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$100$22.00
x 3
x + 22 = 100
$100
$100 $22
x = 100− 22
(100− 22)
22
x = 100− 22
x + 22 = 100− 22 + 22 x + 22 = 100
x− 3 = 9
x x
x 3 3
x− 3 = 9
x− 3 + 3 = 9 + 3 + 3 − 3 9 + 3
x = 12
x− 3 = 11
x 3
x− 3 = 11
+3 = +3
x = 14
x
z − 9.7 = −1.026
z
z − 9.7 = −1.026
+9.7 = +9.7
zb = 8.674
x + 6 = 26
x
x + 6 = 26
−6 = −6
x = 20
x + 20 = −11
x 20
x + 20 = −11
−20 = −20
x = −31
x + 47 = 9
5
x 47
x +4
7=
9
5
−4
7= −
4
7
x =9
5−
4
7
x
5 7 5 · 7 = 35
x =9
5−
4
7
x =7 · 9
35−
4 · 5
35
x =63− 20
35
x =43
35
$1.508× 12.00.
x
18 · x = 1.5
x
8
8
(
1
8· x
)
= 8(1.5)
x = 12
x
9x5 = 5
9x5
95 · x x 9
559
5
9
(
9x
5
)
=5
9· 5
x =25
9
0.25x = 5.250.25 0.25 4
4(0.25x) = 4(5.25)
x = 21
x
$3.25$3.25 5 325 5
5x = 3.25 5 5
5x
5=
3.25
5
x = 0.65
7x = 511 7
x =5
7 · 11
x =5
77
1.375x = 1.2 1.375
x =1.2
1.375x = 0.872
0.872727272 . . .
2017 45x
x + 45 = 2017
−45 = −45
x = 1972
1972.
x 5× 1 lb = 5lb
2x = 5 2
2x
2=
5
2
x = 2.5x 2.5
1911
19 19x
x + 11 = 19
−11 = −11
x = 8
8
2004 53 12 12
2002
53.5 12x
53.5x = 12 53.5
x =12
53.5= 0.224 60
0.224 13.5
0.1
12x y
53.5
12y = 53.5 12
y =53.5
12= 4.458
4.5
53.5z
x + 3 = 53.5
−3 = −3
x = 50.5
50 12 12
x
x + 11 = 7
x− 1.1 = 3.2
7x = 21
4x = 15x12 = 2
3
x + 52 = 2
3
x− 56 = 3
8
0.01x = 11
q − 13 = −13
z + 1.1 = 3.0001
21 = 3
t + 12 = 1
3
7f11 = 7
1134 = −1
2 · y
6r = 38
9b16 = 3
8
10 25
n
w
r
12
$8.50 $1.25
(u)
(c)
(w)
x = −4
x = 4.3
x = 3
x = 0.25
x = 1.6
x = − 116
x = 2924
x = 1100
q = 0
z = 1.9001
s = 1/7
t = −16
f = 1
y = −1.5
r = 116
b = 23
n + 10 = 25 n = 15
8w = 10 w = 1.25
r · w = 15 1.25r = 15 r = 12
u = 3(8.5 + 1.25)
12v = u
w = u− (8.5 + 1.25)
29
x x
29
3x + 2 = 29
29x x
x
x
3x + 2 = 29
−2 = −2 2 :
3x = 27
3x
3=
27
33
x = 9
6(x + 4) = 12
x
6(x + 4) = 12 6.
6(x + 4)
6=
12
6
x + A4 = 2 4 .
− A4− 4
x = −2
x = −2
x−35 = 7
(
x−3
5
)
= 7
5
(
x− 3
5
)
= 5 · 7 5
x− 3 = 35
+ 3 = +3 3
x = 38
x = 38
59 (x + 1) = 2
7
9
5·5
9(x + 1) =
9
5·2
7
x + 1 =18
351
(
1 =35
35
)
.
x =18
35−
35
35
x =18− 35
35
x = −17
35
x
x−
17x, 12x,−1.2x 17x9
3y, 19y, y99
xy, 6xy 0.0001xy
3x 2y
12xy 2x
0.001x 0.001
3x + 4x = (3 + 4)x = 7x
0.03xy − 0.01xy = (0.03− 0.01)xy = 0.02xy
−y + 16y − 5y = (−1 + 16− 5)y = 10y
5z + 2z − 7z = (5 + 2− 7)z = 0z = 0
(x + 5)− (2x− 3) = 6x −2x
x −2x
(x− 2x) + (5− (−3)) = 6 .
