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Beginning and Intermediate Algebra
Student Solutions ManualComplete worked solutions to odd problems
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Beginning Algebra Student Solutions Manual by Tyler Wallace is licensed under a Creative Commons Attribution 3.0 Unported License. Permissions beyond the scope of this license may be available at http://wallace.ccfaculty.org/book/book.html.
1
Beginning and Intermediate Algebra
Student Solutions Manual Table of Contents
Chapter 0: Arithmetic .....................................3
Chapter 1: Solving Linear Equations ..............11
Chapter 2: Graphing Linear Equations ...........40
Chapter 3: Inequalities ..................................55
Chapter 4: Systems of Equations ...................62
Chapter 5: Polynomials .................................88
Chapter 6: Factoring ......................................99
Chapter 7: Rational Expressions ..................109
Chapter 8: Radical Expressions ....................130
Chapter 9: Quadratics .................................144
Chapter 10: Functions .................................189
2
Chapter 0: Arithmetic
3
0.1
1) 1−3 1+(−3 )=−2
3) (−6 )−(−8) −6+8=2
5) (−3 )−3 (−3 )+(−3 )=−6
7) 3−(−5) 3+5=8
9) (−7 )−(−5) −7+5=−2
11) 3−(−1) 3+1=4
13) 6−3 6−3=3
15) (−5 )+3=2
17) 2−3 2+(−3 )=−1
19) (−8 )−(−5)
−8+5=−3
21) (−2 )+(−5 )=−7
23) 5−¿ 5+6=11
25) −6+3=−3
27) 4−7 4+(−7 )=−3
29) (−7 )+7=0
31) (4 ) (−1 )=−4
33) (10 ) (−8 )=−80
35) (−4 ) (−2 )=8
37) (−7 ) (8 )=−56
39) (9 ) (−4 )=−36
41) (−5 ) (2 )=−10
43) (−5 ) (4 )=−20
45) (4 ) (−6 )=−24
47)−49−7
=7
49)−2−1
=2
51)2010
=2
53)−35−5
=7
55)−8−2
=4
57)−162
=−8
59)60
−10=−6
0.2
1)4212
=216
=72
3)3525
=75
5)5436
=2718
=96=32
7)4536
=1512
=54
9)2718
=96=32
11)4016
=208
=104
=52
13)6318
=216
=72
15)8060
=4030
=2015
=43
4
17)7260
=3630
=1815
=65
19)3624
=1812
=96=32
21)(9 )1∙( 89 )=81=8
23)(2 )1∙−29=−49
25) (−2 )( 138 )=−134
27) (−65 )(−118 )=3320
29)(8 )1 (12 )=41=4
31) ( 23 )( 34 )=12
33)(2 )1 ( 32 )=31=3
35) ( 12 )(−75 )=−710
37)−21
÷ 74
(−21 )( 47 )=−87
39)−19
÷−12
(+19 )( 2+1 )=29
41)−32
÷ 137
(−32 )( 713 )=−2126
43)−11
÷ 23
(−11 )( 32 )=−32
45)89÷ 15
( 89 )( 51 )=409
47)−97
÷ 15
(−97 )(51 )=−457
49)−29
÷−32
(+29 )( 2+3 )= 427
51)110
÷ 32
( 110 )( 23 )= 115
53)13+(−43 )=−3
3=−1
55)37−17=27
57)116
+ 76=186
=3
59) ( 44 )35 + 54 ( 44 )
1220
+ 2520
=3720
61) ( 44 )25 + 54 ( 55 )
820
+ 2520
=3320
63) ( 77 ) 98+(−27 )( 88 ) 6356
+−1656
= 4756
65) ( 33 ) 11 +(−13 )
33+−13
=23
67) (−12 )+ 32=22=1
69) ( 44 )15 + 34 ( 55 )
420
+ 1520
=1920
71) ( 88 )−57 −158 ( 77 )
−4056
+−10556
=−14556
73) ( 77 ) 61−87
427
−87=347
75) ( 44 ) 32−158
5
-1 4
1 1
-3 4
4 1
1 1
1 1 1 2
15
128
−158
=−38
77) ( 33 )(−158 )+ 53 (88 )
−4524
+ 4024
=−524
79) ( 66 )−11 −(−16 )
−66
+ 16=−56
81)53−(−13 )
53+ 13=63=2
0.3
1) −6 ∙4 (−1) −24 (−1) 24
3) 3+(8)÷|4| 3+8÷(4) 3+2 5
5) 8÷4 ∙2 2 ∙2 4
7) [−9−(2−5)]÷(−6) [−9− (−3 )]÷(−6) [(−6)]÷(−6) 1
9) −6+ (−3−3 )2÷∨3∨¿
−6+(6)2÷3 −6+36÷3 −6+12 6
11) 4−2|32−16| 4−2∨9−16∨¿ 4−2∨−7∨¿ 4−2(7)
4−14 −10
13) [−1− (−5 ) ]∨3+2∨¿ [4 ](5) 20
15) 2+4|7+22|
4 ∙2+5∙3 Numerator :2+4∨7+22∨¿
2+4∨7+4∨¿2+4∨11∨¿ 2+4(11) 2+44 46
Denominator :4 ∙2+5 ∙3 8+15 23
Fraction: 4623=2
6
17) [6 ∙2+2−(−6 ) ](−5+|(−186 )|) [12+2−(−6 ) ] (−5+|−3|) [14−(−6 ) ] (−5+ (3 ) ) [20](−2) −40
19)−13−2
2−(−1 )3+ (−6 )−[−1− (−3 ) ] Numerator :−13−2
