 – , 2.56813… 0.75319…, 0.16 — , 1.95
a 49 b 55 a 72 b 64
a b c d e
−9, 0, 0.361...., 8 9 , 16,9
−5, − 2 1 —
, − 4, 0.25 , 13, 4
4 5

6 + 4 = ____ −14 · 5 = ____ 3n = ____
a + 8 = ____
(13 · 5) · 2 = _____ (22 + 7) + 3 = _____ (4 + 9x)+x = _____
1
2
(22y) = _____
y = 11
12 ,
a y + 0.7 + ( − y)
b y +(−y) + 0.7
z =− 5 3 ,
a z + 5.39 + ( − z)
b z +(−z) + 5.39
k = 65,
a 4 ⎛
9⎝ 9 4 k⎞ ⎠
b ⎛ ⎝ 4 9 · 94
⎞
⎠ k
m = −13,
a − 2 ⎛
5⎝ 5 2 m⎞ ⎠
b ⎛ ⎝ − 2 5 · 52
⎞
⎠ m
6y +37+(−6y)
1 + 11
4 15 + ⎛ ⎝ −1 ⎞
4⎠
14
11 · 35 9 · 11
14
−18 · 15 · 29
⎛
⎝12 7 + 4 ⎞
5⎠ + 1 5
(3.98d + 0.75d) + 1.25d
−12(4m)
30 ⎛ ⎝ 5 6 q⎞ ⎠
11x +8y +16x +15y
52m + (−20n) + (−18m)+(−5n)
7(x +9) 9(u −4) −3(6m −1)
−8(−7a − 12) 1(15n −6) 3
(y + 10) · p
(a −4)−(6a +9) 4(x +3)−8(x −7)

u =2,
n = 7 8 ,
a 3(8u +9) and
b 3·8u +3·9
3(8u +9) =3·8u +3·9
a 8 ⎛ ⎝ n + 1 ⎞
4⎠
b 8·n +8·14 8⎛ ⎝ n + 1 ⎞
4⎠ =8·n +8·14
d = 14,
y = −18,
a −100(0.1d + 0.35)
b −100 · (0.1d)+(−100)(0.35)
−100(0.1d + 0.35) = −100 · (0.1d)+(−100)(0.35)
a − ⎛ ⎝y −18 ⎞ ⎠
b −y +18 − ⎛ ⎝y −18 ⎞ ⎠ =−y +18
−35(1) = −35 29 + 0 = 29 (6x +0)+4x =6x +4x
9·1+(−3) =9+(−3)
−32 19.4 3
5
− 7 15
9
2
−5 1
10
− 4 9
83 · 0 0 5
9 0
0÷ 2 3
43 + 39 + (−43) (n + 6.75) + 0.25
5
13 ·57·13 5
1
6 ·17·12 2
3 ·28·37
9(6x −11)+15
7 42 5 4

45
1 3 4
14,179
4.5
5
14
30
4
1
9 1 12
50, 25, 83, 45, 32, 60, and 135
4 2
2 15
6 2
8 10
30
4
1.7
8,850
488
13
2.9
650
2
88
2
5
1-liter
145
30
30
69 2.5
84
78
55
25

95°F 23°F 20°F
64°F
30°C −5°C −12°C
24°C

0.18349…, 0.2 – , 1.67,
a b
144
−4, − 1 1 2 , 0, 5, 2, 7,
8
a b c
d
x · 14 = _________
(y + 6) + 3 = _______________
(8 · 2) · 5 = ___________
3
16
⎛
⎝
16
3
n ⎞ ⎠
2 5
n = 42.
a b
3(−29) ⎛ 4 ⎝ 4 ⎞
−3 + 15y +3
3⎠
⎛
⎝1.27q + 0.25q ⎞ ⎠ + 0.75q
⎛
⎝15 8 + 2 ⎞
9⎠ + 7 9
−18 ⎛ ⎝ 3 2 n⎞ ⎠
14y +(−6z)+16y +2z
9 ⎛ ⎝q +9 ⎞ ⎠ 6(5x −4) −10(0.4n + 0.7)
1
4
(8a +12) m(n +2) 8 ⎛ ⎝6p −1 ⎞ ⎠ +2 ⎛ ⎝9p +3 ⎞ ⎠
(12a +4)−(9a +6) 0
8
4.5
0
0÷ ⎛ ⎝ 2 ⎞
3⎠
4 1
2 276
(1 mile = 5,280 feet).
(1 milligram = 0.001 gram)
5
28, 44, 9, 75, and 55
(1 hour = 60 minutes)
15
(1 mile = 1.61 kilometers)
63
(1 inch = 2.54 centimeters)
F = 9 C +32 35°C
5
F

n − 12 = 16.
x.’
x = 2 5x − 3 = 7?

y = 3 4
4y +3=8y.
y = 3 4 3
4
4y +3=8y.
y = 2 3
9y + 2 = 6y?
y = 2 5
5y − 3 = 10y?

a, b, c, a = b, a − c = b − c.
a, b, c, a = b, a + c = b + c.
x +3=8.
3
x =5.
x + 11 = −3.
x, 11

x = −14
x = −14 x +11=−3
x + 9 = −7.
x +16 = −4.
11 x 11
m +4=−5.
m =−9
m −4=−5 m =−1
n − 6 = −7.
x − 5 = −9.
n − 3 8 = 1 2 .

p − 1 3 = 5 6 .
q − 1 2 = 1 6 .
a − 3.7 = 4.3.
a =8
b − 2.8 = 3.6.
c − 6.9 =7.1.

3x −7−2x −4=1.
x
x =12
8y − 4 − 7y − 7 = 4.
6z + 5 − 5z − 4 = 3.
3(n −4)−2n =−3.

n =9
5(p − 3) − 4p = −10.
4(q + 2) − 3q = −8.
2(3k −1)−5k =−2−7.
k =−7.

4(2h − 3) − 7h = −6 − 7.
2(5x + 2) − 9x = −2 + 7.
x 26.
26 21
x 41.
y 51.

5p 4p 23.
4x 3x 14.
7a 6a −8.
71
28

=
b
43 + 28 = ? 71
71 = 71 ✓
13 6
68
26

$24,575 $875
s =
s
25,450 − 875 = ? 24,575
24,575 = 24,575 ✓
$19,875 $1,025
$7.75.
$3.25

y = 1 3
x = 3 4
u =− 1 2
4y +2=10y?
5x +3=9x?
8u −1=6u?
v =− 1 3
9v −2=3v?
x +7=12 y +5=−6 b + 1 4 = 3 4
a + 2 5 = 4 5
p + 2.4 = −9.3 m + 7.9 = 11.6
a −3=7 m − 8 = −20 x − 1 3 =2
x − 1 5 =4 y − 3.8 = 10 y − 7.2 = 5
x − 15 = −42 z + 5.2 = −8.5 q + 3 4 = 1 2
p − 2 5 = 2 3
y − 3 4 = 3 5
c +3−10=18 m +6−8=15 9x +5−8x +14=20
6x +8−5x +16=32 −6x −11+7x − 5 = −16 −8n −17+9n − 4 = −41
3(y −5)−2y =−7 4(y −2)−3y =−6 8(u +1.5)−7u = 4.9
5(w + 2.2) − 4w = 9.3 −5(y −2)+6y =−7+4 −8(x −1)+9x =−3+9
3(5n −1)−14n +9=1−2
2(8m +3)−15m −4=3−5
−(j +2)+2j −1=5
−(k +7)+2k +8=7 6a −5(a −2)+9=−11 8c −7(c −3)+4=−16
8(4x + 5) − 5(6x)−x =53
6(9y − 1) − 10(5y)−3y =22

x
21.
x −5 33. m −14.
y −19. y −3 40. p
52.
9x 8x
17.
5c 4c
60.
n 1 6
1
2 .
f 1 3 −4n 5n
−32.
1
12 .
−9m 10m
−25.
18
7
54
41
3
42
5
12
4.2
31.6
$17.43
$103.76.
8
124
0.7
101.2
5
16
$22.85
$93.75.

2
3
1
4
s
1 4 + s = 2
3
5
8 -inch 12 1
d
d +
12 1 = 5
8
−18 3x =16−5x?
y −18=41
a
b

−7 ⎛ 1 ⎞ ⎝ −7⎠
.
− 3 8 ?
9x + 2 x = −3.
a, b, c, c ≠0, a = b, a c = b c .
a, b, c, a = b, ac = bc.
4, 4
4x = −28.
4.

x =−7
x =−7 4x = −28.
3y = −48.
4z = −52.
a
−7 = −42.
a −7. −7 a.
−7
a = 294
b
−6 = −24.
c
−8 = −16.

−r =2.
−r −1r.
−r −1r
−1
r =−2
−r =2.
−1.
−k = 8.
−g = 3.
2 3 x = 18.
1, x
2
3 .

2 3
x =27
2 3 x =18 2 3
x.
2 5 n = 14.
5 6 y = 15.
8x +9x −5x = −3 + 15.
x =1

7x + 6x − 4x = −8 + 26.
11n − 3n − 6n = 7 − 17.
11 − 20 = 17y −8y −6y.
y =−3
18 − 27 = 15c − 9c − 3c.
18 − 22 = 12x − x − 4x.
−3(n −2)−6=21.

n =−7
−4(n − 2) − 8 = 24.
−6(n − 2) − 12 = 30.

8x =32 7p =63 −5c =55
−9x = −27 −90=6y −72 = 12y
−16p = −64 −8m = −56 0.25z = 3.25
0.75a = 11.25 −3x =0 4x =0
x =15
z
4 2 =14 −20 = q
−5
c
−3 = −12
y
9 =−6 q
6 =−8
m =5 −12 −4 = p
−20
2
3
y =18
3
5
r =15 − 5 8 w =40 24 = − 3 4 x
− 2 5 = 1 10 a −1 3 q =−5 6
8a +3a −6a = −17 + 27 6y −3y +12y =−43+28 −9x −9x +2x =50−2
−5m +7m −8m =−6+36 100 − 16 = 4p −10p − p −18−7=5t −9t −6t
7
8
n − 3 4 n =9+2 5
12
q + 1 2 q =25−3 0.25d + 0.10d = 6 − 0.75
0.05p − 0.01p = 2 + 0.24
18
3
b,
3b = 18.
24
4
g,
4g = 24.

$36.25 5
p, 5p = 36.25.
18 mpg. m,
1 2 m = 18.
$12.96 12
b,
12b = 12.96.
14
f ,
1 3 f = 14.
−3x =36
3
x =40
1 x = 80.
2
a
b

4y − 9 + 9.
y + 12 = 16.
−3y = 63.
ax = b.
4x + 6 = −14.

x
x =−5
3x + 4 = −8.
5a + 3 = −37.
2y − 7 = 15.
7 2y,
7
y =11

5y − 9 = 16.
3m − 8 = 19.
5x =4x +7.
x,
4x
x
6n = 5n + 10.
−6c = −7c + 1.
5y −8=7y.
−8, y,

5y
y =−4
3p −14= 5p.
8m + 9 = 5m.
7x =−x + 24.
24,
−x x
x =3
12j = −4j + 32.

8h = −4h + 12.
7x +5=6x +2.
7x
6x. 7 6,
6x
x =−3
x =−3
12x + 8 = 6x + 2.
9y + 4 = 7y + 12.
1,

6n −2=−3n +7.
6n −3n 6> −3,
3n
n
n
8q − 5 = −4q + 7.
7n − 3 = n + 3.
2a −7=5a +8.
2a 5a 5>2,

2a
a
a =−5
2a − 2 = 6a + 18.
4k − 1 = 7k + 17.
3 2 x +5=1 2 x −3.
3 2 > 1 2 ,

1 2 x
x =−8
7 8 x − 12 = − 1 8 x − 2.
7 6 y + 11 = 1 6 y + 8.
3.4x + 4 = 1.6x −5.
3.4 > 1.6,
1.6x
x =−5

2.8x + 12 = −1.4x − 9.
3.6y + 8 = 1.2y − 4.
1.
3(x + 2) = 18.
x
x =4

5(x + 3) = 35.
6(y − 4) = −18.
−(x +5)=7.
x
x = −12
−(y + 8) = −2.
−(z + 4) = −12.
4(x −2)+5=−3.

x
x =0
2(a − 4) + 3 = −1.
7(n − 3) − 8 = −15.
8 − 2(3y +5)=0.

y =− 1 3
12 − 3(4j + 3) = −17.
−6 − 8(k − 2) = −10.
3(x −2)−5=4(2x +1)+5.

3x 8>3
−4 = x
6(p − 3) − 7 = 5(4p + 3) − 12.
8(q + 1) − 5 = 3(2q − 4) − 1.
1 (6x −2)=5−x.
2

x
x = 3 2
1 (6u + 3) = 7 − u.
3
2 (9x − 12) = 8 + 2x.
3
0.24(100x + 5) = 0.4(30x + 15).

12x x
x = 0.4
0.55(100n + 8) = 0.6(85n + 14).
0.15(40m − 120) = 0.5(60m + 12).

6x −2=40 7x −8=34 11w +6=93
14y +7=91 3a + 8 = −46 4m + 9 = −23
−50 = 7n −1 −47 = 6b +1 25 = −9y +7
29 = −8x −3 −12p −3=15 −14q − 15 = 13
8z =7z −7 9k =8k −11 4x +36=10x
6x +27=9x c =−3c −20 b =−4b −15
5q =44−6q 7z =39−6z 3y + 1 2 =2y
8x + 3 4 =7x −12a − 8 = −16a −15r − 8 = −11r
6x −15=5x +3 4x −17=3x +2 26 + 8d =9d +11
21 + 6 f =7f +14 3p −1=5p −33 8q −5=5q −20
4a +5= −a −40 9c +7=−2c −37 8y −30=−2y +30
12x −17=−3x +13 2z−4=23−z 3y −4=12−y
5
4
c −3= 1 4 c −16 4
3
m −7= 1 3 m −13 8−2 5 q = 3 5 q +6
11 − 1 4 a = 3 4 a +4 4
3
n +9= 1 3 n −9 5
4
a +15= 3 4 a −5
1
4
y +7= 3 4 y −3 3
5
p +2= 4 5 p −1 14n + 8.25 = 9n + 19.60
13z + 6.45 = 8z + 23.75 2.4w − 100 = 0.8w +28 2.7w − 80 = 1.2w +10
5.6r + 13.1 = 3.5r + 57.2 6.6x − 18.9 = 3.4x + 54.7

5(x +3)=75 4(y +7)=64 8=4(x −3)
9=3(x −3) 20(y − 8) = −60 14(y − 6) = −42
−4(2n +1)=16 −7(3n +4)=14 3(10 + 5r) =0
8(3 + 3p) = 0 2
3
(9c −3)=22 3
5
(10x −5)=27
5(1.2u − 4.8) = −12 4(2.5v − 0.6) = 7.6 0.2(30n + 50) = 28
0.5(16m + 34) = −15 −(w −6)=24 −(t −8)=17
9(3a +5)+9=54 8(6b −7)+23=63 10 + 3(z +4)=19
13 + 2(m −4)=17 7+5(4−q) =12 −9+6(5−k) =12
15 − (3r +8)=28 18 − (9r + 7) = −16 11 − 4(y −8)=43
18 − 2(y −3)=32 9(p −1)=6(2p −1) 3(4n −1)−2=8n +3
9(2m −3)−8=4m +7 5(x −4)−4x =14 8(x −4)−7x =14
5 + 6(3s −5)=−3+2(8s −1)
−12+8(x −5)=−4+3(5x −2)
4(x − 1) − 8 = 6(3x −2)−7
7(2x − 5) = 8(4x −1)−9
150 15
w,
150 = 2(w +15)+2w.
$1,506.
$6 $9.
5 3
s,
6s +9(3s − 5) = 1506.
$1.90
n,
0.05n + 0.10(2n − 1) = 1.90.
74
25 L,
2L +2(L − 25) = 74.
10x +14=−2x + 38,

3−7(y − 4) = 38?
1
4
(8x + 20) = 3x −4
a
b

8 · 38 .
5 6 and 1 4 .
4.78 100.
1 8 x + 1 2 = 1 4 .
x 1 2
1 8

1 8 x + 1 2 = 1 4 .
x =−2
1 4 x + 1 2 = 5 8 .
1 6 y − 1 3 = 1 6 .

7= 1 2 x + 3 4 x − 2 3 x.
x = 12.
6 = 1 2 v + 2 5 v − 3 4 v.
−1 = 1 2 u + 1 4 u − 2 3 u.
x + 1 3 = 1 6 x − 1 2 .

x
x =−1.
a + 3 4 = 3 8 a − 1 2 .
c + 3 4 = 1 2 c − 1 4 .
1= 1 (4x +2).
2

x =0.
−11 = 1 (6p + 2).
2
8 = 1 (9q + 6).
3
1 2 (y −5)=1 (y −1).
4

y
9 y.
1 5 (n + 3) = 1 (n + 2).
4
1 2 (m − 3) = 1 (m − 7).
4
0.3 =
10 3 0.17 = 17 .
100
0.8x −5=7.
0.8. 0.8 = 8 10 ,
10. 10

x = 15.
0.6x − 1 = 11.
1.2x − 3 = 9.
0.06x + 0.02 = 0.25x − 1.5.
0.06 =
100 6 , 0.02 = 100 2 , 0.25 = 25
100 , 1.5 = 1 10
5
100.

x =8.
0.14h + 0.12 = 0.35h − 2.4.
0.65k − 0.1 = 0.4k − 0.35.
0.25x + 0.05(x + 3) = 2.85.

x =9.
0.25n + 0.05(n + 5) = 2.95.
0.10d + 0.05(d − 5) = 2.15.

1
4
x − 1 2 =−3 4
3
4
x − 1 2 = 1 4
5
6
y − 2 3 =−3 2
5
6
y − 1 3 =−7 6
1
2
a + 3 8 = 3 4
5
8
b + 1 2 =−3 4
2= 1 3 x − 1 2 x + 2 3 x 2=3 5 x − 1 3 x + 2 5 x 1
4
m − 4 5 m + 1 2 m =−1
5
6
n − 1 4 n − 1 2 n =−2 x + 1 2 = 2 3 x − 1 2
x + 3 4 = 1 2 x − 5 4
1
3
w + 5 4 = w − 1 4
3
2
z + 1 3 = z − 2 3
1
2
x − 1 4 = 1 12 x + 1 6
1
2
a − 1 4 = 1 6 a + 1 12
1
3
b + 1 5 = 2 5 b − 3 5
1
3
x + 2 5 = 1 5 x − 2 5
1= 1 6 (12x −6) 1=1 5 (15x − 10) 1(p −7)= 1 (p +5)
4 3
1
5
(q +3)= 1 2 (q −3) 1
2
(x +4)= 3 4
1
3
(x +5)= 5 6
0.6y +3=9 0.4y −4=2 3.6j − 2 = 5.2
2.1k + 3 = 7.2 0.4x + 0.6 = 0.5x − 1.2 0.7x + 0.4 = 0.6x + 2.4
0.23x + 1.47 = 0.37x − 1.05
0.48x + 1.56 = 0.58x − 0.64
0.9x − 1.25 = 0.75x + 1.75
1.2x − 0.91 = 0.8x + 2.29 0.05n + 0.10(n + 8) = 2.15 0.05n + 0.10(n + 7) = 3.55
0.10d + 0.25(d + 5) = 4.05 0.10d + 0.25(d + 7) = 5.25 0.05(q − 5) + 0.25q = 3.05
0.05(q − 8) + 0.25q = 4.10

$2.00
2
0.10d + 0.01(d +2)=2
d,
$9.45 49-cent
21-cent 21-cent
5 49-cent
0.49s + 0.21(s − 5) = 9.45 s,
49-cent
3 8 , 1, and 2 6 3 .
0.35x + 2.1 = 3.85,
a
b

a - c = b - c
a + c = b + c
c ≠0 a = b a c = b c
a = b ac = bc

x + 16 = 31, x =15 w −8=5, w =3 −9n = 45, n =54
4a = 72, a =18
x +7=19 y +2=−6 a + 1 3 = 5 3
n + 3.6 = 5.1
u −7=10 x −9=−4 c − 3 11 = 9 11
p − 4.8 = 14
n −12=32 y +16=−9 f + 2 3 =4
d − 3.9 = 8.2 y +8−15=−3 7x +10−6x +3=5
6(n −1)−5n = −14
8(3p + 5) − 23(p −1)=35
−6 m n 13.
25.