− x + 8 = 6
−8 = −8 8
− x = −2 − 1
x = 2
x2 −
x3 = 6
3x
6−
2x
6= 6 .
x
6= 6 .
x = 36
$65 $759 : 30 $196.25
x
x x
$65 65
$75 +75x
= 65 + 75x
$196.25
196.25 = 65 + 75x
x
196.25 = 65 + 75x
− 65 = −65 65 :
131.25 = 75x 75
131.25
75= x = 1.75 .
75h m
h =
(m + 75) 75.
1
3(m + 75) .
h = 13 (m + 75)
m = 24 (2 =24 ) h
h =1
3(24 + 75) .
h =1
3(99) .
h = 33
33h = 36
m
36 =1
3(m + 75) 3.
108 = m + 75 75 .
33 = m
33
32 1.8
5025−40
f c
(F − 32) 32.
F − 32
1.81.8.
C =F − 32
1.8.
50 F = 50
C =50− 32
1.8.
C =18
1.8.
C = 10 50 1025 C = 25
25 =F − 32
1.81.8
45 = F − 32
+32 = +32 32 .
77 = F
25 77−40 C = −40
−40 =F − 32
1.81.8.
−72 = F − 32
+32 = +32 32 .
−40 = F
−40 −40
1.3x− 0.7x = 12
6x− 1.3 = 3.2
5x− (3x + 2) = 1
4(x + 3) = 1
5q − 7 = 23
35x + 5
2 = 23
s− 3s8 = 5
6
0.1y + 11 = 0
5q−712 = 2
3
5(q−7)12 = 2
3
33t− 99 = 0
5p− 2 = 32
$10 $0.75$2.35
$150$3.00 $300
x = 20
x = 0.75
x = 1.5
x = −2.75
q = 2315
= −5518
s = 43
y = −110
q = 3
q = 435
t = 3
p = 345
0.75x + 2.35 = 10 x = 10.2
3x + 150 = 300 x = 50
3x+43 − 5x = 6
3x + 4
3−
15x
3= 6 .
3x + 4− 15x
3= 6 .
4− 12x
3= 6 3.
4− 12x = 18 4 .
−12x = 14 − 12
−12
−12x = −
14
12
x = − 76
17(3x + 4) = 7x
17(3x + 4) = 7 17.
51x + 68 = 7
−68 = −68 68 .
51x = −61 51.
x = −61
51
4(3x− 4)− 7(2x + 3) = 3
12x− 16− 14x− 21 = 3 (12x − 14x).
(12x− 14x) + (−16− 21) = 3 .
−2x− 37 = 3
+37 + 37 37 .
−2x = 40
−2x
−2=
40
−2− 2.
x = −20
x
0.1(3.2 + 2x) +1
2
(
3−x
5
)
= 0
0.1(3.2 + 2x) + 0.5(3− 0.2x) = 0
0.32 + 0.2x + 1.5− 0.1x = 0
(0.2x− 0.1x) + (0.32 + 1.5) = 0
0.1x + 1.82 = 0 1.82
−1.82 − 1.82
0.1x = −1.82 − 0.1
0.1x
0.1=−1.82
0.1
x = 18.2
8.5
$150$175
x
8.5 91.50.915x
(0.915x− 150)
$150
0.915x− 150
7
7$175
0.915x− 150
7= 175 7.
0.915x− 150 = 1225 150 .
0.915x = 1375 0.915.
0.915x
0.915=
1373
0.915= 1502.7322 · · · .
$175 $1, 502.73
1200
1200x
(x + 12) 16 .
16(x + 12) 16 .