−15 Denominator :2−(−1 )3+(−6 )−[−1−(−3 ) ]
2− (−1 )3+ (−6 )− [2 ]2−(−1)+ (−6 )−[2] 3+(−6 )−[2] −3−[2]
−5
Fraction:− 15−5
=3
21) 6 ∙−8−4+ (−4 )−[−4−(−3 ) ]
(42+32 )÷5Numerator :−8−4+(−4 )−[−4−(−3 ) ]
−8−4+(−4 )− [−1 ] −12+(−4 )−[−1] −16−[−1]
−15 Denominator : (42+32 )÷5
(16+9)÷5 (25)÷5 5
Fraction: 6∙−155=6 ∙−3=−18
7
23)23+4
−18−6+ (−4 )−[−5 (−1 ) (−5 ) ]
Numerator :23+4 8+4
12 Denominator :−18−6+(−4 )−[−5 (−1 ) (−5 ) ]
−18−6+(−4 )−[5 (−5 ) ] −18−6+(−4 )−[−25 ] −24+ (−4 )−[−25]
−28−[−25] −3
Fraction: 12−3=−4
25)5+32−24÷6∙2
[5+3 (22−5 ) ]+|22−5|2 Numerator :5+32−24÷6 ∙2
5+9−24÷6 ∙2 5+9−4 ∙2 5+9−8 14−8 6
Denominator : [5+3 (22−5 ) ]+|22−5|2
[5+3 (4−5 ) ]+|4−5|2
[5+3 (−1 ) ]+|−1|2
[5−3 ]+ (−1 )2 [2 ]+ (−1 )2
2+1 3
Fraction: 63=2
0.4
1) p+1+q−m ,use m=1 , p=3 , q=4 (3 )+1+(4 )−(1) 4+(4 )−(1) 8−(1)
7
3) p− pq6
,use p=6 , q=5
8
(6 )− (6 ) (5 )6
(6 )−(5) 1
5) c2−(a−1 ) , usea=3 , c=5
(5 )2− [ (3 )−1 ] (5 )2−[2] 25−[2] 23
7) 5 j+ kh2,use h=5 , j=4 , k=2
5 (4 )+ (2 ) (5 )2
5 (4 )+5 20+5 25
9)4−( p−m )
2+q ,use m−4 , p=6 , q=6
4−[ (6 )−(4 ) ]
2+(6)
4−[2 ]2
+(6)
22+6
1+6=7
11) m+n+m+ n2, use m=1, n=2
(1 )+(2 )+(1 )+(2)2
(1 )+(2 )+(1 )+1 3+ (1 )+1 4+1 5
13) q−p− (q−1−3 ) , use p=3 , q=6
(6 )−(3 )−[ (6 )−1−3] (6 )−(3 )−[5−3] (6 )−(3 )−[2]
3−[2] 1
15) y− [4− y−( z−x ) ] , use x=3 , y=1 , z=6 (1 )−{4−(1 )− [ (6 )−(3 ) ]} (1 )−{4−(1 )− [3 ] } (1 )−{3− [3 ]} (1 )−{0 }
1
17) k ×32−( j+k )−5 , use j=4 , k=5 (5 )32−[ (4 )+(5 ) ]−5 (5 )32−[9 ]−5 (5 )9−[9 ]−5 45−[9 ]−5 36−5 31
19) zx−(z−4+x6 ) ,use x=2 , z=6
(6 ) (2 )−[ (6 )− 4+(2 )6 ]
(6 ) (2 )−[ (6 )−66 ]
(6 ) (2 )− [ (6 )−1 ] (6 ) (2 )−[5] 12−[5] 7
21) r−9+10 r+1
23) n+n 2n
25) 8 v+7 v
9
15v
27) −7 x−2 x −9 x
29) k−2+7 k+5
31) x−10−6 x+1 −5 x−9
33) m−2m −m
35) 9n−1+n+4 10n+3
37) −8( x−4) −8 x+32
39) 8n(n+9) 8n2+72n
41) 7k (−k+6) −7 k2+42k
43) −6 (1+6 x) −6−36 x
45) 8m(5−m) 40m−8m2
47) −9 x (4−x )
−36 x+9 x2
49) −9b (b−10) −9b2+90b
51) −8n(5+10n) −40n−80n2
53) 9 (b+10 )+5b 9b+90+5b 14b+90
55) −3 x (1−4 x )−4 x2 −3 x+12x2−4 x2
8 x2−3 x
57) −4k2−8 k (8k+1) −4k2−64 k2−8k −68k2−8k
59) 1−7 (5+7 p) 1−35−49 p −49−49 p
61) −10−4 (n−5) −10−4n+20 −4n+10
63) 4 ( x+7 )+8(x+4 ) 4 x+28+8 x+32 12 x+60
65) −8 (n+6 )−8n (n+8) −8n−48−8n2−64 n −8n2−72n−48
67) 7 (7+3 v )+10(3−10 v) 49+21v+30−100 v −79v+79
69) 2n (−10n+5 )−7 (6−10n) −20n2+10n−42+70n −20n2+80n−42
71) 5 (1−6 k )+10(k−8) 5−30k+10k−80 −20k−75
10
73) (8n2−3n )−(5+4 n2 ) 8n2−3n−5−4 n2
4 n2−3n−5
75) (5 p−6 )+(1−p) 5 p−6+1−p 4 p−5
77) (2−4 v2 )+(3 v2+2 v) 2−4 v2+3 v2+2v −v2+2 v+2
79) (4−2k2 )+(8−2k2) 4−2k 2+8−2k2
−4k2+12
81) (x2−8 )+(2 x2−7) x2−8+2 x2−7 3 x2−15
11
Chapter 1: Solving Linear Equations
12
1.1
1) v+9=16 −9−9 v=7
3) x−11=−16 +11+11 x=−5
5) 30=a+20 −20−20 10=a
7) x−7=−26 +7+7 x=−19
9) 13=n−5 +5+5 18=n
11)340−17
=−17 x−17
−20=x
13) (12 ) (−9 )= n12
(12 )
−108=n
15)20v20
=−16020
v=−8
17)34020
=20n20
17=n
19)16 x16
=32016
x=20
21) −16+n=−13 +16+16
n=3
23) p−8=−21 +8+8 p=−13
25)18012
=12 x12
15=x
27)20b20
=−20020
b=−10
29) (14 ) r14
= 514
(14 )
r=5
31) −7=a+4 −4−4 −11=a
33) 10=x−4 +4+4 14=x
35)13a13
=−14313
a=−11
37) (20 ) p20
=−12(20)
p=−240
39) 9+m=−7 −9−9 m=−16
1.2
1) 5+ n4=4
−5−5
(4 ) n4=−1(4)
n=−4
3) 102=−7 r+4 −4−4
98−7
=(−7 r )/−7
−14=r
5) −8n+3=−77 −3−3
−8n−8
=−80−8
n=10
7)0
−6=−6v
−6 0=v
9) −8= x5−6
13
+6+6
(5 ) (−2 )= x5
(5 )
−10=x
11) 0=−7+ k2
+7+7
(2 )7= k2
(2 )
14=k
13) −12+3 x=0 +12+12
3x3
=123