11 146
3
15
$9.75
$46.25
$152.84
$21.65
8x =72 13a = −65 0.25p = 5.25
−y =4
n =18 y
6
−10 =30 36 = 3 4 x
5
8
u = 15
16
−18m = −72 c =36 0.45x = 6.75
9
11
12 = 2 3 y 5r −3r +9r =35−2 24x +8x −11x = −7−14
8p +7=47 10w −5=65 3x + 19 = −47
32=−4−9n
7y =6y −13 5a +21=2a k =−6k −35
4x − 3 8 =3x
12x −9=3x +45 5n −20=−7n −80 4u + 16 = −19 − u
5
8
c −4= 3 8 c +4

6(x +6)=24 9(2p −5)=72 −(s +4)=18
8+3(n −9)=17 23 − 3(y −7)=8 1
3
(6m + 21) = m −7
8(r −2)=6(r + 10)
4(3.5y + 0.25) = 365
5 + 7(2 − 5x) = 2(9x +1)−(13x − 57)
0.25(q − 8) = 0.1(q +7)
2
5
n − 1 10 = 7 10
1
3
x + 1 5 x =8 3
4
a − 1 3 = 1 2 a + 5 6
1(k +3)= 1 (k + 16)
2 3
0.8x − 0.3 = 0.7x + 0.2 0.36u + 2.55 = 0.41u + 6.8 0.6p − 1.9 = 0.78p + 1.7
0.10d + 0.05(d − 4) = 2.05

3x + 5 = 23.
a 6 b 23 5
n −18=31 9c = 144 4y −8=16
−8x −15+9x − 1 = −21 −15a = 120 2
3
x =6
x + 3.8 = 8.2 10y =−5y +60 8n +2=6n +12
9m −2−4m + m =42−8 −5(2x +1)=45 −(d +9)=23
1
3
(6m + 21) = m −7 2(6x +5)−8=−22
8(3a + 5) − 7(4a −3)=20−3a
1
4
p + 1 3 = 1 2
0.1d + 0.25(d + 8) = 4.1
x 4 16.
$25.82
$3.47

“6 x”
2 3 x = 24.

3x + 8 = 14.

$18,
p=

p = 36, $36. $36
18 36,
$36.”
$120,
16

11
b = number of bananas
3
5
14

$60,
8%
c = the original cost
c = 750
4%. $38.
12% $504
13.

n = the number
17.
−7.
15.
n = the number

14.
25.
n = 1st number
x +5=2 nd number

n =−5
7

40

...1, 2, 3, 4,...
...−10, −9, −8, −7,...
...150, 151, 152, 153,...
n,
n +1. n +1, n +1+1, n +2.
n
n + 1
n + 2
1st integer
2nd consecutive integer
3rd consecutive integer
47.
95.
−31.
42.

96.
−36.

25
20 24
23
8
9
6%
$30
$72. 60%
16
4
6
$9 $50
$40.50.
45%
18
645.
3
3
12%
21
$15 $75
12.
17.
3.
4.
17.
30.
23.
21.
14.
12.

85.
71.
77.
89.
−23.
−37.
78.
60.
−36.
−3.
$35
$10
$6.25 5
$25 $60
8
$3.20.
$1,166.40
$86.40
$2,279
$129
a
b

14(0.25).
100(0.2 + 0.05n).
0.25x + 0.10(x + 4) = 2.5

17
$0.10 17 17 $0.10
$1.70. 17
17 · $0.10 = $1.70
number · value = total value
number · value = total value
14 17 21 39
Number Value ($) Total Value ($)
14 0.25 3.50
17 0.10 1.70
21 0.05 1.05
39 0.01 0.39
6.64
$6.64.
$2.25
$0.10 $0.05. $2.25.

Number Value ($) Total Value ($)
0.10
0.05
2.25
Let d = number of dimes.
d + 9 = number of nickels
Number Value ($) Total Value ($)
d 0.10
d +9 0.05
2.25
number · value
Number Value ($) Total Value ($)
d 0.10 0.10d
d +9 0.05 0.05(d +9)
2.25

12 dimes: 12(0.10) = 1.20
21 nickels: 21(0.05) = _____ 1.05
$2.25 ✓
Adalberto has twelve dimes and twenty-one nickels.
12 dimes 12(0.10) = 1.20
21 nickels 21(0.05) = _____ 1.05
$2.25
$2.05
$2.10 12

$2.43
Number Value ($) Total Value ($)
0.25
0.01
2.43
We are looking for the number of quarters and pennies.

We know Maria has twice as many pennies as quarters. The number of pennies is defined in terms of quarters.
Let q represent the number of quarters.
Then the number of pennies is 2q.
Number Value ($) Total Value ($)
q 0.25
2q 0.01
2.43
Number Value ($) Total Value ($)
q 0.25 0.25q
2q 0.01 0.01(2q)
2.43
9 18 $2.43?
9 quarters 9(0.25) = 2.25
18 pennies 18(0.01) = _____ 0.18
Total
$2.43 ✓
$4.20

$9.35
$2.14
$0.01.
$0.05.
p = number of pennies
10p + 2 = number of nickels
Number Value ($) Total Value ($)
p 0.01 0.01p
10p +2 0.05 0.05 ⎛ ⎝10p +2 ⎞ ⎠
$2.14
4 42 $2.14?

4(0.01) +42(0.05) = ? 2.14
2.14 = 2.14 ✓
4 42
$6.55
$7.00
$1,506. $6
$9 5
Number Value ($) Total Value ($)
6
9
1,506
We are looking for the number of student and adult tickets.
We know the number of adult tickets sold was 5 less than three times the number of student tickets sold.
Let s be the number of student tickets.
Then 3s − 5 is the number of adult tickets.
Multiply the number times the value to get the total value of each type of ticket.

Number Value ($) Total Value ($)
s 6 6s
3s −5 9 9(3s −5)
1,506
6s +9(3s − 5) = 1506
6s +27s − 45 = 1506
33s − 45 = 1506
33s = 1551
s = 47 students
47 $6 136 $9 $1506?
47 · 6 = 282
136 · 9 = _____ 1224
1506 ✓
47 136
$17,610.
$20 $30. 37
$2,612.50. $10
$7.50 25
$10.44 49-cent
8-cent 49-cent 8-cent
49-cent 8-cent
49 8

Let x = number of 8-cent stamps
2x + 4 = number of 49-cent stamps
Number Value ($) Total Value ($)
49-cent 2x +4 0.49 0.49(2x +4)
8-cent x 0.08 0.08x
10.44
0.49(2x +4) + 0.08x = 10.44
0.98x + 1.96 + 0.08x = 10.44
1.06x + 1.96 = 10.44
1.06x = 8.48
x =8
2x +4forx =8.
8(0.08) + 20(0.49) = ? 10.44
0.64 + 9.80 = ? 10.44
10.44 = 10.44 ✓
2x +4
2⋅8+4
16 + 4
20
8-cent 49-cent
$16.64 49-cent
8-cent 49-cent
8-cent
$14.84 49-cent
21-cent 49-cent 21-cent

$2.60 $1.75
14
11
$3.50.
7
$6.30.
14
$11.30
3
$1
$5 $47.
$1 5
$5
$2.55
9
$9.70
8
$125
$5 $10
$10
$5
$2.75
10
$1 $5
$45. $1
3
$5
$175 $5
$10
$5
$10
$3.75
$4.70
$550 $8
10
$5
$12
$584,
$1,220
$7
$9

$1,340
$12
15
$5
$0.49
$0.34
$62.60.
20
$0.49
$0.34
$12.52.
4
$16.80
$0.49
$0.21
$43.40
$0.49 $0.21
$210 $10
$12
$10 5
$12
$475 $45
$25
$25 5
$45
3-day
$75 $30.
$765.
$24
$16
$400
6 9 4
a

x + 3 + 6 = 11.
a
45
= 4 3 .
36 + 64.
180’?
180
∠A A.
∠A m ∠ A.
point A.
∠ A
° m
∠A 27°, m ∠ A = 27.
180°,
180°.

180°.
90°,
90°.
90°.
180°,
∠A ∠B m∠A + m∠B = 180°.
90°,
∠A ∠B m∠A + m∠B = 90°.

40°. a b
a
b

25°. a b
77°. a b
90 180
30°
100°

40°
ΔABC, ABC
ΔABC
A, B, and C a, b, and c.
180°.
m∠A + m∠B + m∠C = 180°
ΔABC, 180°.
m∠A + m∠B + m∠C = 180°
55° 82°.

31° 128°.
49° 75°.
90°
90°
28°.

56°.
45°.
20°

50°
30°

ΔABC
ΔXYZ
ΔABC ΔXYZY
ΔABC:
the length a can also be written BC
the length b can also be written AC
the length c can also be written AB
ΔABC ΔXYZ

Δ ABC
Δ XYZ
AB
XY = BC
YZ = AC
XZ
AB =4 XY =3 AB
XY = 4 3
Δ ABC
Δ XYZ
ΔABC ΔXYZY . a.

ΔABC ΔXYZY . y.
500
90°
90°
90°
ΔABC,
a 2 + b 2 = c 2
c a b
m
25 5 5 2 = 25.
If m = n 2 , then m = n for n ≥0

c = the length of the hypotenuse

= the leg of the triangle

10-inch

13-ft
5
17-ft
15-ft

a b
53° 16° 29°
72°
135°
38°
27.5°
109.5°
56°
36°
34°
52°
26° 98°.
47° 72°.
22.5 ° .
61° 84°.
33°.
36.5 ° .
105° 31°.
51°.
20°
20°

ΔABC ΔXYZ.
b
x
270

13-foot
12-foot
12 16
10
20-foot
6

30 35
2.5
X Y,
XYZ,
X. X
Z, 800 Y Z, 1,000
3 4.
a
b

3 w
1 2 (6h).
5 (10.3 − 7.9).
2

1
2 3 1 1 1
6 6
3
27
1 27

a b c
d e f
a
b
c
d
e
f
a b c d e
f
a b c d e
f
1 4
1

Perimeter = 4 inches
Area = 1 square inch
1
a
b
a 6
b 2 2
a b

a b
L, W.
P,
L + W + L + W
P = L + W + L + W
or
P =2L +2W
2
3 6 A =2⋅3, A,
L, W, A = L ⋅ W.
6
(90°)
P,
P =2L +2W
A,
A = L ⋅ W

32 20 a b
a

120 50 a b
62 48 a b
50 10

80 25
30 6
52

52 = 2L +2L −4
52 = 4L−4
56 = 4L
56
4
= 4L 4
14 = L
14 + 12 + 14 + 12 = 52
58
60

32
64
18
168 14

598 23
21 609
150 15

W = width
W + 15 = length
200 40
30 300
b h, bh.

, h.
1 2 bh.
A = 1 2 bh.
A = 1 bh, b h
2
90°
h
90°
ΔABC, 180°.
m∠A + m∠B + m∠C = 180°
P = a + b + c
b, h.
A = 1 2 bh

11 8
13 2
14 7
48 4 9

48 18 22
7 5 18
90 15

126 18
15 5

93

39
51
156
60

20 48
8 22
b, B.
h,
B, b. h

Area trapezoid = 1 h(b + B)
2
A, A= 1 h(b + B)
2
14 11

A = the area
B h,
b h,
84 66
84 66
75
14 7 16
18 17 8
5 10.3 13.7

7 4.6 7.4
9 6.2 7.8
3.4 8.2
5.6

10.8 6.7
4.6
18
14 15

a b 1
a b
85 45
26 58
15
14
20 35
124
38
20.2
7.8
92
19
16.2
3.2
414
18
782
17

9
46
62 7
150
5
34
2310
42
306
17
8
52
0.7
52.6
72
24
624
132
12
40
224
58
5
1.1
49.4
3
36
28
1316
160
10
7
46
12
5
45
30
8.3 6.1
24.2
20.5
1 1.5
8 6
6
9
23
893
38
14
18
49
36
10
15
207
18
120
30
50

20
76
54
42
12
1.5
39
1
2
3
13
25
95
18
83
24
36 91
35
5
3
51
42
8
0.75
2
5
17
12 9
15
24 18
30
51
43 67
62
58 75
4.2
8.1 5.5
15
12.5 18.3
32.5
54.6 41.4
48 38.6
60.2
3
4 5

5
5 8
18.5
62 50
8
48.2
56.2
50
10
6
8 10 $10
48
8
20 12
$0.05.
a
b
5
50
a
b

a
b

x 2 x = 5.
3.14
π , 8
22 7 (0.25)2
r

d
d =2r
C =2πr
A = πr 2
π 3.14 22 7
π
π
2.5 a b
a
C =2πr
C =2π(2.5)
C ≈2(3.14)(2.5)
C ≈ 15ft

A =πr 2
A = π(2.5) 2
A ≈ (3.14)(2.5) 2
a
A ≈ 19.625 sq. ft
5 a b
4.5 a b
r
C =2πr
of d.
C =2πr
Using the commutative property, we get C = π ·2r
Then substituting d =2r C = π ·d
So
C = π d

π.
C = πd
C = π(4)
C ≈ (3.14)(4)
C ≈ 12.56 feet
5.5
12
47.1

π
47.1 = ? (3.14)(15)
47.1 = 47.1 ✓
94.2
345.4

12 4. 2,
4 6
60

8 4
4, 3 7−4
8, 3 8−4
36.5
105
68

105 68 68
34

7
a b
8 a
b
8.5
a
b
3.5 a
b
12
9.5
20
3
163.28
10
2.2
59.66
17.27
80.07
150.72
251.2
40.82
78.5

250
12 20
3
120
100
50
40
20
30

20-ft 12-ft
6
1-ft 3-ft
a
b
8-ft 10-ft
8 3-ft 6.5-ft
3-ft 6.5-ft
4-ft 3.5-ft
14
π. c
a 3.14 π b 22 7
a
b

x 3 x = 5.
2 x x = 5.
7 2 .

4 2 3
4 2 3
24
24 24 length × width × height.
V,
V = LWH
B, length×width.
B = L · W
B L · W
4×2×3

A front = L×W A side = L×W A top = L×W
A front =4·3 A side =2·3 A top =4·2
A front =12 A side =6 A top =8
S = (front + back)+(left side + right side) + (top + bottom)
S = (2 · front) + (2 · left side) + (2 · top)
S =2·12+2·6+2·8
S = 24 + 12 + 16
S = 52 sq. units
S 52
S =2LH +2LW +2WH
6
L, W, H :
14 17 9 a b
a b

a
V
V = LWH
V =14⋅9⋅17
V = 2,142
2,142
b
S
S =2LH +2LW +2WH
S =2(14 ⋅ 17) +2(14 ⋅ 9) +2(9 ⋅17)
S = 1,034
a b 8 9 11
a b 15 12
8
30 25 20 a b

a b
a
V
V = LWH
V =30⋅25⋅20
V = 15,000
b
S
S =2LH +2LW +2WH
S =2(30 ⋅ 20) +2(30 ⋅ 25) +2(25 ⋅ 20)
S = 3,700
9 4 6 a b
22 14 9 a b

V = LWH S =2LH +2LW +2WH
V = s · s · s S =2s · s +2s · s +2s · s
V = s 3 S =2s 2 +2s 2 +2s 2
S =6s 2
V = s 3 S =6s 2 .
s,
2.5 a b
a b
a
V = s 3
V = (2.5) 3
V = 15.625

S =6s 2
S = 6 ⋅ (2.5) 2
S = 37.5
a b
a b
2 a b
a
V = s 3
V =2 3
V =8

S =6s 2
S =6⋅2 2
S =24
4 a b
16 a b
π 3.14.
r:
6 a b
a b

a
V = 4 3 πr3
V ≈ 4 3 (3.14)63
V ≈ 904.32 cubic inches
b
S =4πr 2
S ≈ 4(3.14)6 2
S ≈ 452.16
a b
a b 1
14 a b

a
π
V = 4 3 πr3
V ≈ 4 3 (3.14)143
V ≈ 11,488.21
b
π
S =4πr 2
S ≈ 4(3.14)14 2
S ≈ 2461.76
9 a b
1.5 a b

h
h
V = Bh
B
B, πr 2 . V = Bh
L

W
r h,
S =2πr 2 +2πrh
r h :
5 3 a b

a
π
V = πr 2 h
V ≈ (3.14)3 2 ⋅5
V ≈ 141.3
b
π
S =2πr 2 +2πrh
S ≈ 2(3.14)3 2 + 2(3.14)(3)5
S ≈ 150.72
a b
a b
a b 4 13

a
π
V = πr 2 h
V ≈ (3.14)4 2 ⋅13
V ≈ 653.12
b
π
S =2πr 2 +2πrh
S ≈ 2(3.14)4 2 + 2(3.14)(4)13
S ≈ 427.04
a b
a b

V =πr 2 h.
πr 2 B
r h
6 2

π
V = 1 3
π r 2 h
V ≈ 1 3 3.14 (2)2 (6)
V ≈ 25.12
7 3
9 5
8 5

π
V = 1 3
π r 2 h
V ≈ 1 3 3.14 (2.5)2 (8)
V ≈ 52.33
18 12
10 7

a b
2 1.5 5 8 3.5 2.1
3
2.5
2.4
8.8
6.5
4.2
16
8 8
a b
22.8 8.5
8.2
a b
26
16 4
a b
21.3 24.2
6.5 a
b
a b
5 6 10.4
12.5
64
a b
a b
45
a b
2.8
a
b
a b
3 9 7.5
2.1

15
a b
36
a b
4.5
a b
2.9 a
b
a b
3 9
1.3 2.8
5
15
1.5
4.2
5
13 a
b
4 3
a b
6
24 a
b
8
28 a
b
9 2
8
6
12.4
5
15.2
4
10 10
8 6
50
70
12 30

30 1
10 0.5
a
b
c
2.5
7 3.25
a
b
c
8
8
a
b

35
6x + 24 =96.
5 $1,000 4%.
60 2
120
distance = ⎛ 60 miles⎞
⎝ 1 hour ⎠ (2 hours)
distance = 120 miles
distance = rate · time
d = rt
d = r = t =
miles .
hour
12 3 1 2

d =?
r =12mph
t =3 1 2 hours
d = rt
d =12⋅3 1 2
d = 42 miles
5 1 2
60
2 1 3
3
520
65

d = 520 miles
r =65mph
t =?
d = rt
520 = 65t
t =8
d = rt
520 = ? 65 ⋅ 8
520 = 520> ✓
770
70
168 3
d = rt. d
r t. t. d r
t.
t d r. d = rt
t.
1

d = rt t:
a d = 520 r =65
b
a
b
a t = d r t.
d = rt r:
a d = 180 t = 4
a
d = rt r:
a d = 780 t = 12
b
A = 1 bh
2
A = 1 bh. h:
2
a A =90 b =15
b

a
b
h = 2A b
A = 1 bh h:
2
a A = 170 b = 17
b
A = 1 bh b:
2
a A = 62 h = 31
b
I = Prt I P
r t
I = Prt P:
a I = $5,600, r =4%, t = 7 years
b

I = Prt.
t: a I = $2,160, r = 6%, P = $12,000; b
I = Prt.
r: a I = $5,400, P = $9,000, t = 5 years b
x y.
y x,
x y y.
3x +2y =18 y:
a x =4
b

3x + 4y = 10 y:
a x = 2
b
5x + 2y =18 y:
a x =4
b
P = a + b + c a.
a
P = a + b + c
P − b − c = a
a = P − b − c
P = a + b + c b.
P = a + b + c c.
3x + y =10 y.
y
3x + y =10
y =10−3x
7x + y = 11 y.
11x + y = 8 y.

6x +5y =13 y.
y
4x + 7y = 9 y.
5x + 8y = 1 y.