16(x + 12) = 1200 x 16.
x + 12 = 75 12 .
x = 63
63
I
V
V = I ·R R Ω
xΩ (5x+20)Ω120 2.5
V = 120 I = 2.5 R = 5x + 20 V = I ·R
120 = 2.5(5x + 20) 2.5.
120 = 12.5x + 50 50 .
−50 = −50
70 = 12.5x 12.5.
70
12.5=
12.5x
12.55.6Ω = x
5.6Ω
= ×
1093 1 30
x
93 = 2x− 10× 1.5 1.5.
62 = 2x− 10
+10 = +10 10 .
72 = 2x
72
2=
2x
22.
36 = x
36
93 1 30 y x
93 = y · 1.5
93
1.5=
1.5y
1.51.5.
y = 62
10
y = 2x− 10
y x
62 = (2x−ZZ10)
+ 10 +ZZ10 10 .
72 = 2x
72 = 2x 2.
72
2=
2x
2x = 36 .
36 .
v
v = 331 + 0.6T T v
250
2.46
= (331 + 0.6T )×
= (2.46− 1)
= 2× 250
2(250) = (331 + 0.6T ) · (2.46− 1) .
500
1.46=
1.46(331 + 0.6T )
1.461.46.
342.47− 331 = 331 + 0.6T − 331 .
11.47
0.6=
0.6T
0.60.6.
19.1 = T
19.1
3(x + 2) = 3x + 6
3(x− 1)− 2(x + 3) = 0
7(w + 20)− w = 5
9(x− 2) = 3x + 3
2(
5a− 13
)
= 27
29
(
i + 23
)
= 25
4(
v + 14
)
= 352
s−411 = 2
5
p16 −
2p3 = 1
9
200 40 250
xΩ(3x + 4)Ω 12
1.2
10$500 20
5010 94
x = 9
w = −22.5
x = 3.5
a = 221
i = 1715
v = 338
s = 425
p = −1687
2(250) = 200 + 40x x = 7.5 → 7
2Ω
$1, 176.50
35
x
x + 4 = 4x + 3
4 4 3
x
x
x + 4 = 4x + 3 3 .
− 3 = −3
x + 1 = 4x
− x = −x .
1 = 3x
1
3= 3x
33.
x =1
3
x + 1 = 4x
1 = 3x
x 1 (x + 2)
3(x + 2) = x + 3(5) 3.
3x + 6 = Ax + 5
− x = −Ax .
2x + A6 = 15
− A6 = −6 6 .
2x = 9 2.
x = 4.5
4.5
3x + 4 = 5x
x
x
A3x + 4 = 5x 3x .
− 3Ax − 3x
4 = 2x
2
4
2=
2x
2
x = 2
9x = 4− 5x
x
9x = 4−ZZ5x
+ 5x + 5Ax 5x .
14x = 4
14x = 4 14.
14x
14=
4
14
x =2
7
x =2
7
3x + 2 = 5x3
x
3x + 2 =5x
33(3x + 2) = 5x 3.
3.
A9x + 6 = 5x
− 9Ax − 9x 9x :
− 4.
6
−4=−4x
−46
−4= x
−3
2= x
x = −1.5
7x + 2 = 5x−36
7x + 2 =5x− 3
6
6(7x + 2) =5x− 3
6· 6 6.
6(7x + 2) = 5x− 3 6.
42x + 12 = 5x− 3 5x :
− 5x −5x
37x +12 = −3 12 .
−12 − 12
37x = −15
37x
x=−15
37− 37.
x = −15
37
x
14x
(x + 3)= 7
(14x) (x + 3) x
14x
(x + 3)(x + 3) = 7(x + 3) (x + 3).
14x = 7(x + 3) 7.
14x = ZZ7x + 21
− 7x = −ZZ7x 7x .
7x = 21
7x
7=
21
77
x = 3
I
V
V = I ·R
R R =r1 + r2 + r3 + r4 + . . .
4.8 15Ω50Ω
2.0
Ωx V
15Ω 4.8V = I ·R
V = 4.8(x + 15)
50Ω 2.0
V = 2(x + 50)
V V
4.8(x + 15) = 2(x + 50) .
4.8x + 72 = ZZ2x + 100
− 2x −ZZ2x 2x .