x=4
15) 24=2n−8 +8+8
322
=2n2
16=n
17) 2=−12+2r +12+12
142
=2r2
7=r
19)b3+7=10
−7−7
(3 ) b3=3(3)
b=9
21) 152=8 n+64 −64−64
888
=8n8
11=n
23) −16=8a+64 −64−64
−808
=8a8
−10=a
25) 56+8 k=64 −56−56
8k8
=88
k=1
27) −2 x+4=22 −4−4
−2x−2
= 18−2
x=−9
29) −20=4 p+4 −4−4
−244
=4 p4
−6=p
31) −5=3+ n2
−3−3
(2 ) (−8 )=n2
(2 )
−16=n
33)r8−6=−5
+6+6
(8 ) r8=1(8)
r=8
35) −40=4n−32 +32+32
−84
=4 n4
−2=n
37) 87=3−7v −3−3
84−7
=−7v−7
−12=v
39) −x+1=−11 −1−1
−x−1
=−12−1
x=12
1.3
1) 2− (−3a−8 )=1 2+3a+8=1 10+3a=1
−10−10
3a3
=−93
14
a=−3
3) −5 (−4+2 v )=−50 20−10v=−50 −20−20
−10v−10
=−70−10
v=7
5) 66=6(6+5 x) 66=36+30 x −36−36
3030
=30 x30
1=x
7) 0=−8( p−5) 0=−8 p+40 −40−40
−40−8
=−8 p−8
5=p
9) −2+2 (8 x−7 )=−16 −2+16 x−14=−16 16 x−16=−16
+16+16
16 x16
= 016
x=0
11) −21 x+12=−6−3x +21 x+21x
12=−6+18 x+6+6
1818
=18 x18
1=x
13) −1−7m=−8m+7
+8m+8m −1+m=7 +1+1
m=8
15) 1−12 r=29−8 r +12 r+12 r 1=29+4 r −29−29
−284
=4 r4
−7=r
17) 20−7b=−12b+30 +12b+12b 20+5b=30 −20−20
5b5
=105
b=2
19) −32−24 v=34−2v +24 v+24 v
−32=34+22v−34−34
−6622
=22v22
−3=v
21) −2−5 (2−4m)=33+5m −2−10+20m=33+5m −12+20m=33+5m
−5m−5m −12+15m=33 +12+12
15m15
=4515
m=3
23) −4n+11=2 (1−8n )+3n −4n+11=2−16n+3n −4n+11=2−13n
15
+13n+13n 9n+11=2 −11−11
9n9
=−99
n=−1
25) −6 v−29=−4 v−5(v+1) −6 v−29=−4 v−5v−5 −6 v−29=−9v−5 +9 v+9v 3v−29=−5 +29+29
3v3
=243
v=8
27) 2 (4 x−4 )=−20−4 x 8 x−8=−20−4 x +4 x+4 x 12 x−8=−20 +8+8
12x12
=−1212
x=−1
29) −a−5 (8a−1 )=39−7a −a−40 a+5=39−7a −41a+5=39−7a +41a+41a
5=39+34a −39−39
−3434
=34a34
−1=a
31) −57=−(−p+1 )+2(6+8 p) −57=p−1+12+16 p −57=17 p+11 −11−11
−6817
=17 p17
−4=p33) −2 (m−2 )+7 (m−8 )=−67
−2m+4+7m−56=−67 5m−52=−67
+52+52
5m5
=−155
m=−3
35) 50=8 (7+7 r )−(4 r+6) 50=56+56 r−4 r−6 50=52r+50 −50−50
052
=52 r52
0=r
37) −8 (n−7 )+3 (3n−3 )=41 −8n+56+9n−27=41
n+47=41 −47−47 n=−6
39) −61=−5 (5 r−4 )+4 (3 r−4) −61=−25 r+20+12 r−16 −61=−13 r+4 −4−4
−65−13
=−13 r−13
5=r41) −2 (8n−4 )=8(1−n)
−16n+8=8−8n +16n +16n
8=8+8n −8−8
16
08=8n8
0=n
43) −3 (−7 v+3 )+8v=5 v−4(1−6v ) 21v−9+8 v=5v−4+24 v
29v−9=29v−4 −29v−29v
−9=−4false , No Solution∅
45) −7 ( x−2 )=−4−6 (x−1) −7 x+14=−4−6 x+6 −7 x+14=2−6 x +7 x+7 x 14=2+x
−2−2 12=x
47) −6 (8 k+4 )=−8 (6k+3 )−2 −48k−24=−48k−24−2 −48k−24=−48k−26 +48k+48k
−24=−26 falseNoSolution∅
49) −2 (1−7 p )=8( p−7) −2+14 p=8 p−56
−8 p−8 p −2+6 p=−56 +2+2
6 p6
=−546
p=−9
1. 4
1)35
(1+ p )=2120
(20 ) 35+(20 ) 3
5p=2120
(20)
12+12 p=21 −12−12
12 p12
= 912
p=34
3) 0=−54
x−65
(4 )0= (4 )(−54 x)+ 32 (4)
0=−5 x+6 −6−6
−6−5
=−5 x−5
65=x
5) (24 ) 34−(24 ) 5
4m=113
24(24 )
18−30m=113 −18−18
−30m−30
= 95−30
m=−196
7)63572
=−52 (−114 +x )
(72 ) 63572
= (72 ) 558
−52x (72 )
635=495−180x −495−495
17
4 4
2
6 6
9 36
140
−180=−180 x
−180
−79
=x
9) (5 )2b+ (5 ) 95=−115
(5)
10b+9=−11 −9−9
10b10
=−2010
b=−2
11)32 ( 73 n+1)=32 (2 ) 72n+(2 ) 3
2=32
(2 )
7n+3=3 −3−3
7n7
=07
n=0
13) −a−54 (−83 a+1)=−19
4
−a (12 )+(12 ) 103
a− (12 ) 54=−194
(12 )
−12a+40a−15=−57 28a−15=−57
+15+1528a28
=−4228
a=−32
15)556
=−52 ( 32 p−53 )
(12 ) 556
=(12 )−154
p+ 256
(12 )
110=−45 p+50 −50 −50
60
−45=−45 p
−45
−43
=p
17)169
=−43 (−43 n−4
3 )
(9 ) 169
=(9 ) 169n+ 16
9(9)
16=16n+16 −16−16
016
=16 n16
0=n
19)−58
=54 (r−32 )
−58
(8 )=(8 ) 54r−15
8(8 )
−5=10r−15 +15+15
1010
=10 r10
1=r
21) −113
+ 32b=52 (b−53 )
−113
(6 )+ (6 ) 32b=(6 ) 5
2b−25
6(6)
−22+9b=15b−25 −9b−9b −22=6b−25 +25+25
36=6b6
12=b
18
4 4 3
2 3 2
2
2 3 3
23) −(−52 x−32 )=−3
2+x