8 1 2
4 5 6
1 3 4
72
2 1 2
12
65
235
450
6
60
338
52
180
16
280 3.5
4
384
64
240
4
750
10
d=rt.
t:
a d = 350
r =70
b
t:
a d = 240
r =60
b
t:
a d = 510
r =60
b
t:
a d = 175
r =50
b
r:
a d = 160
t = 2.5
b
r:
a d = 204 t =3
b
r:
a d = 180
t = 4.5
b
r:
a d = 420 t =6
b

A= 1 2 bh.
b:
a A = 126
h =18
b
b:
a A =65
h =13
b
h:
a A = 176
b =22
b
h:
a A = 375
b =25
b
I=Prt.
P
a I = $5,480 r =4%
t = 7 years
b
P
a I = $3,950 r =6%
t = 5 years
b
t
a I = $2,376 P = $9,000
r = 4.4%
b
t
a I = $624
r = 5.2%
b
P = $6,000
2x +3y =12 y:
a x =3
b
5x +2y =10 y:
a x =4
b
3x + y =7 y:
a x =−2
b
4x + y =5 y:
a x =−3
b
a + b =90 b. a + b =90 a.
180 = a + b + c
a.
9x + y =13 y.
180 = a + b + c
c.
−4x + y =−6 y.
8x + y =15 y.
−5x + y =−1 y.
4x +3y =7 y.
3x +2y =11 y.
x − y =−4 y.
x − y =−3 y.
P =2L +2W L.
P =2L +2W W.
d.
C = πd
π.
C = πd
V = LWH
L.

V = LWH
H.
40°
F C = 5 (F −32)
9
50° C
F = 9 C +32
5
2x +3y =6 y:
a x =−3 b
c
5x −2y =10 x:
a y =10 b
c
a
b

90°
90°
180°
number · value = total value

Number Value ($) Total Value ($)
∠A ∠B m∠A + m∠B = 180
∠A ∠B m∠A + m∠B =90
ΔABC,
m∠A + m∠B = 180

⦜
P
P =2L +2W
A
A = L ⋅ W
ΔABC
m ∠ A + m ∠ B + m ∠ C = 180°
P = a + b + c
A = 1 2 bh
d =2r
C = 2πr C = πd
A = πr 2

V = LWH
S =2LH +2LW +2WH
V = s 3
S =6s 2
V = 4 3 πr3
S =4πr 2
V = πr 2 h
S =2πr 2 +2πrh
r h
V = 1 3 πr2 h
d = rt
9
15%
87 27
$350. 40%

−135.
$4.35 $0.39
5
8
$140 $5
$10 $10
$5
$3.69
3
$842. $10
$6
12
$2 $5
3 10
$619.
$4.06
$0.41
5
$0.26
$34.15
$0.56
10
4
$0.41
48°
61°
24°

45°
22 85 41.5
30°
60°
ΔABC ΔXYZ.
x b

36
8
10
15
a
b
a b
42 28
16
12
36 19
4 45

220
85
39 11.
2356
38
45 2700
12
74
3
96
18 33
15
21
30 40
9 12
32
59
3
5
6
20
9
39
81
8
11 14
5 7
10
25
32.5 21.5
62
91.5 78.1
150.72
3
a
b
345.4
8
a
b

a b
3
14 4.5
10
2.5
32 18
10
a b
4 12
1.45
22
a
b

2
6
0.8 2.5
18
40
8
15
5 24
1
8
2.6
2.6
6
5
4 380
1 1 2
8
68
107
305
5
d=rt.
t:
a d = 403
r =65
b
r:
a d = 750
t =15
b
A= 1 2 bh.
b:
a A = 416
h =32
b
h:
a A =48
b =8
b

I=Prt.
P,
a I = $720
t = 3 years
b
r =4%
t
a I = $3630
P = $11,000
r = 5.5%
b
6x +5y =20 y:
a x =0
b
180 = a + b + c
a.
P =2L +2W W.
2x + y =15 y:
a x =−5
b
4x + y =17 y.
V = LWH H.
a + b =90 a.
−3x + y =−6 y.

12
$2.95
5
14
13
−19.
$1600
$9
$4
30
145
3
65.
52°
ΔABC ΔXYZ.
c.
90
2
40
90
80 55
14
20
23
12
15 8
7
5
75
20
63
4

1 1
2
22
I = Prt
P,
a
I = $1380, r =5%, t =3
b
h:
A = 1 2 bh
a A = 1716
b =66
b
x +5y =14 y.

8x + 3x.
(5n + 8) − (2n − 1).

4y 2 y = 5
ax m , a m
b +1 4y 2 −7y +2 5x 5 −4x 4 + x 3 +8x 2 −9x +1
5 4b 2 −9x 3
3a −7 y 2 −9 17x 3 +14x 2
x 2 −5x +6 4y 2 −7y +2 5a 4 −3a 3 + a
a 8x 2 −7x −9 b −5a 4 c x 4 −7x 3 −6x 2 +5x +2 d 11 − 4y 3 e n

a 8x 2 −7x −9
b −5a 4
c x 4 −7x 3 −6x 2 +5x +2
d 11 − 4y 3
e n
a z b 2x 3 − 4x 2 − x − 8 c 6x 2 − 4x + 1 d 9 − 4y 2 e 3x 7
a y 3 − 8 b 9x 3 − 5x 2 − x c x 4 − 3x 2 − 4x − 7 d −y 4 e w
0
0.
1.

a 4x b 3x 3 −5x +7 c −11 d −6x 2 +9x −3 e 8x +2
a
x x = x 1
4x
b 3x 3 −5x +7
c
d
−11
−6x 2 +9x −3
e 8x +2
a −6y b 4x − 1 c 3x 4 + 4x 2 − 8 d 2y 2 + 3y + 9 e −18

a 47 b 2x 2 − 8x + 2 c x 4 − 16 d y 5 − 5y 3 + y e 9a 3
17x 2 +6x 2 .
23x 2
17x 2 +6x 2
12x 2 + 5x 2 .
−11y 2 + 8y 2 .
11n −(−8n).
11n − (−8n)
19n
9n − (−5n).
−7a 3 − ⎛ ⎝ ⎛ −5a 3⎞ ⎠ ⎞ .
a 2 +4b 2 −7a 2 .

a 2 +4b 2 −7a 2
−6a 2 +4b 2
−6a 2 4b 2
3x 2 + 3y 2 − 5x 2 .
2a 2 + b 2 − 4a 2 .
⎛ ⎝4x 2 −5x +1 ⎞ ⎠ + ⎛ ⎝3x 2 −8x −9 ⎞ ⎠.
⎛ ⎝ 3x 2 − 2x + 8 ⎞ ⎠ + ⎛
⎝x
2 − 6x + 2 ⎞ ⎠ .
⎛ ⎝ ⎛ 7y 2 + 4y − 6 ⎞ ⎠ ⎞ + ⎛ ⎝ ⎛ 4y 2 + 5y + 1 ⎞ ⎠ ⎞ .
⎛ ⎝7u 2 −5u +3 ⎞ ⎠ − ⎛ ⎝4u 2 −2 ⎞ ⎠.

⎛ ⎝ ⎛ 6y 2 + 3y − 1 ⎞ ⎠ ⎞ − ⎛ ⎝ ⎛ 3y 2 − 4 ⎞ ⎠ ⎞ .
⎛ ⎝ ⎛ 8u 2 − 7u − 2 ⎞ ⎠ ⎞ − ⎛ ⎝ ⎛ 5u 2 − 6u − 4 ⎞ ⎠ ⎞ .
⎛ ⎝m 2 −3m +8 ⎞ ⎠ ⎛ ⎝9m 2 −7m +4 ⎞ ⎠.
⎛ ⎝ ⎛ 4n 2 − 7n − 3 ⎞ ⎠ ⎞
⎛
⎝ ⎛ 8n 2 + 5n − 3 ⎞ ⎠ ⎞ .
⎛ ⎝ ⎛ a 2 − 4a − 9 ⎞ ⎠ ⎞
⎛
⎝ ⎛ 6a 2 + 4a − 1 ⎞ ⎠ ⎞ .
3x 2 −9x +7
a x =3 b x =−1

a x =3
3x 2 −9x +7
x 3(3) 2 −9(3) +7
3·9−9(3) +7
27 − 27 + 7
7
b x =−1
3x 2 −9x +7
x 3(−1) 2 −9(−1) +7
3·1−9(−1) +7
3+9+7
19
2x 2 + 4x − 3
a x = 2 b x =−3
7y 2 − y − 2
a y = −4 b y = 0
−16t 2 + 300 t 300
t =3
t

−8t 2 + 24t + 4 t
4 t = 3
−8t 2 + 24t + 4 x
4 t = 2

5x +2 z 2 −5z −6 a 2 +9a +18
−12p 4 y 3 −8y 2 +2y −16 10 − 9x
23y 2 m 4 +4m 3 +6m 2 +4m +1
8a 5 −2a 3 +1 5c 3 +11c 2 − c −8 3x −12
4y +17 −13 −22
6x 2 +9x 2 4y 3 +6y 3 −12u +4u
−3m +9m 5a +7b 8y +6z
4a , −3b, −8a 4x , 3y , −3x 18x −2x
13a −3a 5x 6 from − 12x 6 2p 4 from − 7p 4
(4y 2 +10y +3)+(8y 2 −6y +5)
(7x 2 −9x +2)+(6x 2 −4x +3)
(x 2 +6x +8)+(−4x 2 +11x −9)
(y 2 +9y +4)+(−2y 2 −5y −1)
(3a 2 +7)+(a 2 −7a − 18) (p 2 −5p − 11) + (3p 2 +9)
(6m 2 −9m −3)−(2m 2 + m −5)
(3n 2 −4n +1)−(4n 2 − n −2)
(z 2 +8z +9)−(z 2 −3z +1)
(z 2 −7z +5)−(z 2 −8z +6) (12s 2 −15s)−(s −9) (10r 2 −20r)−(r −8)
⎛ ⎝2p 3 −8 ⎞ ⎠ ⎛ ⎝p 2 +9p +18 ⎞ ⎠
⎛ ⎝q 2 +4q +13 ⎞ ⎠ ⎛ ⎝7q 3 −3 ⎞ ⎠

⎛ ⎝7x 2 −4x +2 ⎞ ⎠ ⎛ ⎝8x 2 − x +6 ⎞ ⎠
⎛ ⎝5x 2 − x +12 ⎞ ⎠ ⎛ ⎝9x 2 −6x −20 ⎞ ⎠
⎛
⎝w 2 + w −42 ⎞ ⎠
⎛
⎝z 2 −3z −18 ⎞ ⎠
⎛
⎝w 2 −10w +24 ⎞ ⎠
⎛
⎝z 2 +5z −20 ⎞ ⎠
Evaluate 8y 2 −3y +2
Evaluate 5y 2 − y − 7 when:
a y =5 b y =−2 a y =−4 b y =1
c y =0
c y =0
Evaluate 4 − 36x when:
a x =3 b x =0
c x =−1
Evaluate 16 − 36x 2 when:
a x =−1 b x =0
c x =2
275
−16t 2 + 275
t
t =4
−5p 2 + 350p.
p =50
x
−
160 1 x2 + 1 x.
2
x = 40 mph.
x
0.06x 2 + 1.1x.
x = 60 mph.
5x 2 +3x 4 8x 6 .
a
b

3 4 · 34 .
(−2)(−2)(−2).
2 4
2, 2 4 2·2·2·2.
a m th
a m , a
a 5 3 b 9 1

a
5 3
5·5·5
125
b
9 1
9
a 4 3 b 11 1
a 3 4 b 21 1
a ⎛ ⎝ 7 2
⎞
8⎠
b (0.74) 2
a
⎛
⎝ 7 2
⎞
8⎠
⎛
⎝ 7 ⎞
8⎠ ⎛ ⎝ 7 ⎞
8⎠
49
64

(0.74) 2
(0.74)(0.74)
0.5476
a ⎛5⎞ 2
⎝ 8⎠
b (0.67) 2
a ⎛2⎞ 3
⎝ 5⎠
b (0.127) 2
a
a (−3) 4 b −3 4
(−3) 4
(−3)(−3)(−3)(−3)
81
b
−3 4
−(3·3·3·3)
−81
a b a
b
a (−2) 4 b −2 4

a (−8) 2 b −8 2
x 2 ⋅ x 3
x 2+3
x 5
a m, n
a m · a n = a m + n
2 2 ·2 3 = ? 2 2+3
4·8 = ? 2 5
32 = 32 ✓
x 5 · x 7 .
a m · a n = a m + n .
x 5 · x 7
x 12

x 7 · x 8 .
x 5 · x 11 .
b 4 · b.
b 4 · b
b = b 1 . b 4 · b 1
a m · a n = a m + n .
b 5
p 9 · p.
m · m 7 .
2 7 ·2 9 .
2 7 ·2 9
a m · a n = a m + n .
2 16
6 · 6 9 .
9 6 · 9 9 .
y 17 · y 23 .

y 17 · y 23
y 40
y 24 · y 19 .
z 15 · z 24 .
x 3 · x 4 · x 2 .
x 3 · x 4 · x 2
x 9
x 7 · x 5 · x 9 .
y 3 · y 8 · y 4 .

(x 2 ) 3
x 2⋅3
x 6
a m, n
(a m ) n = a m · n
(5 2 ) 3 = ? 5 2·3
a
a (x 5 ) 7 a (3 6 ) 8
(a m ) n = a m · n .
(25) 3 = ? 5 6
15,625 = 15,625 ✓
⎛
⎝x 5⎞ 7
⎠
x 35

⎛
⎝3 6⎞ 8
⎠
(a m ) n = a m · n .
3 48
a (x 7 ) 4 b (7 4 ) 8
a (x 6 ) 9 b (8 6 ) 7
(2x) 3
2x ·2x ·2x
2·2·2·x · x · x
x? 2 3 · x 3
(2x) 3 is 2 3 · x 3
(2x) 3
2 3 · x 3
a b m
(ab) m = a m b m
(2 · 3) 2 = ? 2 2 ·3 2
(−11x) 2 .
6 2 = ? 4·9
36 = 36 ✓

(−11x) 2
(ab) m = a m b m .
121x 2
(−14x) 2 .
(−12a) 2 .
(3xy) 3 .
⎛
⎝3xy ⎞ 3
⎠
27x 3 y 3
(−4xyx ) 4 .
(6xyx ) 3 .
a, b m, n
Product Property
Power Property
Product to a Power Property
a m · a n = a m + n
(a m ) n = a m · n
(ab) m = a m b m
(x 2 ) 6 (x 5 ) 4 .

(x 2 ) 6 (x 5 ) 4
x 12 · x 20
x 32
(x 4 ) 3 (x 7 ) 4 .
(y 9 ) 2 (y 8 ) 3 .
(−7x 3 y 4 ) 2 .
(−7x 3 y 4 ) 2
(−7) 2 (x 3 ) 2 (y 4 ) 2
49x 6 y 8
(−8x 4 y 7 ) 3 .
(−3a 5 b 6 ) 4 .
(6n) 2 (4n 3 ).
(6n) 2 (4n 3 )
6n 6 2 n 2 ·4n 3
36n 2 ·4n 3
36·4·n 2 · n 3
144n 5

(7n) 2 (2n 12 ).
(4m) 2 (3m 3 ).
(3p 2 q) 4 (2pq 2 ) 3 .
(3p 2 q) 4 (2pq 2 ) 3
3 4 (p 2 ) 4 q 4 ·2 3 p 3 (q 2 ) 3
81p 8 q 4 ·8p 3 q 6
81 · 8 · p 8 · p 3 · q 4 · q 6
648p 11 q 10
(u 3 v 2 ) 5 (4uv 4 ) 3 .
(5x 2 y 3 ) 2 (3xyx 4 ) 3 .
(4x 2 )(−5x 3 ).
(4x 2 )( − 5x 3 )
4 · (−5) · x 2 · x 3
−20x 5
(7x 7 )(−8x 4 ).
(−9y 4 )(−6y 5 ).

⎛ ⎝ 3 4 c3 d ⎞ ⎠ (12cd2 ).
⎛
⎝ 3 4 c3 d ⎞ ⎠ (12cd2 )
3
4 ·12·c3 · c · d · d 2
9c 4 d 3
⎛4 ⎝ m 4 n 3 ⎞ 5 ⎠(15mn 3 ).
⎛2 ⎝ p 5 q ⎞ 3 ⎠ (18p 6 q 7 ).

4 5 10 3 ⎛ ⎝ 1 2
2⎞
⎠
⎛ ⎝ 3 2
⎞
5⎠
(0.2) 3 (0.4) 3
(−5) 4 (−3) 5 −5 4
−3 5 −10 4 −2 6
⎛ 3
⎝ −2 ⎞
3⎠
⎛ ⎝ −1 4
⎞
4⎠
−0.5 2
−0.1 4
x 3 · x 6 m 4 · m 2 a · a 4
y 12 · y 3 5 ·3 9 5 10 ·5 6
z · z 2 · z 3 a · a 3 · a 5 x a · x 2
y p · y 3 y a · y b x p · x q
(u 4 ) 2 (x 2 ) 7 (y 5 ) 4
(a 3 ) 2 (10 2 ) 6 (2 8 ) 3
(x 15 ) 6 (y 12 ) 8 (x 2 ) y
(y 3 ) x (5 x ) y (7 a ) b
(5a) 2 (7x) 2 (−6m) 3

(−9n) 3 (4rs) 2 (5ab) 3
(4xyz) 4 (−5abc) 3
(x 2 ) 4 ·(x 3 ) 2 (y 4 ) 3 ·(y 5 ) 2 (a 2 ) 6 ·(a 3 ) 8
(b 7 ) 5 ·(b 2 ) 6 (3x) 2 (5x) (2y) 3 (6y)
(5a) 2 (2a) 3 (4b) 2 (3b) 3 (2m 6 ) 3
(3y 2 ) 4 (10x 2 y) 3 (2mn 4 ) 5
(−2a 3 b 2 ) 4 (−10u 2 v 4 ) 3 ⎛ ⎝ 2 3 x2 y ⎞ 3
⎠
⎛ ⎝ 7 9 pq4 ⎞
⎠
2
(8a 3 ) 2 (2a) 4 (5r 2 ) 3 (3r) 2
(10p 4 ) 3 (5p 6 ) 2 (4x 3 ) 3 (2x 5 ) 4 ⎛ ⎝ 1 2 x2 y 3⎞ ⎠4
(4x 5 y 3 ) 2
⎛ ⎝ 1 3 m3 n 2 ⎞
⎠
4
(9m 8 n 3 ) 2 (3m 2 n) 2 (2mn 5 ) 4 (2pq 4 ) 3 (5p 6 q) 2
(12x 2 )(−5x 4 ) (−10y 3 )(7y 2 ) (−8u 6 )(−9u)
(−6c 4 )(−12c) ⎛ ⎝ 1 5 r8 ⎞
⎠ (20r 3 ) ⎛ ⎝ 1 4 a5⎞ ⎠ (36a2 )
(4a 3 b)(9a 2 b 6 ) (6m 4 n 3 )(7mn 5 ) ⎛ ⎝ 4 7 xy2 ⎞
⎠ (14xy 3 )
⎛ ⎝ 5 8 u3 v ⎞ ⎠ (24u5 v) ⎛ ⎝ 2 3 x2 y ⎞ ⎠ ⎛ ⎝ 3 4 xy2 ⎞
⎠ ⎛ ⎝ 3 5 m3 n 2 ⎞
⎠
⎛
⎝
5
9
m 2 n 3 ⎞
⎠

6 2 , 6 3 ,
1 6
2 6 2
3 6 3
…
…
8 ?
5%
1.05
$40,000 1
$40,000(1.05), 2 $40,000(1.05) 2 ,
3 $40,000(1.05) 3 ,
10
1 $40,000(1.05)
2 $40,000(1.05) 2
3 $40,000(1.05) 3
…
…
10 ?
x · x = x 2 .
−5 3 = (−5) 3 −5 4 ≠ (−5) 4 .
⎛ ⎝ 1 2
⎞
2⎠
1.
x 3 · x 5 x 8 , x 15 .
a
b

2(x + 3).
−11(4 − 3a).
x 2 + 9x + 7x + 63.
2(x −3).
x and 3, 2, 2x −6.
x −3, 2.
3(x +7).
3(x +7)
3·x +3·7
3x +21
6(x + 8).
2(y + 12).

x(x −8).
y(y − 9).
p(p − 13).
10x(4x + y).
8x(x + 3y).
3r(6r + s).
−2x(5x 2 +7x −3).

−2x(5x 2 +7x −3)
−2x ⋅5x 2 + ⎛ ⎝−2x ⎞ ⎠ ⋅7x − ⎛ ⎝−2x ⎞ ⎠ ⋅3
−10x 3 −14x 2 +6x
−4y(8y 2 + 5y − 9).
−6x(9x 2 + x − 1).
4y 3 (y 2 −8y +1).
4y 3 (y 2 −8y +1)
4y 3 ⋅ y 2 −4y 3 ⋅8y +4y 3 ⋅1
4y 5 −32y 4 +4y 3
3x 2 (4x 2 − 3x + 9).
8y 2 (3y 2 − 2y − 4).
(x +3)p.