2.8x +ZZ72 = 100
− 72 −ZZ72 72 .
2.8x = 28 2.8.
2.8x
2.8=
28
2.8x = 10
10Ω
3(x− 1) = 2(x + 3)
7(x + 20) = x + 5
9(x− 2) = 3x + 3
2(
a− 13
)
= 25
(
a + 23
)
27
(
t + 23
)
= 15
(
t− 23
)
17
(
v + 14
)
= 2(
3v2 −
52
)
y−411 = 2
5 ·2y+1
3
z16 = 2(3z+1)
9
q16 + q
6 = (3q+1)9 + 3
2
$6$4 $2
2.310Ω 1.9
20Ω 70Ω
1.5 15Ω 3.0
18Ω 150Ω
1.5Ω 3.67 12Ω 0.2
x = 9
x = −22.5
x = 3.5
a = 712
t = −349
v = 14180
y = −8229
z = −3287
q = − 23215
9
$7
6.55Ω
= 25Ω
= 30Ω
= 94Ω
= 1.213Ω
2510 5
4 + 6
15 + 5=
10
20
10
20= 2.5
2.2.5=
1
2
(4× 25) + (6× 10) = 160 (15× 5) + (5× 1) = 80
160
80=
2
1
$10.00$6.50
= 10.00− 3.50
new price
used price=
$10.00
$6.50$ .
10
6.50=
1000
650=
20
13
$3.502013
48 36
48 feet
36 feet=
48
36=
4
3
30 14
14 feet30 inches
14 feet
30 inches=
14× 12 inches
30 inches=
168
30=
28
5
100 , 3.2
Ratio =100 miles
3.2 gallons
3.2
Unit Rate(100
3.2 ) miles
( 3.23.2 ) gallons
=31.25 miles
1 gallon
31.251 31.25
10
15=
6
9
23
x
60 $1.2
=$1, 200, 000
60
x 250250
=$x
250
$1,200,00060 = $x
2501,200,000
60 = x250
1, 200, 000
60=
x
250.
20, 000 =x
250250.
5, 000, 000 = x
250 5
2501000
350 (0.35 )
=250
1000:
1
4
x
0.35
Ratio =x kg
0.35 liters
1 kg4 liters
= x kg0.35 liters
14 = x
0.35
x
1
4=
x
0.350.35.
0.35 ·1
4=
x
0.35· 0.35
x = 0.0875
0.0875 87.5
16
4=
20
5
45
4 · 5 ·16
4= 4 · 5 ·
20
55 · 16 = 4 · 20
5 · 16 = 4 · 20
x
4
3=
9
x
x · 4 = 9 · 3
4x
4=
27
44.
x = 6.75
x
0.5
3=
56
x
x · 0.5 = 56 · 3
0.5x
0.5=
168
0.50.5.
x = 336
x
15 207
= ×
=15
20=
1513
15miles13
=x miles
7 hours
7 · 15 =1
3· x 3.
3 · 7 · 15 = 3 ·1
3· x
315 = x
315 7
1 1.53
= ×x distance
time
1
1.5=
x
3.
3(1)
1.5=
1.5x
1.51.5
2 = x
2 3
65250000 65 65
1 50 (x)65
1
50=
250000
x:
1 · x = 250000 · 50
x = 12, 500, 000
12.5 65
23
321.4
x13
11 · 3 = 5 · x
$150 $3
150 175
200 1
10 2
54 12
5000 250 in2
20 80
180 6
12 4
18 15
136 = 5
x
1.257 = 3.6
x
619 = x
11
1x
= 0.015
3004 = x
99
2.759 = x
( 2
9)
1.34 = x
1.3
0.11.01 = 1.9
x
100 $908 250
29867
22067
4448 4800
501
67
103
57
4.5
20 in2
0.25
30
3
1.2
x = 3013
x = 20.16
x = 6619
x = 500
x = 7425
x = 11162
x = 0.4225
x = 1001919
$2270
5960
310 30
1 : 20(
120
)
2 : 3(
23
)
1 : 1000(
11000
)
1 : 10001 1 1000
1 : 1
1 : 100, 000 1 18.8
.