(2) 52x+(2) 3
2=−32
(2)+(2) x
5 x+3=−3+2 x −2 x −2 x
3 x+3=−3 −3−3
3x3
=−63
x=−2
25) (16 ) 4516
+(16 ) 32n= (16 ) 7
4n−1916
(16 )
45+24n=28n−19 −24n−24n 45=4n−19 +19 +19
644
=4 n4
16=n
27)32 (v+ 32 )=−7
4v−19
6
(12 ) 32v+(12 ) 9
4=−74
v (12 )−196
(12 )
18v+27=−21v−38 +21 v+21 v 39v+27=−38 −27−27
39v39
=−6539
v=−53
29)479
+ 32x=53 (52 x+1)
(18 ) 479
+(18 ) 32x=(18 ) 25
6x+(18 ) 5
3 94+27 x=75 x+30 −27 x−27 x
94=48x+30 −30−30
6448
=48 x48
43=x
1.5
1)aba
= cafor b
b= ca
3) (g ) fgx=b (g ) for x
fxf=bg
f
x=bgf
5) (b )3 x=ab
(b ) for x
3bx3b
= a3b
x= a3b
7)Ec2
=mc2
c2for m
Ec2
=m
19
8 4
6 3 3 2
2 9 3 6
9) (3 )V=(3 ) 43π r3 for π
3v4 r 3
=4 π r3
4 r3
3v4 r 3
=π
11) a+c=b for c −a−a c=b−a
13) c (m+n )= 4 ym+n
(m+n ) for y
c (m+n )4
= 4 y4
c (m+n )4
= y
15) (12 )V=πDn12
(12 ) for D
12vπn
=πDnπn
12vπn
=D
17) pp−c
=n (p−c )p−c
for n
p
p−c=n
19) (L )T=D−dL
(L ) for D
¿=D−d +d+d ¿+d=D
21) L1+at
=L0 (1+at )1+at
L1+at
=Lo
23) 2m+ p=4m+q for m
−2m=2m p=2m+q −q−q
p−q2
=2m2
p−q2
=m
25) (r ) k−mr
=q (r ) for k
k−m=qr +m+m k=qr+m
27) h=vt−16 t 2 for v +16 t 2+16 t2
h+16 t2
t= vt
t
h+16 t2
2=v
29) Q1=P (Q2−Q1 ) forQ2
Q1=PQ2−PQ1
+PQ1+PQ1
Q1+PQ1
P=PQ2
P
Q1+PQ1
P=Q2
31) (d ) R=kA (T1+T 2 )
d(d ) for T 1
dR=kA (T 1+T2) dR=kA T1+kA T2 −kA T2−kAT 2
dR−kAT 2
kA=kAT 1kA
20
dR−kAT 2
kA=T1
33) ax+b=c for a −b−b
axx
= c−bx
a= c−bx
35)lwhlh
=Vlh
forw
w=Vlh
37)1a
(a )+b (a )= ca
(a ) for a
1+ba=c −1−1
bab
= c−1b
a= c−1b
39) at−bw=5 for t +bw+bw
ata
=5+bwa
t=5+bwa41) ax+bx=c for a
−bx−bx
axx
= c−bxx
a= c−bxx
43) x+5 y=3 for y −x−x
5 y5
=3−x5
y=3−x5
45) 3 x+2 y=7 for y −3 x−3 x
2 y2
=7−3x2
y=7−3 x2
47) 5a−7b=4 for b −5a−5a
−7b−7
=4−5a−7
b=4−5a−7
49) 4 x−5 y=8 for y −4 x−4 x
−5 y−5
=8−4 x−5
y=8−4 x−5
1.6
1) |m|=−6 false NoSolution∅
3) |n|=4 n=4 ,−4
5) |b|=7 b=7 ,−7
7) (7 ) ¿ x∨¿7=5 (7 ) ¿
21
|x|=35 x=35 ,−35
9) −10+|k|=−15 +10+10
|k|=−5false NoSolution∅
11) 10|x|+7=57 −7−7
10|x|10
=5010
|x|=5x=5 ,−5
13) 10−5|m|=70 −10−10
−5|m|−5
= 60−5
|m|=−12false
NoSolution∅
15) 9|x|−4=5 +4+4
9|x|9
=99
|x|=1 x=1 ,−1
17) | n10|=1 (10 ) n10
=1 (10 ) ,−1 (10 )
n=10 ,−10
19) |v+10|=2 v+10=2 ,−2
−10−10−10 v=−8 ,−12
21) −4−|a−5|=−13 +4+4
−|a−5|
−1=−9
−1 |a−5|=9a−5=9 ,−9 +5+5+5 a=14 ,−4
23)10|−6 x|10
=6010
|−6 x|=6
−6 x−6
= 6−6
,− 6−6
x=−1 ,1
25)−7|n7|−7
=−2−7
|n7|=27 (7 ) n
7=27
(7 ) ,−27
(7 )
n=2 ,−2
27) −8|−7+p|−6=−14 +6+6
−8|−7+ p|
−8=−8
−8|−7+ p|=1−7+ p=1 ,−1+7+7+7 p=8 ,6
22
29) −3|7+x|−7=−1 +7+7
−3|7+x|−3
= 6−3
|7+x|=−2false
No Solution∅
31) |−7−5 r|=32 −7−5 r=32 ,−32 +7+7+7
−5 r−5
= 39−5
,−25−5
r=−395
,5
33) |8n−6|=66 8n−6=66 ,−66 +6+6+6
8n8
=728
,−608
n=9 ,−152
35) |2v+7|=11 2v+7=11 ,−11 −7−7−7
2v2
=42,−182
v=2 ,−9
37)9|10+6 x|
9=729
|10+6 x|=8 10+6 x=8 ,−8 −10−10−10
6 x6
=−26
,−186
x=−13
,−3
39) −3+|6+6k|=−45 +3 +3 |6+6k|=−42
false NoSolution∅
41) |2n+5|+5=0 −5−5
|2n+5|=−5false No Solution∅
43) 3−2|5−m|=9
−3−3
−2|5−m|
−2= 6
−2 |5−m|=−3false NoSolution∅
45) |−10 x−4|−10=66
+10+10 |−10 x−4|=76
−10 x−4=76 ,−76 +4+4+4
−10x−10
= 80−10
,−72−10
x=−8 , 365
47) |2+3 x|=¿ 4−2x∨¿ 2+3 x=4−2x ,2+3 x=−(4−2 x) +2 x+2x 2+3 x=−4+2 x 2+5 x=4 −2 x−2x −2−2 2+x=−4
5x5
=25 −2−2
x=25 x=−6
23
x=25,−6
49) |2x−53 |=|3 x+42 | (6 ) 2x−5
3=3 x+4
2(6 ), 2x−5
3=
−(3x+4)2
4 x−10=9x+12 (6 ) 2 x−53
=−3 x−42
(6 )
−4 x−4 x 4 x−10=−9x−12
−10=5x+12+9 x+9 x −12−12 13 x−10=−12
−225
=5 x5 +10+10
−225
=x 13x13
=−213
x=−213
x=−225
,− 213
1.7
1) c varies directly as a
ca=k