(x +3)p
x ⋅ p +3⋅ p
xp +3p
(x + 8)p.
(a + 4)p.
p
(x +7) p
(x +7)
x 2 +7x +3x +21
x 2 +10x +21
Sum
Product
x + x
x· x
2x x 2
combine like terms add exponents of like bases
(x +6)(x +8).

(x +6)(x +8)
(x +8)
x 2 +8x +6x +48
x 2 +14x +48
(x + 8)(x + 9).
(a +4)(a + 5).
(2x +9)(3x +4).
(2x +9)(3x +4)
(3x +4)
6x 2 +8x +27x +36
6x 2 +35x +36
(5x + 9)(4x + 3).
(10m + 9)(8m + 7).
(4y + 3)(6y −5).

(4y + 3)(6y −5)
24y 2 −20y +18y −15
24y 2 −2y −15
(7y + 1)(8y − 3).
(3x + 2)(5x − 8).
(x +2)(x − y).
(x + 5)(x − y).
(x + 2y)(x − 1).
(x +2)(x − y)
x 2 ,
x 2 − xy +2x −2y
x and x, (x + 2) and (x − y).
−xy, x and − y,

+2x, 2 and x,
−2y,
(x +3)(x +7)
(x +6)(x +9).
(x + 7)(x + 8).

(y + 14)(y + 2).
(y −8)(y +6).
(y − 3)(y + 8).
(q − 4)(q + 5).
(2a +3)(3a −1).

(4a + 9)(5a − 2).
(7x + 4)(7x − 8).
⎛ ⎝5x − y ⎞ ⎠(2x −7).
(12x − y)(x − 5).

(6a − b)(2a − 9).
23 6 138.
23 4,
(5x − 1)(2x −7).
23(46).
2x −7 −1
2x −7 5x
(4m − 9)(3m − 7).
(6n − 5)(7n − 2).

(x + 3)(2x 2 −5x +8).
2x 3 −5x 2 +8x +6x 2 −15x +24
2x 3 + x 2 −7x +24
(y − 1)(y 2 − 7y + 2).
(x + 2)(3x 2 − 4x + 5).
(x + 3)(2x 2 −5x +8).
(2x 2 −5x +8)
(2x 2 −5x +8) x
(y − 1)(y 2 − 7y + 2).
(x + 2)(3x 2 − 4x + 5).

4(x + 10) 6(y +8) 15(r − 24)
12(v − 30) −3(m + 11) −4(p + 15)
−8(z −5) −3(x −9) u(u +5)
q(q +7) n(n 2 −3n) s(s 2 −6s)
12x(x − 10) 9m(m − 11) −9a(3a +5)
−4p(2p +7) 6x(4x + y) 5a(9a + b)
5p(11p −5q) 12u(3u −4v) 3(v 2 +10v + 25)
6(x 2 +8x + 16) 2n(4n 2 −4n +1) 3r(2r 2 −6r +2)
−8y(y 2 +2y − 15) −5m(m 2 +3m − 18) 5q 3 (q 2 −2q +6)
9r 3 (r 2 −3r +5) −4z 2 (3z 2 +12z −1) −3x 2 (7x 2 +10x −1)
(2y −9)y (8b −1)b (w −6) · 8
(k −4) · 5
a b c
(x +4)(x +6) (u +8)(u +2) (n + 12)(n −3)
(y +3)(y −9)
(y +8)(y +3) (x +5)(x +9) (a +6)(a + 16)
(q +8)(q + 12) (u −5)(u −9) (r −6)(r −2)
(z − 10)(z − 22) (b −5)(b − 24) (x −4)(x +7)
(s −3)(s +8) (v + 12)(v −5) (d + 15)(d −4)
(6n +5)(n +1) (7y +1)(y +3) (2m − 9)(10m +1)

(5r − 4)(12r +1) (4c − 1)(4c +1) (8n − 1)(8n +1)
(3u − 8)(5u − 14) (2q − 5)(7q − 11) (a + b)(2a +3b)
(r + s)(3r +2s) (5x − y)(x −4) (4z − y)(z −6)
a b
(u +4)(u 2 +3u +2) (x +5)(x 2 +8x +3) (a + 10)(3a 2 + a −5)
(n + 8)(4n 2 + n −7)
(y −6)(y 2 −10y +9) (k −3)(k 2 −8k +7) (2x +1)(x 2 −5x −6)
(5v +1)(v 2 −6v − 10)
13 15. 13 10 + 3
15 10 + 5.
a (10 + 3)(10 + 5)
b 13 · 15
c
18 17. 18 20 − 2
17 20 − 3.
a (20 − 2)(20 − 3)
b 18 · 17
c
a
b

8
24
.
⎛ ⎝ ⎛ 2m 3⎞ ⎠ ⎞5 .
12x
12y .
a, b m, n
Product Property
Power Property
Product to a Power
a m ⋅ a n = a m + n
(a m ) n = a m ⋅ n
(ab) m = a m b m
a, b, c b ≠0, c ≠0,
a
b = a b · · c
c and a b · · c
c = a b

Consider x 5
x 2 and x 2
x 3
What do they mean?
x ⋅ x ⋅ x ⋅ x ⋅ x
x ⋅ x
x ⋅ x
x ⋅ x ⋅ x
Use the Equivalent Fractions Property.
x ⋅ x ⋅ x ⋅ x ⋅ x x ⋅ x ⋅1
x ⋅ x ⋅1
x ⋅ x ⋅ x
Simplify. x 3 1 x
1
1
x 5 x 2
x 2 x 3
x 5−2 1
x 3−2
x 3
1 x
a a ≠0, m, n
a m
a n = am − n , m > n and a m
a n = 1
a n − m,
3 4
3 2 =? 4−2
3 52
5 3 =? 1
5 3−2
81 = ? 3 2 25
9 125 =? 1
5 1
9=9✓ 1 = 1 5 5 ✓
n > m
3,
3
a x10 b 29
x 8 2 2

a
x
x 10
x 8
m > n, am
a n = am − n
x 2
b
2 9
2 2
m > n, am
a n = am − n .
2 7
a x12
x 9 b 714
7 5
a y23 b 815
y 17 8 7
a b10 b 33
b 15 3 5

a
b
b 10
b 15
n > m, am
a n = 1
a n − m.
1
b 5
b
3 3
3 5
n > m, am
a n = 1
a n − m.
1
3 2
1
9
1
a x8 b 1211
x 15 12 21
a m17
m 26 b 78
7 14
a a5
a 9
b x11
x 7

a
a
a 5
a 9
n > m, am
a n = 1
a n − m.
1
a 4
b
x
x 11
x 7
m > n, am
a n = an − m .
x 4
a b19
b 11
b z5
z 11
a p9
p 17
b w13
w 9
am
a
m.
2
2 =1 17
17 =1 −43
−43 =1
1. x =1, x x ≠0 1.
m = n
am
a n
m > n n < m
am m
a
8 , 1.
8

8
8 =1
2 3
2 3
2 3 =1
2 3−3 =1
2 0 =1
am
a n a0 1 a 0 =1
a a 0 =1.
1.
a 12 0 b y 0
1.
a
12 0
b
y 0
a 17 0 b m 0

a k 0 b 29 0
(2x) 0 .
(2x) 0
2 0 x 0
1⋅1
(7z) 0 .
(7z) 0
⎛ ⎝ ⎛ −4y ⎞ ⎠ ⎞0 .
⎛2 ⎞ 0
⎝ x 3 ⎠ .
a ⎛ ⎝−3x 2 y ⎞ 0
⎠
b −3x 2 y 0
a
⎛
⎝−3x 2 y ⎞ 0
⎠
1

y −3x 2 y 0
−3x 2 ⋅1
−3x 2
a ⎛ ⎝ ⎛ 7x 2 y ⎞ ⎠ ⎞0 b 7x 2 y 0
a −23x 2 y 0
b ⎛ ⎝ ⎛ −23x 2 y ⎞ ⎠ ⎞0
⎛ x ⎝y ⎞ 3
⎠
x
y ⋅ x y ⋅ x y
x ⋅ x ⋅ x
y ⋅ y ⋅ y
x 3
y 3
⎛x ⎝y ⎞ 3
⎠ x3
y 3.
We write:
⎛ x ⎝y ⎞ 3
⎠
x 3
y 3
a b b ≠0, m
⎛
⎝ a m
⎞
b⎠
= am
b m

⎛
⎝ 2 3
⎞
3⎠
= ? 2 3
3 3
2
3 ⋅ 2 3 ⋅ 2 3 =? 8
27
8
27 = 27 8 ✓
a ⎛ ⎝ 5 2
⎞
8⎠
b ⎛ ⎝3
x 4
⎞
⎠
c ⎛ y
⎝m ⎞ 3
⎠
a
⎛ ⎝ a m
⎞
b⎠
= am
b m
b
⎛ ⎝ a m
⎞
b⎠
= am
b m
c
a ⎛7⎞ 2
⎝ 9⎠
b ⎛y
⎛ ⎞ 3
⎝8
⎠
c ⎛p ⎛ ⎞ 6
⎝ q ⎠
a ⎛1⎞ 2
⎝ 8⎠
b ⎛−5⎞ 3
⎝ m ⎠ c ⎛r ⎛
⎝ s
⎛ ⎞
⎠
⎞ 4

a, b m, n
Product Property
Power Property
Product to a Power Property
Quotient Property
Zero Exponent Definition
Quotient to a Power Property
a m ⋅ a n = a m + n
(a m ) n = a m ⋅ n
(ab) m = a m b m
a m
a n = am − n , a ≠0, m > n
a m
a n = 1 n −
a
m, a ≠0, n > m
a 0 =1, a ≠0
⎛
⎝ a m
⎞
b⎠
= am
b
m, b ≠0
(x 2 ) 3
x 5 .
(x 2 ) 3
x 5
x 6
x 5
x
(a4 ) 5
a 9 .
⎝
⎛ b 5⎞6
⎠
b 11 .
m 8
(m 2 ) 4.

m 8
⎛
⎝m 2⎞ 4
⎠
m 8
m 8
m 0
1
k 11
(k 3 ) 3.
d 23
⎝
⎛ d 4⎞6.
⎠
⎛ ⎝ x7
x 3 ⎞
⎠
2
.
2
⎛
⎝ x7 ⎞
x 3 ⎠
⎛
⎝x 7−3⎞ 2
⎠
⎛
⎝x 4⎞ 2
⎠
x 8
⎛
f
⎜
14 2
⎞
⎝ f 8 ⎟ .
⎠
⎛ b
6 ⎞ 2
⎝ ⎠
.
b 11

3
⎛
⎜ p2 ⎞
⎝q 5 ⎟ .
⎠
3
⎛
⎜ p2 ⎞
⎝q 5 ⎟
⎠
⎛ ⎝ a m
⎞
b⎠
= am
b m
(a m ) n = a m ⋅ n .
⎛
⎝ p 2⎞ 3
⎠
⎛
⎝q 5⎞ 3
⎠
p 6
q 15
⎛ m
3⎞ 5
⎝ ⎠
.
n 8
⎛ t
10⎞ 2
⎝ ⎠
.
u 7
⎛ ⎝ 2x3
3y
4
⎞
⎠
.
⎛
⎝ 2x3 ⎞
3y ⎠
4
⎛
⎝2x 3⎞ 4
⎠
4
⎛
⎝3y ⎞ ⎠
2 4 ⎛ ⎝x 3⎞ 4
⎠
3 4 y 4
16x 12
81y 4
⎛ 5b ⎞ 2
⎝9c 3 ⎠
.

⎛
4p
⎜
4 3
⎞
⎝7q 5 ⎟ .
⎠
(y 2 ) 3 (y 2 ) 4
(y 5 ) 4 .
(y 2 ) 3 (y 2 ) 4
(y 5 ) 4
⎛
⎝y 6⎞ ⎠ ⎛ ⎝y 8⎞ ⎠
y 20
y 14
y 20
1
y 6
(y4 ) 4 (y 3 ) 5
(y 7 ) 6 .
(3x4 ) 2 (x 3 ) 4
(x 5 ) 3 .
56x 5 ÷7x 2 .
56x 5 ÷7x 2
56x 5
7x 2
56
7
⋅ x5
x 2
8x 3

63x 8 ÷ 9x 4 .
96y 11 ÷ 6y 8 .
42x 2 y 3
−7xy 5 .
42x 2 y 3
−7xy 5
42
−7 ⋅ x2 x ⋅ y3
y 5
−6 ⋅ x ⋅ 1 y 2
− 6x
y 2
−84x 8 y 3
7x 10 y 2 .
−72a 4 b 5
−8a 9 b 5 .
24a 5 b 3
48ab 4 .
24a 5 b 3
48ab 4
24
48 ⋅ a5 a ⋅ b3
b 4
1
2 ⋅ a4 ⋅ 1 b
a 4
2b

16a 7 b 6
24ab 8 .
27p 4 q 7
−45p 12 q .
14x 7 y 12
21x 11 y 6.
14x 7 y 12
21x 11 y 6
2y 6
3x 4
14
21
28x 5 y 14
49x 9 y 12.
30m 5 n 11
48m 10 n 14.
(3x 3 y 2 )(10x 2 y 3 )
6x 4 y 5 .
(3x 3 y 2 )(10x 2 y 3 )
6x 4 y 5
30x 5 y 5
6x 4 y 5
5x

(3x 4 y 5 )(8x 2 y 5 )
12x 5 y 8 .
(−6a 6 b 9 )(−8a 5 b 8 )
−12a 10 b 12 .

4 8
4 2 3 12
3 4 x 12
x 3
u 9
u 3 r 5
r
y 4
y
y 4
y 20 x 10
x 30 10 3
10 15
r 2
r 8 a
a 9 2
2 5
5 0 10 0 a 0
x 0 −7 0 −4 0
a ⎛ ⎝10p ⎞ 0
⎠ b 10p 0 a (3a) 0 b 3a 0 a ⎛ ⎝−27x 5 y ⎞ 0
⎠
b −27x 5 y 0
a ⎛ ⎝−92y 8 z ⎞ 0
⎠
b −92y 8 z 0
a 15 0 b 15 1 a −6 0 b −6 1
2·x 0 +5 · y 0 8·m 0 −4 · n 0
⎛ ⎝ 3 5
⎞
2⎠
⎛ ⎝ 4 3
⎞
5⎠
⎛ ⎝ m 3
⎞
6 ⎠
⎛ 5
p⎞
⎝2⎠
⎛ ⎝3b
a 2
⎞
⎠
⎛x ⎝y ⎞ 10
⎠
⎛2x
⎝ y ⎞ 4
⎠
⎛ ⎝ a 8
⎞
b⎠

⎛
⎝x 2⎞ 4
⎠
x 5
⎛
⎝y 4⎞ 3
⎠
y 7
⎛
⎝u 3⎞ 4
⎠
u 10
⎛
⎝y 2⎞ 5
⎠
y 6
y 8
⎛
⎝y 5⎞ 2
⎠
p 11
⎛
⎝p 5⎞ 3
⎠
r 5
r 4 · r
⎛ ⎝ u
u 10 ⎞
⎠
2
⎛
⎝y 3⎞ 5
⎠
⎛
⎝y 4⎞ 3
⎠
a 3 · a 4
a 7
⎛ ⎝ a4 · a 6
a 3 ⎞
⎠
2
⎛
⎝z 6⎞ 2
⎠
⎛
⎝z 2⎞ 4
⎠
⎛ ⎝ x2
x 8 ⎞
⎠
3
⎛ ⎝ x3 · x 8
x 4 ⎞
⎠
3
⎛
⎝x 3⎞ 6
⎠
⎛
⎝x 4⎞ 7
⎠
⎛
⎝x 4⎞ 8
⎠
⎛
⎝x 5⎞ 7
⎠
⎛ ⎝ 2r3
5s
⎞
⎠
4
⎛ ⎝ 3m2
4n
⎞
⎠
3
⎛
⎜ 3y2 · y 5 0
⎞
⎝y 15 · y 8 ⎟
⎠
⎛ ⎝ 15z4 · z 9
0.3z 2 ⎞
⎠
0
⎛
⎝r 2⎞ ⎛⎝
⎠5
r 4⎞ 2
⎠
7
⎛
⎝r 3⎞ ⎠
⎛
⎝p 4⎞ ⎛⎝
⎠2
p 3⎞ ⎠
9
⎛
⎝p 2⎞ ⎠
5
⎛
⎝3x 4⎞ ⎛⎝
⎠3
2x 3⎞ 2
⎠
⎛
⎝6x 5⎞ 2
⎠
⎛
⎝−2y 3⎞ ⎛⎝
⎠4
3y 4⎞ 2
⎠
⎛
⎝−6y 3⎞ 2
⎠
48b 8 ÷6b 2 42a 14 ÷6a 2 36x 3 ÷ ⎛ ⎝−2x 9⎞ ⎠
20u 8 ÷ ⎛ ⎝−4u 6⎞ ⎠ 18x 3
9x 2
36y 9
4y 7
−35x 7
−42x 13 18x 5
−27x 9
18r 5 s
3r 3 s 9
24p 7 q
6p 2 q 5 8mn 10
64mn 4 10a 4 b
50a 2 b 6

−12x 4 y 9
15x 6 y 3
48x 11 y 9 z 3
36x 6 y 8 z 5
64x 5 y 9 z 7
48x 7 y 12 z 6
(10u 2 v)(4u 3 v 6 )
5u 9 v 2
(6m 2 n)(5m 4 n 3 )
3m 10 n 2
(6a 4 b 3 )(4ab 5 )
(12a 8 b)(a 3 b)
(4u 5 v 4 )(15u 8 v)
(12u 3 v)(u 6 v)
a 24a 5 +2a 5 b 24a 5 −2a 5 c 24a 5 ⋅2a 5 d 24a 5 ÷2a 5
a 15n 10 +3n 10 b 15n 10 −3n 10 c 15n 10 ⋅3n 10 d 15n 10 ÷3n 10
a p 4 ⋅ p 6
b ⎛ ⎝p 4⎞ 6
⎠
a q 5 ⋅ q 3
b ⎛ ⎝q 5⎞ 3
⎠
a y3
y
b y
y 3
a z6
z 5
b z5
z 6
⎛ ⎝8x 5⎞ ⎠ ⎛ ⎝9x ⎞ ⎠ ÷6x 3 ⎛ ⎝4y ⎞ ⎠ ⎛ ⎝12y 7⎞ ⎠ ÷8y 2
27a 7
3a 3 + 54a9
9a 5 32c 11
4c 5 + 42c9
6c 3
32y 5
8y 2 − 60y10
5y 7 48x 6
6x 4 − 35x9
7x 7
63r 6 s 3
9r 4 s 2 − 72r2 s 2
6s
56y 4 z 5
7y 3 z 3 − 45y2 z 2
5y
10 6
10 9
10 6
10 12
x20
x 4
x5 .
y 3
y
=3.
y 3
y

−3 0 (−3) 0
n 0 0.
a
b

6 64,891?
0.0012.
5 − (−3).
a a ≠0, m, n
a m
a n = am − n , m > n and a m
a n = 1
a n − m,
n > m
x2
x 5.
x 2
x 5
x 2−5
x −3
x2
x 5 x 2
x 5.
x −3 = 1
3
x

n a ≠0, a −n = 1 a n.
a 4 −2
b 10 −3
a
4 −2
a −n = 1 a n. 1
4 2
1
16
b
10 −3
a −n = 1 a n. 1
10 3
1
1000
a 2 −3
b 10 −2
a 3 −2
b 10 −4
a (−3) −2
b −3 −2

a
−3
(−3) −2
1
(−3) 2
1
9
b
−3 −2 3 −2
−3 −2
−1 · 3 −2
−1 · 1
3 2
− 1 9
a (−5) −2
b −5 −2
a (−2) −2
b −2 −2
a b
a 4·2 −1
b (4 · 2) −1
a
4·2 −1
a −n = 1 a
n. 4· 1
2 1
2

(4 · 2) −1
(8) −1
a −n = 1 a n. 1
8 1
1
8
a 6 · 3 −1
b (6 · 3) −1
a 8 · 2 −2
b (8 · 2) −2
x −6 .
x −6
a −n = 1 a n. 1
x 6
y −7 .
z −8 .
a 5y −1 b (5y) −1 c (−5y) −1

a
y
5y −1
y 5· 1
y 1
5 y
b
5y
5y
(5y) −1
1
(5y) 1
1
5y
c
(−5y) −1
−5y −5y
a
−b =−a b .
1
(−5y) 1
1
−5y
− 1
5y
a 8p −1 b (8p) −1 c (−8p) −1
a 11q −1 b (11q) −1 c (−11q) −1
a m
a n = am − n ,
a ≠0
a −n = 1 a n.

a, b m, n
Product Property
Power Property
Product to a Power Property
Quotient Property
Zero Exponent Property
Quotient to a Power Property
Definition of Negative Exponent
a m · a n = a m + n
(a m ) n = a m · n
(ab) m = a m b m
a m
a n = am − n , a ≠0
a 0 =1, a ≠0
⎛
⎝ a m
⎞
b⎠
= am
b
m, b ≠0
a −n = 1 a n
a
a x −4 · x 6 b y −6 · y 4 c z −5 · z −3
x −4 · x 6
a m · a n = a m + n . x −4 + 6
x 2
b
y −6 · y 4
y −6 + 4
y −2
a −n = 1 a n. 1
y 2

c
z −5 · z −3
z −5−3
z −8
a −n = 1 a n. 1
z 8
a x −3 · x 7 b y −7 · y 2 c z −4 · z −5
a a −1 · a 6 b b −8 · b 4 c c −8 · c −7
(m 4 n −3 )(m −5 n −2 ).
(m 4 n −3 )(m −5 n −2 )
m 4 m −5 · n −2 n −3
m −1 · n −5
1
m 1 · 1
n 5
1
mn 5
(p 6 q −2 )(p −9 q −1 ).
(r 5 s −3 )(r −7 s −5 ).