.=
1
100, 000
8.8
.(x)=
1
100, 000.
880000 = x100 = 1 .
x = 8800 1000 = 1 .
8.81 = 1
A
B 183
183
=
a2 + b2 = c2
2= 82 + 142
2= 64 + 196
2= 260
=√
260 = 16.12
=16.12
.
=16.12
1831 = 100 .
=16.12
1830016.12.
≈1
1135.23:
1 : 1100
=
( ) = ( )× ( )
( ) = = ( ) ·
(
1)
8831 : 500
1 : 500 883500 = 1.766
12(1.766) = 21.192
22
880 6951 100
(8.5 × 11 )
( ) = ( )× ( )
= 1 100 =
(
1
100
)
= 880 ×
(
1
100
)
= 8.8
= 695 ×
(
1
100
)
= 6.95
8.8 × 6.95 8.5 × 11
500 3.03 = 500
=
(
3
500
)
!
5.6
=5.6(
3500ft
) = 5.6 ·
(
500
3
)
93010
3.45.5
(3.4 + 5.5) = 8.9
= ≈ 8.9
(
500
3
)
1480
= 6.2
= ≈ 6.2 ·
(
500
3
)
1030
500
(2.75 )(1.8 )
500
2.75=
x
1.8
1.8 ·500
2.75=
x
1.8· 1.8
327.3 = x
327.3
200 3.11.9
200
3.1=
x
1.9
1.9 ·200
3.1=
x
1.9· 1.9
122.6 = x
122.6
=
1 4
22 25
18 13
50
20
6003 1 5 1
4
3
7
12 12
4 12
= 600
= 250
0.3
0.12
= 21
= 10
= 12
= 5 12
= 4
1250
100
(
50100
)
50(
501000
)
=(
0.5100
)
0.5(
10100
)
10(
125
)
=(
4100
)
4(
99100
)
99(
35
)
=(
60100
)
60(
125100
)
12.5(
110,000
)
=(
0.01100
)
0.01
10, 100, 1000, 10000100
10 0.1 0.05 .0005 0 099 0.99 0.25 .0025 100 1
0.2
100 0.2
0.2 = 0.2× 100×
(
1
100
)
100×
(
1
100
)
= 1
0.2 = 20×
(
1
100
)
0.2 =
(
20
100
)
= 20
0.2 = 20
100× 1100
0.07 = 0.07× 100×
(
1
100
)
0.07 = 7×
(
1
100
)
0.07 =
(
7
100
)
= 7
0.07 = 7
100
35
35 =
(
35
100
)
= 0.35
0.5
0.5 =
(
0.5
100
)
=
(
5
1000
)
= 0.005
x100 x x
35
x x100
(
3
5
)
=x
100.
5x = 100 · 3 5 .
5x = 300
x =300
5= 60
(
3
5
)
= 60
1340
x100 x
13
40=
x
10040x = 1300
x =1300
40= 32.5
(
13
40
)
= 32.5
28
28
(
28
100
)
=
(
2 · 2 · 7
5 · 5 · 2 · 2·
)
(
2.2.7
2.2.5.5
)
=7
25
28 =
(
7
25
)
× = R
25 $80$80 R
25 25100 0.25
0.25· 20
25 $80 $20
17 $93$93 17 0.17
0.17 · 93 = 15.81
17 $93 $15.81
$90 $160($90)
($160)
× 160 = 90 160
=
(
90
160
)
= 0.5625 = 0.5625
(
100
100
)
=56.25
100
$90 56.25 160
$50 15$50
15 0.15 x
0.15x = 50 0.15.
x =50
0.15≈ 333.33
$50 15 $333.33
20 35
=
(
−)
× 100
Or
100=
( )
500 20
+20 500
=
(
−)
× 100
20 =
(
− 500
500
)
× 100 100 .
= .
0.2 =x− 500
500500.
100 = x− 500 500 .
600 = x
600
$150 30
x
=
(
−)
× 100
(
x− 150
150
)
· 100 = −30 100 .