3) w varies inversely as x wx=k
5) f varies jointly as x and y
fxy
=k
7) h is directly proportional to b
hb=k
9) a is inversely proportional to b ab=k
11) p is jointly proportional to q and r and p = 12 when q = 8 and r = 3
pqr
=k
12
(8 ) (3 )=1224
=12
13) t varies directly as the square of u and t = 6 when u = 3
tu2
=k
24
(6 )(3 )2
=69=23
15) w is inversely proportional to the cube of x and w is 54 when x = 3 w x3=k (54 ) (3 )3=54 (27 )=1458
17) a is jointly proportional with the square of x and the square root of y anda = 25 when x = 5 and y = 9
a
x2√ y=k
(25 )
(5 )2√ (9 )= 2525 ∙3
=13
19) The electrical current, in amperes, in a circuit varies directly as the voltage. When 15 volts are applied, the current is 5 amperes. What is the current when 18 volts are applied?
av=k
(5 )15
=13
(18 ) a18
= 13
(18 )
a=6amperes
21) Hooke’s law states that the distance that a spring is stretched by hanging object varies directly as the mass of the object. If the distance is 20 cm when the mass is 3 kg, what is the distance when the mass is 5 kg?
dm
=k
20(3 )
=203
x5=203
x=1003=33.3cm.
25
23) The number of aluminum cans used each year varies directly as the number of people using the cans. If 250 people use 60,000 cans in one year, how many cans are used each year in Dallas, which has a population of 1,008,000?
cp=k
60000250
=250
(1,000,000 ) c1,000,000
=240(1,000,000)
c=241,920,000cans
25) According to Fidelity Investment Vision Magazine, the average weekly allowance of children varies directly as their grade level. In a recent year, the average allowance of a 9th-grade student was 9.66 dollars per week. What was the average weekly allowance of a 4th-grade student?
ag=k (9.66 )
(9 )=1.07
(4 ) x4=1.07 (4)
x=$4.28
27) The number of kilograms of water in a human body varies directly as the mass of the body. A 96-kg person contains 64 kg of water. How many kilo grams of water are in a 60-kg person?
wm
=k (64 )(96 )
=23
(60 ) x60
=23
(60 )
x=40kg
29) The weight of an object on Mars varies directly as its weight on Earth. A person weighs 95lb on Earth weighs 38 lb on Mars. How much would a 100-lb person weigh on Mars?
me=k 38
95=25
(100 ) m100
=25
(100 )
m=40 lbs
31) The time required to empty a tank varies inversely as the rate of pumping. If a pump can empty a tank in 45 min at the rate of 600 kL/min, how long will it take the pump to empty the same tank at the rate of 1000 kL/min? tr=k (45 ) (600 )=27000
26
t (1000 )1000
=270001000
t=27min
33) The stopping distance of a car after the brakes have been applied varies directly as the square of the speed r. If a car, traveling 60 mph can stop in 200 ft, how fast can a car go and still stop in 72 ft?
dr2
=k (200 )(60 )2
=200600
= 118
(r2 ) 72r2
= 118
r2
(18 ) (72 )= r2
18(18 )
√1296=√r2 36mph=r
35) The intensity of a light from a light bulb varies inversely as the square of the distance from the bulb. Suppose intensity is 90 W/m2 (watts per square meter) when the distance is 5 m. How much further would it be to a point where the intensity is 40 W/m2?
I d2=k (90 ) (5 )2=k90 (25 )=2250
40d2
40=225040
√d2=√56.25 d=7.5
7.5−5=2.5m further
37) The intensity of a television signal varies inversely as the square of the distance from the transmitter. If the intensity is 25 W/m2 at a distance of 2 km, how far from the transmitter are you when the intensity is 2.56 W/m2?