(2x −6 y 8 )(−5x 5 y −3 ).
(2x −6 y 8 )(−5x 5 y −3 )
2(−5) · (x −6 x 5 )·(y 8 y −3 )
−10 · x −1 · y 5
a −n = 1 a
n. −10 · 1
x 1 · y5
−10y 5
x
(3u −5 v 7 )(−4u 4 v −2 ).
(−6c −6 d 4 )(−5c −2 d −1 ).
⎛ ⎝k 3⎞ ⎠−2
.
⎛
⎝k 3⎞ −2
⎠
(ab) m = a m b m .
k 3(−2)
k −6
1
k 6
⎛ 4⎞⎞ ⎝x
−1 ⎠ .
⎛ 2⎞⎞ ⎝y
−2 ⎠ .
⎛ ⎝5x −3⎞ ⎠2
.

⎛
⎝5x −3⎞ 2
⎠
(ab) m = a m b m .
5 2 ⎛ ⎝x −3⎞ 2
⎠
5 2 x
(a m ) n = a m · n .
25k −6
x −6
a −n = 1 a n. 25 · 1
x 6
25
x 6
⎛ ⎝ ⎛ 8a −4⎞ ⎠ ⎞2 .
⎛ ⎝ ⎛ 2c −4⎞ ⎠ ⎞3 .
r 5
r −4.
a m
a n = am − n
x 8
x −3.
y 7
y −6.
10.

4000 0.004. 4000 4 × 1000 0.004 4×
1000 1 .
1000
4000 0.004
4 × 1000 4 ×
1000
1
4×10 3 4× 1
10 3
4×10 −3
10, 10
a ×10 n
a ≥1 a

n
3.7×10 4
10 n
10 −n
10 4
37,000 = 3.7×10 4
96,000.
48,300.
1 10.
n,
10.
1, 10 10 n .
0 1, 10 10 −n .
0.0052.

5.2×10 −3
5.2 × 10 −3
5.2 × 1
10 3
5.2 × 1
1000
5.2 × 0.001
0.0052
0.0052 = 5.2 × 10 −3
0.0078.
0.0129.
9.12 × 10 4 9.12 × 10 −4
9.12 × 10,000 9.12 × 0.0001
91,200 0.000912
6.2 × 10 3 .

n 6.2×10 3
n
n
|n|
10 3 6.2×10 3 = 6,200
1.3 × 10 3 .
9.25 × 10 4 .
n, 10.
n
n
|n|
8.9 × 10 −2 .
8.9×10 −2
n
8.9×10 −2 = 0.089
1.2 × 10 −4 .
7.5 × 10 −2 .

(4 × 10 5 )(2 × 10 −7 ).
(4 × 10 5 )(2 × 10 −7 )
10 5 10 −7
4·2·10 5 ·10 −7
8×10 −2
0.08
(3 × 10 6 )(2 × 10 −8 ).
(3 × 10 −2 )(3 × 10 −1 ).
9×10 3
3×10 −2.
9×10 3
3×10 −2
9
3 × 103
10 −2
10 3 10 −2 3×10 5
300,000
8 ×10 4
2 × 10 −1.
8 × 10 2
4 × 10 −2.

5 −3 8 −2 3 −4
2 −5 7 −1 10 −1
2 −3 +2 −2 3 −2 +3 −1 3 −1 +4 −1
10 −1 +2 −1 10 0 −10 −1 +10 −2 2 0 −2 −1 +2 −2
a (−6) −2 b −6 −2 a (−8) −2 b −8 −2 a (−10) −4
a (−4) −6 b −4 −6 a 5·2 −1 b (5 · 2) −1 a 10 · 3 −1
b −10 −4
b (10 · 3) −1
n −4
a 4·10 −3 b (4 · 10) −3 a 3·5 −2 b (3 · 5) −2
p −3 c −10 m −5
a 4x −1 b (4x) −1
c (−4x) −1
a 3q −1 b (3q) −1
c (−3q) −1
a 6m −1 b (6m) −1
c (−6m) −1
a 10k −1 b (10k) −1
c (−10k) −1
p −4 · p 8 r −2 · r 5 n −10 · n 2
q −8 · q 3 k −3 · k −2 z −6 · z −2
a · a −4 m · m −2 p 5 · p −2 · p −4
x 4 · x −2 · x −3 a 3 b −3 u 2 v −2
⎛ ⎝x 5 y −1⎞ ⎠ ⎛ ⎝x −10 y −3⎞ ⎠ ⎛ ⎝a 3 b −3⎞ ⎠ ⎛ ⎝a −5 b −1⎞ ⎠ ⎛ ⎝uv −2⎞ ⎠ ⎛ ⎝u −5 v −4⎞ ⎠
⎛ ⎝pq −4⎞ ⎠ ⎛ ⎝p −6 q −3⎞ ⎠ ⎛ ⎝−2r −3 s 9⎞ ⎠ ⎛ ⎝6r 4 s −5⎞ ⎠ ⎛ ⎝−3p −5 q 8⎞ ⎠ ⎛ ⎝7p 2 q −3⎞ ⎠

⎛ ⎝−6m −8 n −5⎞ ⎠ ⎛ ⎝−9m 4 n 2⎞ ⎠ ⎛ ⎝−8a −5 b −4⎞ ⎠ ⎛ ⎝−4a 2 b 3⎞ ⎠ ⎛ ⎝a 3⎞ ⎠
−3
⎛ ⎝q 10⎞ ⎠
−10
⎛ ⎝n 2⎞ −1
⎠
⎛ ⎝x 4⎞ −1
⎠
⎛ ⎝y −5⎞ ⎠
4
⎛ ⎝p −3⎞ ⎠
2
⎛ ⎝q −5⎞ ⎠
−2
⎛ ⎝m −2⎞ ⎠
−3
⎛ ⎝4y −3⎞ ⎠
2
⎛ ⎝3q −5⎞ ⎠
2
⎛ ⎝10p −2⎞ ⎠
−5
⎛ ⎝2n −3⎞ ⎠
−6
u 9
u −2
b 5
b −3
x −6
x 4 m 5
m −2
q 3
q 12 r 6
n −10
r 9 n −4
p −3
p −6
310,000,000.
6,850,000,000.
0.0018
2010
0.0000000057.
4.1 × 10 2 8.3 × 10 2 5.5 × 10 8
1.6 × 10 10 3.5 × 10 −2 2.8 × 10 −2
1.93 × 10 −5 −8
6.15 × 10
2×10 4 .

$1.5 × 10 13 .
3.9 × 10 −4 .
1×10 −5
(2 × 10 5 )(2 × 10 −9 ) (3 × 10 2 )(1 × 10 −5 ) (1.6 × 10 −2 )(5.2 × 10 −6 )
(2.1 × 10 −4 )(3.5 × 10 −2 ) 6×10 4
3×10 −2 8×10 6
4×10 −1
7×10 −2
1×10 −8 5×10 −3
1×10 −10
1.5
a 1.5
b 1.5
1 2,598,960
3.848 × 10 −7 .
0.000125
a
b
6 50
50 · 49 · 48 · 47 · 46 · 45.
1.1441304 × 10 10 .
2 −3
a
2 3
.
b
a

56
−3(6a + 11).
4x 2 ⎛
⎝x
2 + 3x − 1 ⎞ ⎠ .
24 36.

24 36 12.
24 = 12 · 2
36 = 12 · 3
54, 36.
48, 80.
5x and 15.

7y, 14.
22, 11m.
12x 2 18x 3 .
The GCF of 12x 2 and 18x 3 is 6x 2
16x 2 , 24x 3 .
27y 3 , 18y 4 .
14x 3 , 8x 2 , 10x.
The GCF of 14x 3 and 8x 2 , and10x is 2x
21x 3 , 9x 2 , 15x.

25m 4 , 35m 3 , 20m 2 .
12
2·6or3·4),
2 ⎛ ⎝ x + 7 ⎞ ⎠ factors
2·x + 2·7
2x + 14 product
2x + 14,
2(x +7).
a, b, c
a(b + c) = ab + ac and ab + ac = a(b + c)
2x + 14.
2x =2⋅x
14=2⋅7
2(x +7)
4x + 12.
6a + 24.
Noun 7 is a factor of 14
Verb factor 2 from 2x +14

3a +3.
9a + 9.
11x + 11.
12x − 60.

11x − 44.
13y − 52.
3y 2 +6y +9.
4y 2 + 8y + 12.
6x 2 + 42x − 12.

6x 2 +5x.
6x 2 +5x
6x 2 5x
x(6x +5)
x(6x +5)
x ⋅6x + x ⋅5
6x 2 +5x ✓
9x 2 + 7x.
5a 2 − 12a.
4x 3 −20x 2 .
2x 3 + 12x 2 .
6y 3 − 15y 2 .

21y 2 +35y.
21y 2 35y
18y 2 + 63y.
32k 2 + 56k.
14x 3 +8x 2 −10x.
14x 3 , 8x 2 , and 10x 2x.
14x 3 +8x 2 −10x
2x(7x 2 +4x −5)
18y 3 − 6y 2 − 24y.
16x 3 + 8x 2 − 12x.
−9y − 27.

−9y −27
−9 ⎛ ⎝y +3 ⎞ ⎠
−5y − 35.
−16z − 56.
−4a 2 +16a.
−4a 2 +16a
−4a(a −4)
−7a 2 + 21a.
−6x 2 + x.

40, 56 45, 75 72, 162
150, 275 3x, 12 4y, 28
10a, 50 5b, 30 16y, 24y 2
9x, 15x 2 18m 3 , 36m 2 12p 4 , 48p 3
10x, 25x 2 , 15x 3 18a, 6a 2 , 22a 3 24u, 6u 2 , 30u 3
40y, 10y 2 , 90y 3 15a 4 , 9a 5 , 21a 6 35x 3 , 10x 4 , 5x 5
27y 2 , 45y 3 , 9y 4 14b 2 , 35b 3 , 63b 4
2x +8 5y +15 3a −24
4b −20 9y −9 7x −7
5m 2 +20m +35 3n 2 +21n +12 8p 2 +32p +48
6q 2 +30q +42 8q 2 +15q 9c 2 +22c
13k 2 +5k 17x 2 +7x 5c 2 +9c
4q 2 +7q 5p 2 +25p 3r 2 +27r
24q 2 −12q 30u 2 −10u yz +4z
ab +8b 60x −6x 3 55y −11y 4
48r 4 −12r 3 45c 3 −15c 2 4a 3 −4ab 2
6c 3 −6cd 2 30u 3 +80u 2 48x 3 +72x 2
120y 6 +48y 4 144a 6 +90a 3 4q 2 +24q +28
10y 2 +50y +40 15z 2 −30z −90 12u 2 −36u − 108

3a 4 −24a 3 +18a 2 5p 4 −20p 3 −15p 2 11x 6 +44x 5 − 121x 4
8c 5 +40c 4 −56c 3 −3n −24 −7p −84
−15a 2 −40a −18b 2 −66b −10y 3 +60y 2
−8a 3 +32a 2 −4u 5 +56u 3 −9b 5 +63b 3
p
−5p 2 + 150p.
80 4
−16t 2 +80t +4, t =
36 60 12.
y 4 y 5 y 10
y a y b
y c
a
b

0
ax m a m
n a ≠0 a −n = 1 a n
a ×10 n , a ≥1
a < 10, n
a a 0 =1
a m th
a m, n
a m · a n = a m + n
a m, n
(a m ) n = a m ⋅ n
a b m
(ab) m = a m b m

a, b, c b ≠0, c ≠0,
a
b = a b · · c
c and a b · · c
c = a b
a a 0 =1.
1.
a a ≠0, m, n
a m
a n = am − n , m > n and a m
a n = 1
a n − m,
a b b ≠0, m
n > m
⎛
⎝ a m
⎞
b⎠
= am
b m
a, b m, n
Product Property
Power Property
Product to a Power Property
Quotient Property
Zero Exponent Property
Quotient to a Power Property
Definition of Negative Exponent
a m · a n = a m + n
(a m ) n = a m · n
(ab) m = a m b m
a m
a n = am − n , a ≠0
a 0 =1, a ≠0
⎛
⎝ a m
⎞
b⎠
= am
b
m, b ≠0
a −n = 1 a n

n
10 n
10 -n
n
n
n
|n|
a b c
a(b + c) = ab + ac ab + ac = a(b + c)
y 2 +8y −20 −6a 4 9x 3 −1
n 3 −3n 2 +3n −1
16x 2 −40x −25 5m +9 −15
y 2 +6y 3 +9y 4

4p +11p −8y 3 −5y 3 4n 5 , −n 5 , −6n 5
10x 2 3x 2
(4a 2 +9a − 11) + (6a 2 −5a + 10)
(8m 2 +12m −5)−(2m 2 −7m −1) (y2 −3y + 12) + (5y 2 −9)
(5u 2 +8u)−(4u −7) 8q 3 −27
q 2 +6q −2
x 2 +6x +8 x 2 −8x +15
200x − 1 5 x2 x =5 200x − 1 5 x2 x =0 200x − 1 5 x2 x =15
5+40x − 1 2 x2
x =10
5+40x − 1 2 x2
x =−4
5+40x − 1 2 x2 x =0
640
−16t 2 + 640
t
t =6.
x
−
160 1 x2 + 1 2 x.
x =20
6 3 ⎛
⎝ 1 4
⎞
2⎠
(−0.5) 2
−3 2
p 3 · p 10 2·2 6 a · a 2 · a 3
x · x 8

(y 4 ) 3 (r 3 ) 2 (3 2 ) 5
(a 10 ) y
(8n) 2 (−5x) 3 (2ab) 8
(−10mnp) 4
(3a 5 ) 3 (4y) 2 (8y) (x 3 ) 5 (x 2 ) 3
(5st 2 ) 3 (2s 3 t 4 ) 2
(−6p 4 )(9p) ⎛
⎝ 1 3 c2 ⎞⎛ ⎠⎝ 30c 8⎞ ⎠
(8x 2 y 5 )(7xy 6 )
⎛
⎝ 2 3 m3 n 6⎞ ⎠ ⎛ ⎝ 1 6 m4 n 4 ⎞
⎠
7(10 − x) a 2 (a 2 −9a − 36) −5y(125y 3 −1)
(4n − 5)(2n 3 )
(a +5)(a +2) (y −4)(y + 12) (3x + 1)(2x −7)
(6p − 11)(3p − 10) (n +8)(n +1) (k +6)(k −9)
(5u −3)(u +8) (2y − 9)(5y −7) (p +4)(p +7)
(x −8)(x +9) (3c + 1)(9c −4) (10a − 1)(3a −3)

(x +1)(x 2 −3x − 21) (5b − 2)(3b 2 + b −9) (m +6)(m 2 −7m − 30)
(4y − 1)(6y 2 −12y +5)
2 8
2 2 6
a n 3
n 12
x
x 5
3 0 y 0 (14t) 0
12a 0 −15b 0
⎛
⎝ 3 2
⎞
⎛
5⎠
⎝2
x 5
⎞
⎠
⎛ 5m ⎝ n ⎞ 3
⎠
⎛
⎝10t
s 2
⎞
⎠
(a 3 ) 2
a 4 u 3
u 2 · u 4 ⎛
⎝
x
x 9 ⎞
⎠
5
⎛
⎜ p4 · p 5 2
⎞
⎝ p 3 ⎟
⎠
(n 5 ) 3
(n 2 ) 8
⎛
⎝ 5s2 ⎞
4t ⎠
3
72p 12 ÷8p 3 −26a 8 ÷ ⎛ ⎝2a 2⎞ ⎠
45y 6
−15y 10
−30x 8
−36x 9 28a 9 b
7a 4 b 3 11u 6 v 3
55u 2 v 8

⎛
⎝5m 9 n 3⎞ ⎠ ⎛ ⎝8m 3 n 2⎞ ⎠
⎛
⎝10mn 4⎞ ⎠ ⎛ ⎝m 2 n 5⎞ ⎠
42r 2 s 4
6rs 3
− 54rs2
9s
6 −2 (−10) −3 5·2 −4
(8n) −1
x −3 · x 9 r −5 · r −4
⎛
⎝uv −3⎞ ⎠ ⎛ ⎝u −4 v −2⎞ ⎠
⎛
⎝m 5⎞ −1
⎠
⎛
⎝k −2⎞ −3
⎠
q 4
q 20
b 8
b −2 n −3
n −5
5,300,000 0.00814
0.097
21,000,000
2.9 × 10 4 1.5 × 10 8 3.75 × 10 −1
9.413 × 10 −5
⎛
⎝3×10 7⎞ ⎠ ⎛ ⎝2×10 −4⎞ ⎠
⎛
⎝1.5 × 10 −3⎞ ⎠ ⎛ ⎝4.8 × 10 −1⎞ ⎠ 6×10 9
2×10 −1
9×10 −3
1×10 −6

5n, 45 8a, 72 12x 2 , 20x 3 , 36x 4
9y 4 , 21y 5 , 15y 6
16u −24 15r +35 6p 2 +6p
10c 2 −10c −9a 5 −9a 3 −7x 8 −28x 3
5y 2 −55y +45 2q 5 −16q 3 +30q 2

8y 4 −3y 2 +1
a
b
(5a 2 +2a − 12) + (9a 2 +8a −4)
(10x 2 −3x +5)−(4x 2 −6)
⎛
⎝ −3 3
⎞
4⎠
n · n 4 (10p 3 q 5 ) 2 (8xy 3 )(−6x 4 y 6 )
4u(u 2 −9u +1) (s +8)(s +9) (m + 3)(7m −2)
(11a −6)(5a −1) (n −8)(n 2 −4n + 11) (4a +9b)(6a −5b)
5 6
5 8 ⎛
⎝
x 3 · x 9
x 5 ⎞
⎠
2
(47a 18 b 23 c 5 ) 0
24r 3 s
6r 2 s 7
8y 2 −16y +20
4y
(15xy 3 −35x 2 y)÷5xy
4 −1
⎛
⎝2y ⎞ −3
⎠
p −3 · p −8
x 4
x −5
80a 3 + 120a 2 +40a −6x 2 −30x
112,000
5.25 × 10 −4
⎛
⎝2.4 × 10 8⎞⎛ ⎠⎝2×10 −5⎞ ⎠ 9×10 4
3×10 −1 240
−16t 2 + 240
t
t =3.

x + 3 x = −1.
2x − 5y x = 3, y = −2.

y: 40 − 4y = 20.
1, 2, 3, 4
2
a
b
a 4,
b 4 4
a
b
a
b

x-axis 0. 0
0.
x-axis.
y-axis.
x-y
x-axis y-axis
(x, y)
The first number is the x-coordinate.
The second number is the y-coordinate.

x-y
(2, 5) x 2 y 5
x 2, x-axis. x =2,
y 5, y y =5
(2, 5).
(1, 3) (3, 1)
x y (1, 3).
x-coordinate 1 1 x-axis x =1. y-coordinate 3
3 y-axis y =3. (1, 3).