(
x− 150
150
)
=30
100= −0.3 150.
x− 150 = 150(−0.3) = −45 150 .
x = −45 + 150
$105y
100=
( )
−30
100=
y
150150.
150(−0.3) = y
y = −45
−45($) $150− $45 = $105
100
30 2025
25
$1000
$1000
$300 (30 1000 = 0.30 · 1000 = 300)
$1300
$1000 $1300 $300
$1300
20 0.20× $1300 = $260
25 0.25× $1300 = $325
20 $1300− 104025 $1300− 97520 $40
25 $975 $25x
= 0.3x
= 1.3x
20 = 0.80× 1.3x = 1.04x
25 = 0.75× 1.3x = 0.975x
20 4 252.5
2004 112071 87846
11754 6899
× =
112071 87846(112071− 87846) = 24225
× 112071 = 24225 112071.
= 0.216 100 :
= 21.6
21.6 2004
= 112071 = 11754
× 112071 = 11754 112071.
= 0.105 100 :
= 10.5
10.5 2004
24225(24225− 11754− 6899) = 5572
5572
× 24225 = 5572 24225
= 0.230 100 .
= 23
23 2004
1995 18136000 827025 1526832005 19254630 85839 491829
1995 2005 †
=
(
−)
× 100
2005 1995
=
(
19, 254, 630− 18, 136, 000
18, 136, 000
)
× 100
=
(
1, 118, 630
18, 136, 000
)
× 100
= 0.0617× 100
= 6.17
6.17
=
(
491, 829− 827, 025
827, 025
)
× 100
=
(
−335, 196
827, 025
)
× 100
= −0.4053× 100
= −40.53
40.53
=
(
85, 839− 152, 683
152, 683
)
× 100
=
(
−66, 844
152, 683
)
× 100
= −0.4377× 100
= −43.77
43.77 . †
100 13 = 13100
× = R
= − × 100
0.011
0.001
0.91
1.75
20
16524671171397
11
65
16
12.5
87.5
30 90
16.7 199
11.5 10.01
y 3x
35 $195
$9.50 12
40 25060
1.1
0.1
91
175
2000
16.67
20.83
85.71
157.14
−13.40
1110013204
25)
1878
27
33.233
−1.151153xy100
$300
$10.64
722
= = x
= 2×
= 722
= 2x
= x× 2x = 2x2
2x2 = 722
2x2 = 722 2
x2 = 361 .
x =√
361 = 19
2x = 2× 19 = 38
19 38
38 19
2× 19 = 38 − .
38 19 722
38× 19 = 722 −
2000500
20000
=2000
1
= 4
=500
1
= = x
= 20000
= 4 · 2000 feet1 minute
= 8000 feet.
8000(20000− 8000) = 12000
= ×
time = = ·
(
1)
x = 12, 000 ·
(
1
500
)
x = 24
= x + 4
20000 28
4 2000
4× 2000 = 8000 8000 .
24 500
24× 500 = 12000 8000 .
(8000 + 12000) = 20000
time = distancespeed
340 150010
= 10, 000
=340
1
=1500
1
= x
= y
= x− y
x = 10, 000 · 1 second340 meters
y = 10, 000 · 1 second15000 meters
x, y
x = 29.41
y = 6.67
= x− y = (29.41− 6.67)
22.7 .
29.41
= × = 340× 29.41 = 9999
6.67
= times = 1500× 6.67 = 10005
10000
15
16 $7.50
= 15 = 0.15
=1
12
=1
3= 16
= $7.50
= x
= 16× $7.50 = $120= 1− 0.15 = 0.85
= 0.85(
1− 112
)
= 0.85(
1112
)
≈ 0.85 · 0.91667 ≈ 0.779167= 1
3 · 0.779167 ≈ 0.25972
0.25972 · $120 = $31.1664
$31.17
$31.17 3× $31.17 = $93.51$93.51 $93.51 · 12
11 = $102.01$102.01 $102.01 · 100
85 = $120.01
$120.01 $7.50 $102.01$7.50 = 16.002
30
500 1.75
2010
$36
20340
30 $1000
2.3 5.7cm
TM 25
13
6860
$50
0.06
$565
85.45 2
0.293 /2