I d2=k (25 ) (2 )2=k 25 (4 )=100
(2.56 )d2
2.56=1002.56
√d2=√39.0625 d=6.25m
27
1.8
1) When five is added to three more than a certain number, the result is 19. What is the number? x+3+5=19 x+8=19 −8−8
x=11
3) When 18 is subtracted from six times a certain number, the result is − 42. What is the number? 6 x−18=−42
+18+18
6 x6
=−246
x=−4
5) A number plus itself, plus twice itself, plus 4 times itself, is equal to − 104. What is the number? x+x+2x+4 x=−104
8 x8
=−1048
x=−13
7) Eleven less than seven times a number is five more than six times the number. Find the number. 7 x−11=6 x+5
−6 x−6 x x−11=5 +11+11
x=16
9) The sum of three consecutive integers is 108. What are the integers? F : x S : x+1 T : x+2 3 x+3=108 −3−3
3x3
=1053
x=35 35 ,36 ,37
28
11) Find three consecutive integers such that the sum of the first, twice the second, and three times the third is − 76. F : x→x 2S :2 ( x+1 )→2x+2 3T :3 ( x+2 )→3 x+6
6 x+8=−76 −8−8
6 x6
=−846
x=−14 −14 ,−13 ,−12
13) The sum of three consecutive odd integers is 189. What are the integers? F : x S : x+2 T : x+4 3 x+6=189 −6−6
3x3
=1833
x=61 61 ,63 ,65
15) Find three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70. F : x→x 2S :2 ( x+2 )→2 x+4 3T :3 ( x+4 )→3 x+12
6 x+16=70 −16−16
6 x6
=546
x=9 9 ,11 ,13
17) Two angles of a triangle are the same size. The third angle is 12 degrees smaller than the first angle. Find the measure the angles. F : x (64) S : x (64) T : x−12 (64−12=52)
29
3 x−12=180 +12+12
3x3
=1923
x=64 64 ° ,64 ° ,52 °
19) The third angle of a triangle is the same size as the first. The second angle is 4 times the third. Find the measure of the angles. F : x (30) S :4 x (4 ∙30=120) T : x (30)
6 x6
=1806
x=30 30 ° ,120 ° ,30 °
21) The second angle of a triangle is twice as large as the first. The measure of the third angle is 20 degrees greater than the first. How large are the angles? F : x (40) S :2x (2 ∙40=80) T : x+20 (40+20=60) 4 x+20=180 −20−20
4 x4
=1604
x=40 40 ° ,80 ° ,60°
23) The second angle of a triangle is five times as large as the first. The measure of the third angle is 12 degrees greater than that of the first angle. How large are the angles? F : x (24) S :5 x (5 ∙24=120) T : x+12 (24+12=36) 7 x+12=180 −12−12
7 x7
=1687
x=24 24 ° ,120° ,36 °
30
25) The second angle of a triangle is four times the first and the third is 5 degrees more than twice the first. Find the measures of the angles. F : x (25) S :4 x (4 ∙25=100) T :2 x+5 (2 ∙25+4=50+5=55) 7 x+5=180 −5−5
7 x7
=1757
x=25 25 ° ,100 ° ,55 °
27) The perimeter of a rectangle is 304 cm. The length is 40 cm longer than the width. Find the length and width. L : x+40 (56+40=96) W : x (56) 2 (2x+40 )=304 4 x+80=304
−80−80
4 x4
=2244
x=56 56 x 96
29) The perimeter of a rectangle is 280 meters. The width is 26 meters less than the length. Find the length and width. L : x (83) W : x−26 (83−26=57) 2 (2x−26 )=280 4 x−52=280
+52+52
4 x4
=3324
x=83 57 x83
31) A mountain cabin on 1 acre of land costs S30,000. If the land cost 4 times as much as the cabin, what was the cost of each? C : x (6000) L :4 x (4 ∙6000=24000)
5x5
=300005
x=6000 Cabin :$ 6,000 , Land : $24,000
31
33) A bicycle and a bicycle helmet cost S240. How much did each cost, if the bicycle cost 5 times as much as the helmet? B:5 x (5 ∙40=200) H : x (40)
6 x6
=2406
x=40 Bike : $200 ,Helmet : $40
35) If Mr. Brown and his son together had S220, and Mr. Brown had 10 times as much as his son, how much money had each? B :10 x (10 ∙20=200) S : x (20)
11 x11
=22022
x=20 Mr .Brown : $200 , Son :$ 20
37) Aaron had 7 times as many sheep as Beth, and both together had 608. How many sheep had each? A :7 x (7 ∙76=532) B: x (76)
8 x8
=6088
x=76 Aaron :532Sheeep ,Beth :76 Sheep
39) Jamal and Moshe began a business with a capital of S7500. If Jamal furnished half as much capital as Moshe, how much did each furnish? J : x (2500) M : 2x (2 ∙2500=5000)
3x3
=75003
x=2500 Jamal :$2500 , Moshe :$5000
41) A 6 ft board is cut into two pieces, one twice as long as the other. How long are the pieces? L :2x (2 ∙2=4) S : x (2)
3x3
=63
x=2 4 ft∧2 ft
32
43) An electrician cuts a 30 ft piece of wire into two pieces. One piece is 2 ft longer than the other. How long are the pieces? L : x+2 (14+2=16) S : x (14) 2 x+2=30 −2−2
2x2
=282
x=14 16 ft∧14 ft
45) The cost of a private pilot course is $1,275. The flight portion costs $625 more than the ground school portion. What is the cost of each? F : x+625 (325+625=950) G : x (325) 2 x+625=1275 −625−625
2x2
=6502
x=325 Flight :$ 950 ,Ground :$325
1.9
1) rA boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find the present age of each.
H=2B ( x+14 )=2(x+4)
x+14=2x+8
−x−x 14=x+8
6=x Boy :16 ,Brother :63) Pat is 20 years older than his son James. In two years Pat will be twice as old as James. How old
are they now? P=2 J
x+22=2(x+2) x+22=2 x+4−x−x 22=x+4
−4−4
33
Now +4H-boy x+10 x+14
B-Brother
x x+4
Now +2P x+20 x+22J x x+2
18=x Pat :38 , James :18
5) Fred is 4 years older than Barney. Five years ago the sum of their ages was 48. How old are they now?
F+B=48 ( x−1 )+( x−5 )=48
2 x−6=48
+6+62x2
=542
x=27 Fred :31 ,Barney :27
7) Tim is 5 years older than JoAnn. Six years from now the sum of their ages will be 79. How old are they now?T+J=79
( x+11)+( x+6 )=79
34
Now −¿5
F x+4 x-1B x x-5
Now +6T x+5 x+11J x x+6
2 x+17=79 −17−17
2x2
=622
x=31 Tim :36 , JoAnn :31
9) The sum of the ages of John and Mary is 32. Four years ago, John was twice as old as Mary. Find the present age of each.
J=2m ( x−4 )=2(28−x )
x−4=56−2 x+2 x+2x
3 x−4=56
3x3
=603
x=20 John:20 , Mary :12
11) The sum of the ages of a china plate and a glass plate is 16 years. Four years ago the china plate was three times the age of the glass plate. Find the present age of each plate.