(3, 1), 3 x-axis x =3. 1
y-axis y =1. (3, 1).
(1, 3) (3, 1).
(5, 2), (2, 5).
(4, 2), (2, 4).
a (−1, 3) b (−3, −4) c (2, −3) d ⎛ ⎝ 3, 5 ⎞
2⎠
x-coordinate, y-coordinate.
a x =−1,y =3, (−1, 3)
b x =−3,y =−4, (−3, −4)
c x =2,y =−1, (2, −1)
d x =3,y = 5 2 , ⎛ ⎝ 3, 5 ⎞
2⎠ 5 2 21 2 ,
2.5. 2 3 y-axis.

a (−2, 1) b (−3, −1) c (4, −4) d ⎛ ⎝ ⎛ −4, 3
2
⎞
⎠
⎞
a (−4, 1) b (−2, 3) c (2, −5) d ⎛ ⎝ ⎛ −3, 5
2
⎞
⎠
⎞
a (−5, 2) b (−5, −2) c (5, 2) d (5, −2 ⎞ ⎠
x-coordinate y-coordinate,
a (4, −3) b (4, 3) c (−4, −3) d (−4, 3)

a (−1, 4) b (1, 4) c (1, −4) d (−1, −4)

(0, 4) (−2, 0) (0, 4)
( −2,0)
y-coordinate 0 x-axis, (a, 0).
x-coordinate 0 y-axis, (0, b).
(0, 0)
(0, 0)
a (0, 5) b (4, 0) c (−3, 0) d (0, 0) e (0, −1)
a x =0, (0, 5) y-axis.
b y =0, (4, 0) x-axis.
c y =0, (−3, 0) x-axis.
d x =0 y =0, (0, 0)

e x =0, (0, −1) y-axis.
a (4, 0) b (−2, 0) c (0, 0) d (0, 2) e (0, −3)
a (−5, 0) b (3, 0) c (0, 0) d (0, −1) e (0, 4)
(x, y).
−3 x-axis, x-coordinate −3. 3 y-axis,
y-coordinate 3. (−3, 3).
−1 x-axis, x-coordinate −1. −3
y-axis, y-coordinate −3. (−1, −3).
2 x-axis, x-coordinate 2. 4 y-axis,
y-coordinate 4. (2, 4).

4 x - axis, x-coordinate 4. −4 y-axis,
y-coordinate −4. (4, −4).

x =−4
y =−2
x =3
y =1
( −4,0)
(0, − 2)
(3, 0)
(0, 1)
x =4.
3x +5=17
3x =12
x =4
Ax + By = C.
Ax + By = C, A and B

x y:
y =−5x +1 Ax + By = C.
5x
Ax + By = C.
y =−5x +1 5x + y =1,
Ax + By = C.
x,
y
(x, y). x y
(x, y) Ax + By = C,
x- y-values
x +4y =8:
a (0, 2) b (2, −4) c (−4, 3)
x- and y-values
a (0, 2) b (2, −4) c (−4, 3)
(0, 2) (2, −4) (−4, 3)

2x + 3y = 6
a (3, 0) b (2, 0) c (6, −2)
4x − y = 8
a (0, 8) b (2, 0) c (1, −4)
y =5x −1:
a (0, −1) b (1, 4) c (−2, −7)
x- y-values
a (0, −1) b (1, 4) c (−2, −7)
(0, −1) (1, 4) (−2, −7)
y = 4x − 3
a (0, 3) b (1, 1) c (1, 0)
y = −2x + 6
a (0, 6) b (1, 4) c (−2, −2)
x- and y-values
x y. y x.
y =5x −1

y =5x −1
x y (x, y)
0 −1 (0, −1)
1 4 (1, 4)
x =2 y.
y =5x −1
y =10−1
y =9
y =5x -1
y =5x −1
x y (x, y)
0 −1 (0, −1)
1 4 (1, 4)
2 9 (2, 9)
x y
y =4x −2:
y =4x −2
x y (x, y)
0
−1
2
x =0,x =−1, x =2 y =4x −2.

y =4x −2 y =4x −2 y =4x −2
y =0−2 y =−4−2 y =8−2
y =−2 y =−6 y =6
(0, −2) (−1, −6) (2, 6)
y =4x −2
x y (x, y)
0 −2 (0, −2)
−1 −6 (−1, −6)
2 6 (2, 6)
y = 3x − 1.
y =3x −1
x y (x, y)
0
−1
2
y = 6x + 1
y =6x +1
x y (x, y)
0
1
−2

5x −4y = 20:
5x −4y =20
x y (x, y)
0
0
5
5x −4y =20
x y (x, y)
0 −5 (0, −5)
4 0 (4, 0)
8 5 (8, 5)
2x − 5y = 20.
2x −5y =20
x y (x, y)
0
0
−5

3x − 4y = 12.
3x −4y =12
x y (x, y)
0
0
−4
x
y. 1, 100, 1,000,
0
3x +2y =6.
x y.
x =0.
y x =0
0 x.
x =0,y =3.
(0, 3).
4x + 3y = 12.
2x + 4y = 8.
3x +2y =6.

3x +2y =6.
3x +2y =6, x y. x y.
1 x, 0 −3 y.
(2, 0) (1, 3 ) (4, −3)
2
(2, 0) (1, 3 ) (4, −3)
2
(2, 0), ⎛ ⎝ 1, 3 ⎞
2⎠ (4, −3) 3x +2y =6.
(0, 3)
3x +2y =6
x y (x, y)
0 3 (0, 3)
2 0 (2, 0)
1
3
2
(1, 3 2 )
4 −3 (4, −3)
2x + 3y = 6.

4x + 2y = 8.
x −4y =8.
x y.
(0, −2) (8, 0) (20, 3)
(0, −2), (8, 0), (20, 3) x −4y =8.
x −4y =8
x y (x, y)
0 −2 (0, −2)
8 0 (8, 0)
20 3 (20, 3)
4x + y = 8.
x + 5y = 10.

(3, 2) (4, 1) (1, 5)
(3, 4) (4, 1), (1, 4) (3, 2), (2, 3)
(3, 4), (4, 3)
a (−4, 2) b (−1, −2)
c (3, −5) d ⎛ ⎝ 2, 5 ⎞
2⎠
a (−2, −3) b (3, −3)
c (−4, 1) d ⎛ ⎝ 1, 3 ⎞
2⎠
a (−1, 1) b (−2, −1)
c (1, −4) d ⎛ ⎝ 3, 7 ⎞
2⎠
a (3, −2) b (−3, 2)
c (−3, −2) d (3, 2)
a (4, −1) b (−4, 1)
c (−4, −1) d (4, 1)
a (−2, 0)
b (−3, 0)
c (0, 4)
d (0, 2)

2x + y =6
x +3y =9
4x −2y =8
a (1, 4) b (3, 0)
c (2, 3)
a (0, 3) b (6, 1)
c (−3, −3)
a (3, 2) b (1, 4)
c (0, −4)
3x −2y =12
a (4, 0) b (2, −3)
c (1, 6)
y =4x +3
a (4, 3) b (−1, −1)
c ( 1 2 ,5)
y =2x −5
a (0, −5) b (2, 1)
c ( 1 2 ,−4)
y = 1 2 x −1
a (2, 0) b (−6, −4)
c (−4, −1)
y = 1 3 x +1
a (−3, 0) b (9, 4)
c (−6, −1)

y =2x −4
y =3x −1
y =−x +5
x y (x, y)
−1
0
2
x y (x, y)
−1
0
2
x y (x, y)
−2
0
3
y = 1 3 x +1
x y (x, y)
0
3
6
y =− 3 2 x −2
x y (x, y)
−2
0
2
x +2y =8
x y (x, y)
0
4
0
a
a
Age Weight (x, y)
0 7 (0, 7)
2 11 (2, 11)
4 15 (4, 15)
6 16 (6, 16)
8 19 (8, 19)
10 20 (10, 20)
12 21 (12, 21)
Height
x
Weight
y
⎛
⎝x, y)
28 22 (28, 22)
31 27 (31, 27)
33 33 (33, 33)
37 35 (37, 35)
40 41 (40, 41)
42 45 (42, 45)
b
b

a
b

3x + 2 x = −1.
5x + 2y = 20 y.
3 8 (−24).
3x +2y =6
(0, 3) (2, 0) ⎛ ⎝ 1, 3 ⎞
2⎠ (4, − 3) 3x +2y =6
3x +2y =6

( −2,6) x =−2
y =6
(4, 1) 3x +2y =6 (4, 1)
3x +2y =6
Ax + By = C
y =2x −3

a b
(0, 3) (3, − 3) (2, − 3) ( −1, −5)
x y
a
b (0, − 3) (3, 3) (2, − 3) ( −1, −5)
(0, − 3) (3, 3) ( −1, −5) y =2x −3 (2, − 3)
y =2x −3

y = 3x − 1
a
b
(0, − 1)
(2, 2)
(3, − 1)
( − 1, − 4)
y =2x +1
x y,
y x. 0, 1, -2 x
x y.

y =2x +1
x y (x, y)
0 1 (0, 1)
1 3 (1, 3)
−2 −3 (−2, −3)
y =2x +1.

y =−3x.
x, y.
y =−3x
x y (x, y)
0 0 (0, 0)
1 3 (1, −3)
−2 6 (−2, 6)

y = −4x.
y = x.
x, x.
x.
y = 1 2 x +3.
1 2
x,
x 2
y = 1 2 x +3
x y (x, y)
0 3 (0, 3)
2 4 (2, 4)
4 5 (4, 5)

y = 1 3 x − 1.
y = 1 4 x + 2.
y x. x y
x + y =5.
x y.
x + y =5
x y (x, y)
0 5 (0, 5)
1 4 (1, 4)
4 1 (4, 1)

x + y = −2.
x − y = 6.
2x + y =3. y 0, x?
⎛ ⎝ 3 2 ,0⎞ ⎠ . x
y = 1 2 x +3,
x
2x + y =3 y,
2x + y =3
y =−2x +3
x x =0,x =1,
x =−1
y =−2x +3
x y (x, y)
0 5 (−1, 5)
1 3 (0, 3)
−1 1 (1, 1)

3x + y =−1.
y.
3x + y = −1
y = −3x −1
x 0, 1, −1
y =−3x −1
x y (x, y)
0 −1 (0, −1)
1 −4 (1, −4)
−1 2 (−1, 2)

x- y
2x + y = 2.
4x + y = −3.
x y, y x?
x =−3. x −3,
y. y x −3.
−3 x y. x
y, 1, 2, 3 y
x =−3
x y (x, y)
−3 1 (−3, 1)
−3 2 (−3, 2)
−3 3 (−3, 3)
x = a.
x (a, 0)

x =2.
x, x 2. x 2
y.
x =2
x y (x, y)
2 1 (2, 1)
2 2 (2, 2)
2 3 (2, 3)
x 2.
x = 5.
x = −2.
y x y =4. y
y x.
4 y x.
0, 2, 4 x

y =4
x y (x, y)
0 4 (0, 4)
2 4 (2, 4)
4 4 (4, 4)
y-axis
4.
y = b.
y-axis (0, b).
y =−1.
y =−1 y. y
y −1 0, 3, −3 x.
y =−1
x y (x, y)
−3 −1 (−3, −1)
0 −1 (0, −1)
3 −1 (3, −1)

y –1
y = −4.
y = 3.
y =4x.
y =4x y =4?
y =4x x y. y x. y-coordinate
x.
y =4 y y-coordinate 4.
x.
y =4x y =4

y =−3x y =−3
x,
y = −4x y = −4.
y = 3 y = 3x.

a
y = x +2
y = x −4
b
y = 1 2 x −3
(0, 2)
(1, 2)
( −1,1)
( −3,1)
(0, − 4)
(3, − 1)
(2, 2)
(1, − 5)
(0, − 3)
(2, − 2)
( −2, −4)
(4, 1)
y = 1 3 x +2
(0, 2)
(3, 3)
( −3,2)
( −6,0)

y =3x −1 y =2x +3 y =−2x +2
y =−3x +1 y = x +2 y = x −3
y =−x −3 y =−x −2 y =2x
y =3x y =−4x y =−2x
y = 1 2 x +2 y = 1 3 x −1 y = 4 3 x −5
y = 3 2 x −3 y =−2 5 x +1 y =−4 5 x −1
y =− 3 2 x +2 y =−5 3 x +4 x + y =6
x + y =4 x + y =−3 x + y =−2
x − y =2 x − y =1 x − y =−1
x − y =−3 −x + y =4 −x + y =3
−x − y =5 −x − y =1 3x + y =7
5x + y =6 2x + y =−3 4x + y =−5
2x +3y =12 3x −4y =12 1
3
x + y =2
1
2
x + y =3
x =4 x =3 x =−2
x =−5 y =3 y =1
y =−5 y =−2 x = 7 3
x = 5 4
y =− 1 2 x y =−1 y =−
2
1 3 x y =−1 y =2x y =2
3
y =5x y =5

y =4x y =2x y =− 1 2 x +3
y = 1 4 x −2 y =−x y = x
x − y =3 x + y =−5 4x + y =2
2x + y =6 y =−1 y =5
2x +6y =12 5x +2y =10 x =3
x =−4
$594 $0.32
y = 594 + 0.32x
y, x
400, 800, and 1,200
$200 15%
y = 200 + 0.15x
y x
$900, $1,600, and $2,000,
x-values
y = 1 5 x −2.
a
b

3x + 4y = −12 x y = 0.
(0, −5) x-axis y-axis?
2x − y = 6?
a (6, 0) b (0, −6) c (4, −2).
x-axis y-axis

(a, 0)

x-intercept (a, 0), x-axis. x-intercept y
y-intercept (0, b), y-axis.
y-intercept x
x- and y-intercepts

a x +2y =4
b 3x − y =6
c x + y =−5
a

c
x- y-intercepts x − y = 2.
x- y-intercepts 2x + 3y = 6.
x-intercept y y-intercept x
x-intercept, y =0 x.
y-intercept, x =0 y.
y =0
x =0

2x + y =6
y =0
x =0
(3, 0) (0, 6)

3x + y = 12
x + 4y = 8
4x−3y = 12.
x-intercept, y =0.
4x −3y =12
y.
4x −3·0=12
4x −0=12
4x =12
x =3
x-intercept (3, 0).
y-intercept, x =0.
4x −3y =12
x. 4·0−3y =12
0−3y =12
−3y =12
y =−4
y-intercept (0, −4).
(−3, 0) (0, −4).
4x−3y =12
3 0
0 −4
3x−4y = 12.
2x−4y = 8.

−x +2y =6
x-intercept. y =0,
−x +2y =6
−x +2(0)=6
−x =6
x =−6
x-intercept (–6, 0).
y-intercept. x =0.
−x +2y =6
−0+2y =6
2y =6
y =3
y-intercept (0, 3).
x =2,
−x +2y =6
−2+2y =6
2y =8
y =4
(2, 4).
−x +2y =6
−6 0 (−6, 0)
0 3 (0, 3)
2 4 (2, 4)

x−2y = 4.
−x + 3y = 6.
x - y-intercepts
y = 0 x
x = 0 y.
4x−3y =12

4x−3y =12
x y (x, y)
3 0 (3, 0)
0 −4 (0, −4)
6 4 (6, 4)
5x−2y = 10.
3x−4y = 12.
y =5x
(0, 0).
x, x 1 −1.

y =5x
x y (x, y)
0 0 (0, 0)
1 5 (1, 5)
−1 −5 (−1, −5)
y = 4x.
y = − x.
y =2x +1
y = 1 2 x +3
x =−7
y =4
2x + y =6
4x −3y =12

Ax + By = C y =0 x =0
x = a x-axis a
y = b y-axis b.
y
x y-
Ax + By = C,
x- y-
a y =−3 b 4x−6y =12 c x =2 d y = 2 5 x−1
a y =−3
y. y-axis −3.
b 4x−6y =12
Ax + By = C.
c x =2
x. x-axis 2.
d y = 2 5 x−1
y
a 3x + 2y = 12 b y = 4 c y = 1 x−4 d x = −7
5

a x = 6 b y = − 3 x + 1 c y = −8 d 4x−3y = −1
4

x- y-

x y
x + y =4 x + y =3 x + y =−2
x + y =−5 x − y =5 x − y =1
x − y =−3 x − y =−4 x +2y =8
x +2y =10 3x + y =6 3x + y =9
x−3y =12 x−2y =8 4x − y =8
5x − y =5 2x +5y =10 2x +3y =6
3x−2y =12 3x−5y =30 y = 1 3 x−1
y = 1 4 x−1 y = 1 5 x +2 y = 1 3 x +4
y =3x y =−2x y =−4x
y =5x
−x +5y =10 −x +4y =8 x +2y =4
x +2y =6 x + y =2 x + y =5
x + y =3 x + y =−1 x − y =1
x − y =2 x − y =−4 x − y =−3
4x + y =4 3x + y =3 3x − y =−6
2x − y =−8 2x +4y =12 3x +2y =12

3x−2y =6 5x−2y =10 2x−5y = −20
3x−4y = −12 y =−2x y =−4x
y = x y =3x
x =2 y =4 y =5
x =−3 y =−3x +4 y =−5x +2
x − y =5 x − y =1 y = 2 3 x−1
y = 4 5 x−3 y =−3 y =−1
3x−2y = −12 2x−5y = −10 y =− 1 4 x +3
y =− 1 3 x +5
1,000 x-axis
y-axis
x-axis
y-axis
a x- y-
b x- y-
a x- y-
b x- y-

x-intercept
3x−2y =6?
y-intercept
5x − y =10?
4x + y =−4
y = 2 3 x−2
a
b

1 − 4
8 − 2 .
0
4
, 4 0 .
15
−3 , −15
3 , −15
−3 .

90 °
2 2
3 3

2 .
3
m.
m = rise run .
m = rise run
m = 2 3
The line has slope 2 3 .
m = rise run

The rise is 3 units. m = 3 run
The run is 4 units. m = 3 4
The slope is 3 4 .
m = rise run

The rise is −1.
The run is 3.
m = run
−1
m = −1
3
m =− 1 3
The slope is − 1 3 .

1 2 .
m 1 2
m = rise run
1
2
= rise run
1 2
1 2

1 2 .
m = 1 3 .
m = 3 2 .
−1
4 ,
m − 1 4
m = rise run
−1 4 = rise run
−1 4.
1 4
− 1 4 .
m = −2
1 .
m = −1
3 .

(0, −3) (5, 1).
(0, −3), (5, 1).
m = rise run
m = 4 5
4 5

m = rise run

m = rise run
m = −2
3
m =− 2 3
− 2 3 .
(−3, 7) (6, 1).
(−3, 7), (6, 1).

m = rise run
m = −6
9
m =− 2 3
− 2 3 .
y
y

(2, 3) (7, 6)
(2, 3)
(2, 3) (7, 6)
m = rise run
m = 3 5
3 5 .

y = b; y
x = a; x
y =4?
(0, 4)
(3, 4)

m = rise run
m = 0 3
m =0
y =4 0.
0 y 0
y = b, 0.
x =3 (3, 0)
(3, 2)
m = rise run
m = 2 0
0. 0 x =3
0.
x = a,

a x =8 b y =−5
a x =8
b y =−5
0.
x = −4.
y = 7.
(x, y)
(x, y)
(x 1 , y 1 ) read x sub 1, y sub 1
(x 2 , y 2 ) read x sub 2, y sub 2
(x 1 , y 1 ) (x 2 , y 2 )
(x 3 , y 3 ), (x 4 , y 4 ),
(2, 3) (7, 6)

x 1 , y 1 x 2 , y 2
(2, 3) (7, 6)
3. y-coordinates
y 2 − y 1
6−3
3
5. x-coordinates.
x 2 − x 1
7−2
5
m = rise run
m = 3 5
m = 6−3
7−2
y y 2
y y 1
x x 2
x x 2
m = y 2 − y 1
7−2
m = y 2 − y 1
x 2 − x 1
m = y 2 − y 1
x 2 − x m = rise
1 run .
(x 1 , y 1 ) (x 2 , y 2 )
m = y 2 − y 1
x 2 − x 1
Slope is y of the second point minus y of the first point
over
x of the second point minus x of the first point.
(1, 2) (4, 5).

x
(1, 2) (4, 5) 1 , y 1 x 2 , y 2
(1, 2) and (4, 5)
m = y 2 − y 1
x 2 − x 1
y y m = 5−2 x 2 − x 1
x x
m = 5−2
4−1
m = 3 3
m =1
3 3,
m = rise run
m = 3 3
m =1
(8, 5) (6, 3).
(1, 5) (5, 9).

x
(4, 5) (1, 2) 1 , y 1 x 2 , y 2
(4, 5) and (1, 2)
m = y 2 − y 1
x 2 − x 1
y y m = 2−5 x 2 − x 1
x x
m = 2−5
1−4
m = −3
−3
m =1
(−2, −3) (−7, 4).
x
(−2, −3) (−7, 4) 1 , y 1 x 2 , y 2
(−2, −3) and (−7, 4)
m = y 2 − y 1
x 2 − x 1
y y
x x
m = 4−(−3)
x 2 − x 1
m = 4−(−3)
−7 − (−2)
m = 7 −5
m =− 7 5

m = rise run
m = −7
5
m =− 7 5
(−3, 4) (2, −1).
(−2, 6) (−3, −4).
(1, −1) m = 3 4 .
(1, −1).
m = rise run

m = 3 4
rise
run = 3 4
rise = 3
run = 4
3 4
3 4.
(2, −2), m = 4 3
(−2, 3), m = 1 4

y (0, 2) m =− 2 3 .
y (0, 2).
m = rise run
m =− 2 3
rise
run = −2
3
rise = –2
run = 3
(0, 2),

y 4, m = − 5 2
x −3, m = − 3 4
(−1, −3) m =4.
m =4
rise
run = 4 1
rise = 4 run = 1

(−2, 1) m = 3.
(4, −2) m = −2.
m = rise run
m = 9 ft
18 ft
m = 1 2
1 2
= 14 = 24.
= 15 = 36.

1 4
m = rise run
m = − 1 4 in.
1ft
m = − 1 4 in.
1ft
m = − 1 4 in.
12 in.
m =− 1 48
− 1 48 .
1 3
3 4

2
3
3
4
1
4
4
3
− 1 2
− 3 4
− 2 3
− 3 2

y =3 y =1 x =4
x =2 y =−2 y =−3
x =−5 x =−4
(1, 4), (3, 9) (2, 3), (5, 7) (0, 3), (4, 6)
(0, 1), (5, 4) (2, 5), (4, 0) (3, 6), (8, 0)
(−3, 3), (2, −5) (−2, 4), (3, −1) (−1, −2), (2, 5)
(−2, −1), (6, 5) (4, −5), (1, −2) (3, −6), (2, −2)
(1, −2); m = 3 4
(1, −1); m = 1 2
(2, 5); m =− 1 3
(1, 4); m =− 1 2
(−3, 4); m =− 3 2
(−2, 5); m =− 5 4

(−1, −4); m = 4 3
(−3, −5); m = 3 2
(0, 3); m =− 2 5
(0, 5); m =− 4 3
(−2, 0); m = 3 4
(−1, 0); m = 1 5
(−3, 3); m =2 (−4, 2); m =4 (1, 5); m =−3
(2, 3); m =−1
12-inch
6%.
a
b
2 50
a
b
1 12
a
24-inch
b
1-inch 16-inch
a
24-inch
b

m = 1 2
m =2?
a
b

y = b
(0, b).
Ax + By = C, A and B
(x, y)
x y
(x, y)
x-coordinate, y-coordinate
(0, 0) x y
x y
m = rise run
(x, y)
Ax + By = C
x = a
(a, 0).
ax + by = c

(a, 0)
(0, b)
y =0
x =0
y =0
x =0
x = a
y = b
Ax + By = C
m = rise run
y = b
x = a

(x 1 , y 1 ) (x 2 , y 2 ) m = y 2 - y 1
x 2 - x 1
(1, 3), (3, 1) (2, 5), (5, 2)
a (−1, −5)
b (−3, 4)
c (2, −3)
d ⎛ ⎝ 1, 5 ⎞
2⎠
a (3, −2)
b (−4, −1)
c (−5, 4)
d ⎛ ⎝ 2, 10 3
⎞
⎠

5x + y =10
y =6x −2
a (5, 1) b (2, 0)
c (4, −10)
a (1, 4)
c (6, −2)
b ⎛ ⎝ 1 3 ,0⎞ ⎠
y =4x −1
y =− 1 2 x +3
x +2y =5
x y (x, y)
0
1
−2
x y (x, y)
0
1
−2
x y (x, y)
0
1
−1
3x −2y =6
x y (x, y)
0
0
−2
x + y =3 x + y =−4 y =3x +1
y =−x −1
a
b

y =−x +4
y = 2 3 x −1
(0, 4)
(−1, 3)
(2, 2)
(−2, 6)
(0, −1)
(3, 1)
(−3, −3)
(6, 4)
y =4x −3 y =−3x 2x + y =7
y =−2 x =3
x- y-intercepts.
x + y =5 x − y =−1 y = 3 4 x −12
y =3x

−x +3y =3 x + y =−2
x =5 y =−3 2x + y =5
x − y =2 y = 1 2 x +2 y = 3 4 x −1
1
3
3
2
− 2 3
− 1 2

y =2 x =5 x =−3
y =−1
(2, 1), (4, 5) (−1, −1), (0, −5) (3, 5), (4, −1)
(−5, −2), (3, 2)
(2, −2); m = 5 2
(−3, 4); m =− 1 3
10
15

a (2, 5)
b (−1, −3)
c (−4, 0)
d (3, −5)
e (−2, 1)
x-intercept
y-intercept
x-intercept
y-intercept
3x − y =6.
(1, 3)
x +4y =12?
y =−x +1.
x y (x, y)
0
1
3
−2
4x + y =8
x y (x, y)
0
0
3
y =−3x 2x +3y =−6

(0, −4) (5, 2).
y =2.
(1, 1) m = 3 20
2 . 1,000

1. 5(3 + 2 · 6) − 8 2
2. 17 = y −13
3. p +14=23
4. 11 7 x.
5. y 7 84.
6. 72.
7. 132.
8. 12 20.
9. |8−9| − |3−8|
10. −2 + 4(−3 + 7)
11. 27 − (−4 − 7)
12. 28 ÷ (−4) −7
13. −5 13, 11.
14. −11 and 8.
15. 7 −4 and m.
16. −3 −51.
17. −6r =24
18. 7
8
, 5 3 ,31 4 ,5.
19. 21p
57q
20. 3 ⎞ ·
7
⎛⎝ −28
45⎠
21. −6 3 4 ÷ 9 2

22. −3 3 5 ÷6
23. −4 2 ⎛
3 ⎝ −6 ⎞
7⎠
24. −21 4
− 3 8
25. 7·8+4(7−12)
9·6−2·9
26. − 23
36 + 17
20
27.
1
2 + 1 3
3
4 − 1 3
28. 3 5 8 −21 2
29. − 2 3 r =24
30. 24.76 − 7.28
31. 12.9 + 15.633
32. (−5.6)(0.25)
33. $6.29 ÷ 12
34. 3 (13.44 − 9.6)
4
35. 64 + 225
36. 121x 2 y 2
37. 5 8 , 0.75, 8
15
38. −8.6x = 34.4
39. 3.14 8
40. 18, 16, 20, 12
41. 24, 29, 27, 28, 30
42. 6, 4, 4, 5, 6, 6, 4, 4, 4, 3, 5
43. 3 $28.97.
44. 14.7%
45. 63 35%
46. 180 81
47. $510 $3,400

48. $67.50. $90.
49. $23,000 5 $23,000 $9,200
50. 9 p = −6
14
51.
−5, −2 1 4 , − 4, 0.25, 13 5 ,4
52. ⎛ ⎝15 8 + 4 ⎞
7⎠ + 3 7
53. 3(y +3)−8(y −4)
54. 8 ·49·17 17 8
55. 55
56.
48, 26, 81, 54, 43, 62, 106.
57. 2.8 1 1.61
58. y +13=−8
59. p + 2 5 = 8 5
60. 48 = 2 3 x
61. 4(a −3) −6a = −18
62. 7q + 14 = −35
63. 4v −27=7v
64. 7 8 y −6=3 8 y −8
65. 26 − 4(z −2)=6
66. 3 4 x − 2 3 = 1 2 x − 5 6
67. 0.7y + 4.8 = 0.84y − 5.3
68. n 13.
69. 8
70. −95.
71. $3.65 2

72. 24°
73. 20°
74. 12
15
75. 4 96
76. 24 18
6
77. ⎛ ⎝8m 2 +12m −5 ⎞ ⎠ − ⎛ ⎝2m 2 −7m −1 ⎞ ⎠
78. p 3 · p 10
79. ⎛ ⎝y 4⎞ 3
⎠
80. ⎛ ⎝3a 5⎞ 3
⎠
81. ⎛ ⎝x 3⎞ ⎛⎝
⎠5
x 2⎞ 3
⎠
82. ⎛ ⎝ 2 3 m3 n 6⎞ ⎠ ⎛ ⎝ 1 6 m4 n 4 ⎞
⎠
83. ⎛ ⎝y −4 ⎞ ⎠ ⎛ ⎝y +12 ⎞ ⎠
84. (3c +1)(9c −4)
85. (x −1) ⎛ ⎝x 2 −3x −2 ⎞ ⎠
86. (8x) 0
87.
⎛
⎝x 3⎞ 5
⎠
⎛
⎝x 2⎞ 4
⎠
88. 32a7 b 2
12a 3 b 6
89. ⎛ ⎝ab −3⎞ ⎠ ⎛ ⎝a −3 b 6⎞ ⎠

90. (a) 4,800,000 (b) 0.00637
91. 3x 4 −6x 3 −18x 2
92. y =4x −3
93. y =−3x
94. y = 1 2 x +3
95. x − y =6
96. y =−2
97. 2x +3y =12
98. 2x −4y =8
99.
100. (−5, −2), (3, 2).
101. (−3, 4) m =− 1 3 .

n n 2 n n 3 3 n
1 1 1 1 1
2 4 1.414214 8 1.259921
3 9 1.732051 27 1.442250
4 16 2 64 1.587401
5 25 2.236068 125 1.709976
6 36 2.449490 216 1.817121
7 49 2.645751 343 1.912931
8 64 2.828427 512 2
9 81 3 729 2.080084
10 100 3.162278 1,000 2.154435
11 121 3.316625 1,331 2.223980
12 144 3.464102 1,728 2.289428
13 169 3.605551 2,197 2.351335
14 196 3.741657 2,744 2.410142
15 225 3.872983 3,375 2.466212
16 256 4 4,096 2.519842
17 289 4.123106 4,913 2.571282
18 324 4.242641 5,832 2.620741
19 361 4.358899 6,859 2.668402
20 400 4.472136 8,000 2.714418
21 441 4.582576 9,261 2.758924
22 484 4.690416 10,648 2.802039
23 529 4.795832 12,167 2.843867
24 576 4.898979 13,824 2.884499
25 625 5 15,625 2.924018
26 676 5.099020 17,576 2.962496

n n 2 n n 3 3 n
27 729 5.196152 19,683 3
28 784 5.291503 21,952 3.036589
29 841 5.385165 24,389 3.072317
30 900 5.477226 27,000 3.107233
31 961 5.567764 29,791 3.141381
32 1,024 5.656854 32,768 3.174802
33 1,089 5.744563 35,937 3.207534
34 1,156 5.830952 39,304 3.239612
35 1,225 5.916080 42,875 3.271066
36 1,296 6 46,656 3.301927
37 1,369 6.082763 50653 3.332222
38 1,444 6.164414 54,872 3.361975
39 1,521 6.244998 59,319 3.391211
40 1,600 6.324555 64,000 3.419952
41 1,681 6.403124 68,921 3.448217
42 1,764 6.480741 74,088 3.476027
43 1,849 6.557439 79,507 3.503398
44 1,936 6.633250 85,184 3.530348
45 2,025 6.708204 91,125 3.556893
46 2,116 6.782330 97,336 3.583048
47 2,209 6.855655 103,823 3.608826
48 2,304 6.928203 110,592 3.634241
49 2,401 7 117,649 3.659306
50 2,500 7.071068 125,000 3.684031
51 2,601 7.141428 132,651 3.708430
52 2,704 7.211103 140,608 3.732511
53 2,809 7.280110 148,877 3.756286

n n 2 n n 3 3 n
54 2,916 7.348469 157,464 3.779763
55 3,025 7.416198 166,375 3.802952
56 3,136 7.483315 175,616 3.825862
57 3,249 7.549834 185,193 3.848501
58 3,364 7.615773 195,112 3.870877
59 3,481 7.681146 205,379 3.892996
60 3,600 7.745967 216,000 3.914868
61 3,721 7.810250 226,981 3.936497
62 3,844 7.874008 238,328 3.957892
63 3,969 7.937254 250,047 3.979057
64 4,096 8 262,144 4
65 4,225 8.062258 274,625 4.020726
66 4,356 8.124038 287,496 4.041240
67 4,489 8.185353 300,763 4.061548
68 4,624 8.246211 314,432 4.081655
69 4,761 8.306624 328,509 4.101566
70 4,900 8.366600 343,000 4.121285
71 5,041 8.426150 357,911 4.140818
72 5,184 8.485281 373,248 4.160168
73 5,329 8.544004 389,017 4.179339
74 5,476 8.602325 405,224 4.198336
75 5,625 8.660254 421,875 4.217163
76 5,776 8.717798 438,976 4.235824
77 5,929 8.774964 456,533 4.254321
78 6,084 8.831761 474,552 4.272659
79 6,241 8.888194 493,039 4.290840
80 6,400 8.944272 512,000 4.308869

n n 2 n n 3 3 n
81 6,561 9 531,441 4.326749
82 6,724 9.055385 551,368 4.344481
83 6,889 9.110434 571,787 4.362071
84 7,056 9.165151 592,704 4.379519
85 7,225 9.219544 614,125 4.396830
86 7,396 9.273618 636,056 4.414005
87 7,569 9.327379 658,503 4.431048
88 7,744 9.380832 681,472 4.447960
89 7,921 9.433981 704,969 4.464745
90 8,100 9.486833 729,000 4.481405
91 8,281 9.539392 753,571 4.497941
92 8,464 9.591663 778,688 4.514357
93 8,649 9.643651 804,357 4.530655
94 8,836 9.695360 830,584 4.546836
95 9,025 9.746794 857,375 4.562903
96 9,216 9.797959 884,736 4.578857
97 9,409 9.848858 912,673 4.594701
98 9,604 9.899495 941,192 4.610436
99 9,801 9.949874 970,299 4.626065
100 10,000 10 1,000,000 4.641589

Perimeter: P =2l +2w
Area: A = lw
Perimeter: P =4s
Area: A = s 2
Perimeter: P = a + b + c
Area: A = 1 2 bh
Sum of Angles: A + B + C = 180°
Pythagorean Theorem: a 2 + b 2 = c 2
Area: A = 1 2 ab
C =2πr
Circumference: or
C = πd
Area: A = πr 2
Perimeter: P =2a +2b
Area: A = bh

Perimeter: P = a + b + c + B
Area: A = 1 (B + b)h
2
Volume: V = lwh
Surface Area: SA =2lw +2wh +2hl
Volume: V = s 3
Surface Area: SA =6s 2
Volume: V = 1 3 πr2 h
Surface Area: SA = πr 2 + πr h 2 + r 2
Volume: V = 4 3 πr3
Surface Area: SA =4πr 2

Volume: V = πr 2 h
Surface Area: SA =2πr 2 +2πrh

215 342, 006
910 2, 162 284
81 17 5, 394
a
b
a
b
a
b
c
d
e
a
b
c
d
e
a
b
a
b

a 0 + 19 = 19
b 39 + 0 = 39
a 0 + 24 = 24
b 57 + 0 = 57
9 + 7 = 16; 7 + 9 = 16 8 + 6 = 14; 6 + 8 = 14 32 + 54 = 86
25 + 74 = 99 35 + 98 = 133 72 + 89 = 161
456 + 376 = 832 269 + 578 = 847 4,597 + 685 = 5,282
5,837 + 695 = 6,532
17 + 26
43
a
b
46,195 + 397 + 6,281 = 52,873
28 + 14
42
53,762 + 196 + 7,458 = 61,416
29 + 76
105
a
b
a b
a b

a
b
a 0 b 0
a
b
a 0 b 0
a 19 b 39
a 24 b 57
512
342 1,735 3,234
a b
a b
a
b
c
a
b
c
a b a b
a
b

a b
a b
a b
a
b
a
b
a b
c d
e
a
b c
d e
a b
a b a b
a 64,000 b 63,900
a 1,000,000,000
b 1,400,000,000
c 1,356,000,000
a
c
b
d
2+4=6

8+4=12
16 + 25 = 41
14 + 75 = 89
78
a b
a 11 b 11
99 85 142
493 861 1,031
6,850 22,333 1,104,860
29,191 101,531 13 + 18 = 31
90 + 65 = 155 33 + 49 = 82 250 + 599 = 849
628 + 77 = 705 915 + 1,482 = 2,397

a
b

a
b
a
b
a
b
c d
e
a
b
a
b

a
b
a
b
a
b
112 180, 096 807
1, 024 73
19 42 x −8
4 215 15 2, 3, 5, 6, 10
2 4

a
b
c
d
a
b
c
d
a
a
a
b
b
b
a
b
c
d
a b
a
b
a
b
a
b
a
b
c
d
a
b
a
b
a
a b
a b
a b
a b
a b
a
a
b
a
a
a
b
b
a
b
b

a
a
a
b
b
b
a
a
a
b
b
b
a
a
b
b
a b

a
a
b
b
a b
a
b
a
a
b
b

a b
a
b
a
b
a
b
< 14
20 3+(−7) 5
20 − (−15) 11 20 ÷ 13
−15 −3 0
16
x −5
a b c d
a b c d
a
b
a
a
b
b

a
b
a b c
d
a
b
a b c
d
a
b
a b c d
a b c d
a b
a b
a b c
d
a b c
d
a
a
b
b
c
c
d
d
a b
a b
a b
a b
a b
a b

a
a
a
b
b
b
c
d
c
d

a
a
b
a
b
b
a
b
a
b
a
a
b
a
b
a b c
a
b
a
b
a
b
a b
a b c
d
a
a
a
b
b
b
a
a
a
b
b
b
b
a b c
d
a
a
b
b
a b c
a b c

a b c d
a b c d
a
b
a
a
a b
a b
a b
a b c
a
b
b
b
a b c
a
a b
c d
a b c
d
a b
b
a
b
a b
a b
a b
a b

a
b
a
b
a
b
a
b
a b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
−6
a + b

a b c
a b c
a
b
a
b
a
a b
a b
b

a b
a b c
a
b
a
b
11 −2 > − 5 2⋅2⋅2⋅2⋅3
3
10
12 , 15
18 , 50
60 ,
1
12
13
5
5x +5
2
7
2
25
5
6
10
12 , 15
18
2 3 4
1 y =6
a 3 8
b 4 9
a 3 5
b 3 4

5
3 =12 3
13
8
=1 5 8
1 2 7 1 3 4
11
8
11
6 1 6 7 . 1 5 9
3 2 7 4 4 11
23
8
50
11
6
10 , 9 , and 12
15 20 .
8
10 , 12 , and 16
15 20 .
18
21
26
7
34
3
30
100

a b c d
a b c d
2
3
3
4 − 3 4
− 2 3 − 9 7 − 9 5
23
5
x
40
8
y
a
3
5
b
10 12
a
2
15
21
40
5
14
14
27 −
21 4 −
16
3
b
a 24
b 44x
3
a 7 5
b −8 c −
11
4
d 1 14
a 7 3
b −12 c −
14
9
d 1 21
− 9 14 − 10
21
21
5p
15
8q
5
4
4
15
2
3
9
4
− 85 6
5
8
7
8
5
2
9s
14
5y
a − b
p + q
6
cd
r
4
5
11
14 − 2 7
− 10 3
3
4b
4
q
4
5