C=3G ( x−4 )=3(12−x )
x−4=36−3 x +3 x+3 x 4 x−4=36 +4+4
4 x4
=404
x=10 China :10 ,Glass :6
13) A is now 34 years old, and B is 4 years old. In how many years will A be twice as old as B?A=2B
34+ t=2(4+t) 34+ t=8+2 t −t−t 34=8+t −8−8
26=t
35
Now -4J x x-4
M 32-x 28-x
Now -4C x x-4G 16-x 12-x
Now +tA 34 34+tB 4 4+t
15) An Oriental rug is 52 years old and a Persian rug is 16 years old. How many years ago was the Oriental rug four times as old as the Persian Rug?
O=4 P 52−t=4 (16−t )
52−t=64−4 t +4 t+4 t 52+3 t=64−52−52
t=4
17) The age of the older of two boys is twice that of the younger; 5 years ago it was three times that of the younger. Find the age of each.
O=3Y 2 x−5=3 (x−5)
2 x−5=3 x−15−2 x−2x −5=x−15
+15+15
19) Marge is twice as old as Consuelo. The sum of their ages seven years ago was 13. How old are they now?
M +C=13 (2 x−7 )+( x−7 )=13
3 x−14=13 +14+14
3x3
=273
x=9 Marge :18 ,Consuelo :9
21) A silver coin is 28 years older than a bronze coin. In 6 years, the silver coin will be twice as old as the bronze coin. Find the present age of each coin.
S=2B ( x+34 )=2(x+6)
x+34=2x+12−x−x 34=x+12 −12−12
36
Now -tO 52 52-tP 16 16-t
Now -5O 2x 2x-5Y x x-5
Now -7M 2x 2x-7C x x-7
Now +6S x+28 x+34B x x+6
22=x Silver :50 , bronze :22
23) A limestone statue is 56 years older than a marble statue. In 12 years, the limestone will be three times as old as the marble statue. Find the present age of the statues.
L=3M ( x+68 )=3 (x+12)
x+68=3 x+36 −x−x 68=2x+36 −36−36
322
=2 x2
16=x Limestone :72, Marble :16
25) Brandon is 9 years older than Ronda. In four years the sum of their ages will be 91. How old are they now?
B+R=91 ( x+13 )+( x+4 )=91
2 x+17=91 −17−17
2x2
=742
x=37 Brandon: 46 , Ronda :37
27) A father is three times as old as his son, and his daughter is 3 years younger than the son. If the sum of their ages 3 years ago was 63 years, find the present age of the father.
F+S+D+63 (3 x−3 )+( x−3 )+ ( x−6 )=63
5 x−12=63
+12+12
5x5
=755
x=15
37
Now +12L x+56 x+68M x x+12
Now +4B x+9 x+13R x x+4
Now -3F 3x 3x-3S x x-3D x-3 x-6
Father :45 , Son=15 , Daughter :12
29) The sum of the ages of two ships is 12 years. Two years ago, the age of the older ship was three times the age of the newer ship. Find the present age of each ship. O=3Y
( x−2 )=3(10− x) x−2=30−x
+x+x 2 x−2=30
4 x4
=324
x=8 8∧4
31) Ann is eighteen years older than her son. One year ago, she was three times as old as her son. How old are they now?
A=3S ( x+17 )=3 (x−1)
x+17=3 x−3−x−x 17=2x−3 +3+3
202
=2 x2
10=x Ann:28 , Son :10
38
Now -2O x x-2Y 12-x 10-x
Now -1A x+18 x+17S x x-1
33) A mosaic is 74 years older than the engraving. Thirty years ago, the mosaic was three times as old as the engraving. Find the present age of each.
M=3 E ( x+44 )=3( x−30)
x+44=3 x−90 −x−x 44=2x−90 +90+90
1342
=2x2
67=x
35) A wool tapestry is 32 years older than a linen tapestry. Twenty years ago, the wool tapestry was twice as old as the linen tapestry. Find the present age of each.
W=2L ( x+12 )=2(x−20)
x+12=2 x−20−x−x 12=x−20
+20+20
37) Nicole is 26 years old. Emma is 2 years old. In how many years will Nicole be triple Emma’s age? N=3E
(26+t )=3 (2+t) 26+t=6+3 t −t−t 26=6+2 t −6−6
202
=2 t2
10=t
39) Mike is 4 years older than Ron. In two years, the sum of their ages will be 84. How old are they now?
39
Now -30M x+74 x+44E x x-30
Now -20W x+32 x+12L x x-20
Now +tN 26 26+tE 2 2+t
M +R=84 ( x+6 )+ ( x+2 )=84
2 x+8=84 −8−8
2x2
=762
x=38 Mike :42 , Ron :38
1.10
1) A is 60 miles from B. An automobile at A starts for B at the rate of 20 miles an hour at the same time that an automobile at B starts for A at the rate of 25 miles an hour. How long will it be before the automobiles meet?
20 t+25t=60
45 t45
=6045
t=1.33hr
3) Two trains travel toward each other from points which are 195 miles apart. They travel at rate of 25 and 40 miles an hour respectively. If they start at the same time, how soon will they meet?
25 t+40 t=195
65 t65
=19565
t=3hr
5) A passenger and a freight train start toward each other at the same time from two points 300 miles apart. If the rate of the passenger train exceeds the rate of the freight train by 15 miles per hour, and they meet after 4 hours, what must the rate of each be?
4 r+4 r+60=300 8 r+60=300
−60−60
8 r8
=2408
r=30
7) A man having ten hours at his disposal made an excursion, riding out at the rate of 10 miles an hour and returning on foot, at the rate of 3 miles an hour. Find the distance he rode.
10 t=30−3 t +3 t+3 t
40
Now +2M x+4 x+6R x x+2
r t Dr 10 t 10tw 3 10-t 30-3t
r t dP r+15 4 4r+60F r 4 4r
r t d25 t 25t40 t 40t
r t dA 20 t 20tB 25 t 25t
13t13
=3013
t=3013
9) A boy rides away from home in an automobile at the rate of 28 miles an hour and walks back at the rate of 4 miles an hour. The round trip requires 2 hours. How far does he ride?
28 t=8−4 t +4 t+4 t
32t32
= 832
t=.25
11) A family drove to a resort at an average speed of 30 mph and later returned over the same road at an average speed of 50 mph. Find the distance to the resort if the total driving time was 8 hours.