25
49 . 1
2 − 3 5 , 3
−5
−2, 2
7 −7
10
9
1
2
−3
4 − 2 3
2
9
8
5
5
8
5
6
5
6
x +3
y +5
4
8
1
3
9p
d
m
8
9q
5
− 2 3 − 2 3
3
8 1
6 3
7
1
x −2
y −13
2 7
14
− 16 x
− 2 3
− 22 a
− 35
60 , 44
60
7
12
1
2
1
96
− 25
96
a − 11
12
b − 1 8
(24k + 25)
a
30
a 7 6
b − 1 4
− 1 2
b 24k
25
52
96 , 51
96
7
10
79
60
75
224
3y +14
18
a − 10 3
b 13
12
1
90
2
7
9
12 , 10
12
72
224 , 189
224
5
8
103
60
37
210
5x +14
30
(27a − 32)
a
36
a −1 b − 1 2
− 3 4 − 23 8
2
3 − 5 2
b 2a 3

3
2
4 2 3 3 1 2 5 6 7
7
11 9 16 1 2 15 1 3
9 1 3 6 3 5
3
4
4
5
9
5
5
3
2
3
3
4
2
3

2
5 2 2 3 1 5 7
2 2 3 1 5 7 6
12
7
12
10 3 4
10 7 2 11
12
− 41 8 − 757
63
a b c
− 1 2 − 4 15
− 6 7 − 19 3
a b c
23
2
n
= −21; n = −147
7
n
8 = −56; n = −448 q
p
= 72; q = −576
−8 = 81; p = −729
−9
2
5 f = 16; f =40 3
4 f = 21; f =28 n =
2 12 5 ; n = 18
5 3
c
3
8
= 4 9 ; c = 1 6
5
8 + x = 1 4 ; x =−3 8 1 3 4 − x = 5 6 ; x = 11
12
a 1 4
b 3 4
c 3 8
d 5 8
a 5 4 =11 4
b 7 4 =13 4
c 11 8 =13 8

a 11 4 =23 4
b 19 8 =23 8
1 2 3
2 3 5 3 1 9 3
15
2
7
5
17
6
19
7
32
9
2 6 , 3 9 , 4 . 12
10
12 , 15
18 , 20
24 .
10
18 , 15
27 , 20
36 .
a b 6 1 4
3
4 −
11 5 − 12 7
10
21 − 7 x
8
y
− x
4y
2x 2
3y
1
3
1
3

5
18
2
15
27
40
1
4 − 1 6 − 2 9
− 21
50
11
30
20
11
3
8
1296
8 27 625
4
3 − 17 8
5 11
− 1 19
− 3 4
8
7
16
15 − 5 4
18
5y
25
2c
− 1 25
2
9
a 4·23 cups = 8 3 cups = 22 3 cups
14r
15s
1
8 yard
b
44
21
35
8 − 16 3 − 63
16
3
2 − 4 3
7v
a
13
b
A
3−B
3
2 − 15
22

−
10
3
9
2y
1
6
3r
5s
− 1 18
−4, 9
− 4 9
13
−6 , −13
6
12
7
5
2
23
4 − 1 3
4
5 − 1 2
5
6
26
25
11
6
5
2
51
18
20 7
a 4 3 =11 3 cups
b 80 3 =262 3 cups
7
10
7
9
16
15
3
4
y +2
3
14 q
9a
7
−4x
5 − 3 4 − 7 8
8
19 − 13
12
1
2
1
5
5
13
1
6
− 8 x −8
11z −8
3
19 13
− 14 c − 55 v
3d
11

− 14s
3
2
7 −
11
3
− 1 8
6−s
11 − 2 9
49
1
27 4 cup
13
12
5
20 , 4
20
14
24 , 15
24
33
48 , − 20
48
20
60 , 45
60 , 36
60
9
20
11
24
3
8
2
3
23
20
19
24
1
48
1
24
1
12
7
60 − 53
80 − 291
245
1
20
11
40
11
6
7
3y +4
4x −5
10
6
20
a 5 6
b 4
a − 4q
27
a 1 3
b
b −1
4q −75
15q
a − 37
40
b −
10
1
11
24
−
15
2 −33−8x
88
16
39
1
81
36
25
60
7
15
2
3
13
19
30
91
60
11
40
5
6
22
7
a 1 2
b −1
4
15
− 9 16
a 1 6
b 2 3
a 9a
14
b
9a −14
24
7
18
a 0 b − 1 6

She needs 2 3 8 cups
3 2 3
7 5 9
3 2 3
11 1 3 11 1 3
1 1 6 6 2 5 1 7 9
5
6 7 11
12 16 1 6
5 11
20 −1 11
24 6 17
18
7
9
8 3 4
4
15 5 3 5
1
16
13
13 1 7
30 1 10
27
2
45
7 17
24
7
12 yards
13 1 4 inches
a b c
h =− 2 3
a b c
1
2
m = 1 2

p = 37
40 k =− 7 15 k = 18 7
v =− 11 3
k = 17
20
g = 1 9 q =− 3 4 n = 14 5
y =− 25
24 d =−
15
8 n
= −24; n = −144
6
m
=−8;m =56
f
−7 = −20; f =80
−4
g
= 14; g = 126
9
2
5 q = 20; q =50 4
9 p = −28; p = −63 h
= 43; h =86
2
a = 3 2 4 ; a = 1 3
2 4 + x = 1 8 ; x =−5 8
3
y − 1 3 =−1 6 ; y = 1 6
5
3
9 2 =11 2 5
11
8
49
5
− 4 7
4
15
− 3
4
− x y
24m
11
4
15 −
2b 5 − 268
11
V
3
5
− 19
15
h −6 − 2 9
7
36
5
8
x +7
10
1
2
8
15
9
and 20
24 24
20
60 , 45 and 48
60 60
3y −10
30
1
4 − 77
90
14d
11
256
225
a − 8 5
b − 11
20
13 2 3 8 1 11 5 1 10
1 1 3
h =− 51
40
a b c
n =− 5 11
q − 1 10 = 1 2 ; q = 3 5

3y = 24; y =64
8
− 7 13
23
7
1
4
5p
2q
1
4
12
25
−
4x
9
y = 4 5 f = 13
12
750
7
10
27
100
4
0.03 > <
5
12
20
−294
5 50 2.5
2
3
0.24
6
11
81 3.85 960
a
b
c
d
a
b
c
d
a 5 3 10
b 6 7
100
c − 117
500
a 8 7 10
b 1 3
100
c − 3
125
a
b
a
b

a b c
a b c
a b c
—
—
2.45
2.318
4
5 , 0.82, 7 3
8 4 , 13
16 , 0.835
a
a b
a
b
a
b
a b c
a
b
a 220
147 m b 550
a b c
a b c
b
a
b
3087 sq. m a 40
21 in. b 200
693 sq.in.
q
= 4.5; q = −15.3
−3.4
r = 2.5; r = −6.5
−2.6
a
10 1 b 0.1
a 1 20
b 0.05
a 3 8
b 3 2
a 2 9
b 5 1
a 5 8
b 0.625
a 3 8
b 3 2
a 2 8
b 0.25
2
3
a 2 5
b 10 3
9
22

37 102
8
8 19 3
2
123 miles
121 miles
9 2 hours 3 hours
a
b
c
a b
a
a
a
b
b
a
b
c
a
b
a
b
a b
a b
6< 38

a b c
a b c
a
c
b
a
a
0. 5 – —
—
1.36
0.135
b
a b c
d e
b
0.55, 9 , 3 16 5
5
8 , 13
20 , 0.702 −
20 7 , − 1 3 , − 0.3
− 7 9 , − 3 4 , −0.7
a b a b
a
b
a 33
14 yard b 99
a
b
224 sq.yard a 55
21 m b 275
504 sq.m
a b c
a 22 5 mile b 77
50 sq.mile
a b c

q =− 3 4 m =
35 4 y =− 25
24
f =− 11
20
n − 1.9 = 3.4; 5.3
y
−1.7 = −5; 8.5
1
20 , 0.05 1
24 , 0.041 6– ≈ 0.042 2
5 , 0.4
5
7 , 0.714285 ——— ≈ 0.714
a b
c
5
9
7
8
7
3
7
3
1
5
10
17
5
4
2
7
11
4
1
3
6
23
82
15
35
9
9
4
35 calories
3 ounces
41 lbs
488 miles
$119
15 sq. in. 7 hours
8 hours
$3
0.5 lbs.
105 calories
x ounces
793 miles
p hours
y
5x

a
b
c
d
8< 70

a adults
19
45 children 3+n
12 < 155

11. 1 – %, or 11 1 9 % 16. 6 – %, or 16 2 3 %
a b
a b
a b
a b
a
a
b
b
a
b
a
a 5 9 = 20
36
b
11 7 = 28
44
c 2.50
8 = 3.75
12
a 6 7 = 36
42
b
36 8 = 12
54
c 3.75
6 = 2.50
4
a
b
b
a b
n
105 = 100 60
n
85 = 100
40
36
n =
100 25
27
n =
100 36
n
100 = 39
52
n
100 = 23
92
n
40 = 100 65 n =26 n
40 = 100 85 n =34
n
64 = 125
100 ; n =80 n
84 = 175
100 ; n = 147 3.23
n = 100 8.5 n =38
4.64
n = 7.25
100 ; n =64 27
72 = 100 n n = 37.5% 23
92 = 100 n n = 25%
6
100
a
100 42 b 42%
13
32
1000
1
25
5
3
25 4 8
23
19
125 200
a
100 57 b 57%
17
100
4
75

a
200 3 b 0.015
a 7023
10,000
b 0.7023
a 1 4
b 0.25
a 3 5
b 0.6
41 2 3 % or 41. 6– % 266. 6 – %
55. 5 – %
1
2
0.5 50%
a 1 10
b approximately $9.50
17
20 ; 0.85
9
20
9
50
1
3
1
125
0.45 45%
0.18 18%
0.3 33.3%
0.008 0.8%
2 2.0 200%
1
4 , 1 2 , 3 4 ,1.
a
b
a b
a
b

a
b
a b
a b
a b
a b
c
a
a
a
b
b
b
a
b
a
a
a
4
15 = 135
36
12
= 96
5 40
5
7 = 115
161
8
1 = 48 6
9.36
18 = 2.60
5
18.04
11 = 4.92
3
b
b
3
4 cup
n
250 = 100 35
n
47 = 110
100
45
n =
100
30
90
n = 150
17
100 85 = p
340
100 260 = p
100
n
180 = 100 65
n
92 = 100 18
n
26 = 175
100
n
488 = 300
100
14
56 = p
100
32
100
7.65
n = 17
12
96 = p
100
100
13
100 , 13%
13.53
n = 8.25
100
b
12
25

641
1000
a 27
200
b 0.135
a 1 2
b 0.5
a b
a
b
a b
a b
3
8 = 12
32
1
18 = 414
23
n
395 = 100 62
15
1000 = p
100
n
900 = 100 85 3.51
n = 100 4.5
0.24, 6
25 3.5, 7 2 33. 3¯ % or 33 1 3 %
319
100
0.45
12
8y +15 12 −15
0.75 16 12y +15
4
2
5
27
10
4, 290
5
9
5
12
a −24
1
b 357
100
a
b
a −19
1
b 841
100
a
b
a
c
a
b
b
c

a
b
a
b
a
(1 + 0.7) + 0.3 = 1 + (0.7 + 0.3)
b (−9 · 8) · 34 =−9⎛ ⎝ 8·34 ⎞
⎠
a
a
b
a
11
25
b
a
(4 + 0.6) + 0.4 = 4 + (0.6 + 0.4)
b (−2 · 12) · 56 =−2⎛ ⎞
⎝12 · 56 ⎠
a
22
15
b
b
5
49
11
9
2
5 p +4
3
7 u +9
a
b
a
a
b
b
a
a
b
b

a
b
a −18 b − 7 9
c −1.2
a
18 1 b − 5 4
c 5 3
a b c
a
c
b
a
b
a −47 b −
13 7 c −8.4
a b c
a b c
a
b
a 1 5
b −7 c 10 3
a b c
a
c
b
a
b
a
b
a 5 1
b 319
100
—
0.45 3.59
1.919293…
a −12
1
b 9279
1000
a
b
0.75, 0.223 –
1.39174…
a
b
a
b
c

⎛ ⎞
⎝13 · 23 ⎝ 2 3 ·18⎞ ⎠
a b
a b
a b
a b
5
12
25
7
65
23
13
12
23
15
a + 6 5 b
a b c
d e f
1
3 u +3 4
5 m +4
a b
a 9 2
b 9 2
a
b
a
a
b
b
a
b
a
b
1
14 −
19 1 13
8
− 12 5
5
2
a b
c

−5
1
a
b
18
10
a 4 b −5, − 4, 4
c
−5, − 2 1 4 , − 4, 0.25 — , 13
5
, 4
—
0.16 , 1.95
a
b
d
e
−5, − 2 1 4 , − 4, 0.25 — , 13
5
, 4
a
b
1
2 (22y) =⎛ ⎝ 1 2 ·22⎞ ⎠ y
11
15
a b
a b
7
15 − 1 5
− 9 4
144 = 12 therefore rational.

a − 2 5
b 5 2
23
15
28 x −5
1
− 8 3
−25
4y 4 −21
3 12 478
p = 7 6 q = 2 3
x =− 4 7
j = 5 3 k = 5 2
x = 1 2

= 1 2
x = 7 3
q =− 1 4 y = 27
20
f − 1 3 = 12 1 ; f = 12
5
d = 13
24
q = 5 2
t = 5 2
x =− 3 4
k = 3 2
x = 1 2
a = 3 4
w = 9 4
x =− 5 2

a = 4 3
c = 12
11
f = 10 3
a = 14 3
a
b
p = 2 3
2x −6 36 2
3.5 20 + 5n 6
x =2 60 10
w −3 3h 6
25
50.24in. 200.96sq.in 0.196
125 32
77
2
35miles
1gallon
12 $200
a
b
a
b

a b
a b
a b
c d c d
e f
e f
a
b
a
b
a
b
a b
a
b
a b
a
a
b
a
b
a
b
a
b
b
a
a
b
a
b
a
a
b
b
b
a
b
a
a
a
b
a
b
b
a r =45 b r = d t
b

a r =65
b r = d t
a t = 3 years b t = I
Pr
a y =−1
b y =
y = 1−5x
8
18 − 5x
2
a h =20b h = 2A b
a r = 0.12 = 12%
b r = I
Pt
a b =4
a y =1
b b = 2A h
b y =
y = 9−4x
7
10 − 3x
4
a
b
a
b
a
b
a
a
b
b
a
a
b
b

a
b
a
b
a
b
a
b
a
a
b
a
b
a
b
b
a
a
b
a
b
a
b
a
b
b
a
a
b
a
b
a
b
a
b
a
b
b
a b
c
a t =5 b t = d r
a t = 8.5 b t = d r a r =68 b r = d t
a r =64
b r = d t
a P = $19,571.43
b P = I rt
a b =14
a t = 6 years
b b = 2A h
b t = I
Pr
a h =30
a y =2
b h = 2A b
b y =
12 − 2x
3

a
L =
b
V
WH
a
L = P −2W
2
b
y = 7−4x
3
d = C π
a b
a
b
a
b
a b
a b
a
b
a
b
a
b
a
b
a
b
a t = 6.2
a b =26
b b = 2A h
a P = $6000 b P = I
(r ⋅ t)
a y =4
b t = d r
b y =
W = P −2L
2
20 − 6x
5
a height = 52
b h = 2A b

11x 3n +9 100
9
16
−8 2x +6
−44 + 33a x 2 +16x +63 1
3
32m 15
x
y
8 2⋅2⋅2⋅7
−18a −33 4x 4 +12x 3 −4x 2
a
c
e
b
d
a b c d
e
a b
c d
c
a b c d
e
a
b
a
a
a
a
a
b
a 25
64
b 0.4489
a
b
b
b
a 8
125
b 0.016129
a b a b

a b a b
a 1 x 7 b 1
12 10 a 1 m 9 b 1 7 6 a b
8 b 1 z 6
a 1 p 8 b w 4 a b
a
b
a
a
b
a 49
81
b y3
512
c p6
q 6
b
a 1 64
b − 125
m 3
1
1
d
b 10
m 15
n 40
64p 12
343q 15
t 20
u 14
1
y 11
25b 2
81c 6
c r4
s 4
−12y
x 2
9
a 5 2a 6
3b 2
− 3q6
5p 8
4y 2
7x 4
a 1 25
b − 1 25
a 2 b 1
256
a 8 p b 8p 1 c −
8p
1
a a
5 b 1 b 4 c 1
c 15
5
8m 5 n 3
a 1 8
b 1
100
a 1 4
b − 1 4
a 1 9
b 1
10,000
a 2 b 1 18
1
y 7 1
z 8
a 11 q b 11q 1 c −
11q
1
a x
4 b 1 y 5 c 1 z 9
1
p 3 q 3 1
r 2 s 8
− 12v5
30d 3
1
u
c 8
x 4
1
y 4 64
a 8 8
c 12

a b c
a b c
1
4
−
27
8
8
27 x6 y 3
1
2 x3 y 3
a b
c

1
y 16 1
10 12
a
1
a 8
b
a
m 3
216
x 10
y 10
b
a b
243
32
5
6x 6 6r 2
s 8
− 4y6
5x 2
a z
b 1 z
a 2
9b 2
1
y 2
1
x 18
1
x 10 16r 12
625s 4
1
r 3
−18
x 6
n 6
8
4z
3x 2 y 3 10n 2
m 4
a 18n
10 b 12n
10
c 45n
20 d 5
a q 8 b q 15
6y 6 15c 6
3x 2 −yz 2
1
64
1
32
1
10
4
9
3
5
3
4
a 1
4096
b − 1
a 10 3
b 1 30
4096
1
p 3 1
m 5
a 10
k
b
10k 1 c − 1
1
z 8
10k
1
m
a 1 64
b − 1 64
a 3 25
b 1
225
a 3 q b 3q 1 c −
3q
1
1
q 5
1 x

u 2
v 2
− 21q5
p 3
1
1
a 2 b 4 p 5 q 7
32
a 3 b
1
1
x 4 p 6
9
q 10 n 18
64
1
r 3
1
q 100
a a
1
n 9
9
125m 3
25
n 3
1
x 40
− 3 y 4 4a 5
1
n
b 2
4m 9
n 4

1
5
36
16
1
u 3 v 5
a
b
4r
s 6
1
8y 3
2 16 5
−1
y = 20 − 5x
2
−9
−4 y-axis
− 1 2
0, undefined −5, − 5, 5
a
b
a
b

a c
b c b a b

y =3x −1
x y (x, y)
0 −1 (0, −1)
−1 −4 (−1, −4)
2 5 (2, 5)
y =6x +1
x y (x, y)
0 1 (0, 1)
1 7 (1, 7)
−2 −11 (−2, −11)
2x −5y =20
x y (x, y)
0 −4 (0, −4)
10 0 (10, 0)
−5 −6 (−5, −6)
3x −4y =12
x y (x, y)
0 −3 (0, −3)
4 0 (4, 0)
−4 −6 (−4, −6)
a b
a b
a b
a b

(4, 0) (0, 12) (8, 0) (0, 2)

a
b
c
d
a
b
c
d
− 4 3
4
3
1
4 − 2 3

2
5
3
4 − 4 3 − 3 5
5
4
3
2
7
12
5
12 − 1 36 − 1 48

a b a c
b c a b
x y (x, y)
−1 −6 (−1, −6)
0 −4 (0, −4)
2 0 (2, 0)

x y (x, y)
−2 7 (−2, 7)
0 5 (0, 5)
3 2 (3, 2)
x y (x, y)
−2 1 (−2, 1)
0 −2 (0, −2)
2 −5 (2, −5)
a b
a b
a b
a
b
a b
a b
a b
a b
a b

a b
1
4

− 3 2
2
5
5
4 − 1 3 − 3 4
3
4 − 5 2 − 2 3
1
4
5
2
3
4 − 5 2 − 8 5
7
3

1
3 a 3 50 b
rise = 3; run = 50
a
b
a b c a
a b c
d
a b c
d
bc
x y (x, y)
0 −1 (0, −1)
1 3 (1, 3)
−2 −9 (−2, −9)

x y (x, y)
5 0 (5, 0)
1 2 (1, 2)
−1 3 (−1, 3)
a b
a b
a b
a b
4
3

− 2 3
− 1 2
1
2
− 5 2
x y (x, y)
0 8 (0, 8)
2 0 (2, 0)
3 −4 (3, −4)
1
50