30 t=400−50 t +50 t+50t
80 t80
=40080
t=5
13) A, who travels 4 miles an hour starts from a certain place 2 hours in advance of B, who travels 5 miles an hour in the same direction. How many hours must B travel to overtake A?
4 t+8=5 t −4 t−4 t
8hr=t
15) A motorboat leaves a harbor and travels at an average speed of 8 mph toward a small island. Two hours later a cabin cruiser leaves the same harbor and travels at an average speed of 16 mph toward the same island. In how many hours after the cabin cruiser leaves will the cabin cruiser be alongside the motorboat?
8 t+16=16 t −8 t−8 t
168
=8 t8
41
r t dM 8 t+2 8t+16C 16 t 16t
r t DA 4 t+2 4t+8B 5 t 5t
r t dT 30 t 30tR 50 8-t 400-50t
r t dr 28 t 28tw 4 2-t 8-4t
2hr=t
17) A car traveling at 48 mph overtakes a cyclist who, riding at 12 mph, has had a 3 hour head start. How far from the starting point does the car overtake the cyclist?
48 t=12 t+36 −12 t−12t
36 t36
=3636
t=1d=48 (1 )=48mi .
19) Two men are traveling in opposite directions at the rate of 20 and 30 miles an hour at the same time and from the same place. In how many hours will they be 300 miles apart?
20 t+30t=30050t50
=30050 t=6hr
21) A motorboat leaves a harbor and travels at an average speed of 18 mph to an island. The average speed on the return trip was 12 mph. How far was the island from the harbor if the total trip took 5 h?
18 t=60−12 t +12 t+12 t
30t30
=6030
t=2d=18 (2 )=36mi
23) A jet plane traveling at 570 mph overtakes a propeller-driven plane that has had a 2 h head start. The propeller-driven plane is traveling at 190 mph. How far from the starting point does the jet overtake the propeller-driven plane?
570 t=190 t+380 −190 t−190 t
380t380
=380380
t=1 d=570 (1 )=570mi .
42
r t dCar
48 t 48t
Cy 12 t+3 12t+36
r t d20 t 20t30 t 30t
300
r t dT 18 t 18tR 12 5-t 60-12t
r t dJ 570 t 570tP 190 t+2 190t+380
25) As part of flight training, a student pilot was required to fly to an airport and then return. The average speed on the way to the airport was 100 mph, and the average speed returning was 150 mph. Find the distance between the two airports if the total flight time was 5 h.
100 t=750−150 t +150 t+150t
250t250
=750250
t=3
27) A car traveling at 56 mph overtakes a cyclist who, riding at 14 mph, has had a 3 h head start. How far from the starting point does the car overtake the cyclist?
56 t=14 t+42 −14 t−14 t
42 t42
=4242
t=1d=56 (1 )=56mi
29) A bus traveling at a rate of 60 mph overtakes a car traveling at a rate of 45 mph. If the car had a 1 h head start, how far from the starting point does the bus overtake the car?
60 t=45 t+45 −45t−45 t
15t15
=4515
t=3d=60 (3 )=180mi .
31) A truck leaves a depot at 11 A.M. and travels at a speed of 45 mph. At noon, a van leaves the same place and travels the same route at a speed of 65 mph. At what time does the van overtake the truck?
45 t+45=65 t −45t−45 t
4520
=20 t20
43
r t dT 100 t 100tR 150 5-t 750-150t
r t dCar
56 t 56t
Cy 14 t+3 14t+42
r t dB 60 t 60tC 45 t+1 45t+45
r t dT 45 t+1 45t+45V 65 t 65t
2.25=t2.25=2hr 15min .12 :00 pm+2:15=2 :15 pm
33) Three campers left their campsite by canoe and paddled downstream at an average rate of 10 mph. They then turned around and paddled back upstream at an average rate of 5 mph to return to their campsite. How long did it take the campers to canoe downstream if the total trip took 1 hr?
10 t=5−5 t +5 t+5t
15t15
= 515
t=13hr=20min .
35) A student walks and jogs to college each day. The student averages 5 km/hr walking and 9 km/hr
jogging. The distance from home to college is 8 km, and the student makes the trip in one hour. How far does the student jog?
5 t+9−9 t=8 −4 t+9=8
−9−9
−4 t−4
=−1−4
t=.25d=9−9 (.25 )=9−2.25=6.75mi
37) On a 220 mi trip, a car traveled at an average speed of 50 mph and then reduced its average speed to 35 mph for the remainder of the trip. The trip took a total of 5 h. How long did the car travel at each speed?
50 t+175−35 t=220 15 t+175=220
−175−175
15t15
=4515
t=33hr@50mph ,2hr@35mph
44
r t dd 10 t 10tv 5 1-t 5-5t
r t dW 5 t 5tJ 9 1-t 9-9t
8
r t dF 50 t 50tS 35 5-t 175-35t
220
45
Chapter 2: Graphing
46
2.1
1)B (4 ,−3 ) ,C (1 ,2 ) , D (−1 ,4 ) ,E (−5 ,0 ) , F (2 ,−3 ) ,G (1,3 ) , H (−1,−4 ) , I (−2 ,−1 ) , J (0 ,2 ) ,K (−4 ,3 )
3) y=−14
x−3
let x=−4
y=−14
(−4 )−3
¿1−3 ¿−2 let x=0
y=−14
(0 )−3
¿0−3 ¿−3 let x=4
y=−14
(4 )−3
¿−1−3 ¿−4
x y −4−2 0−3 4−4
5) y=−54
x−4
let x=−4
y=−54
(−4 )−4
¿5−4 ¿1 let x=0
y=−54
(0 )−4
¿0−4 ¿−4 let x=4
y=−54
(4 )−4
¿−5−4 ¿−9
x y −41 0−4 4−9
47
7) y=−4 x+2
let x=−1 y=−4 (−1)+2 ¿4+2 ¿6 let x=0 y=−4 (0)+2 ¿0+2 ¿2 let x=1 y=−4 (1)+2 ¿−4+2 ¿−2
x y −16 02 1−2
9) y=32x−5
let x=−2
y=32(−2)−5
¿−3−5 ¿−8 let x=0
y=32(0)−5
¿0−5 ¿−5 let x=2
y=32(2)−5
¿3−5 ¿−2
x y −2−8 0−5 2−2
11) y=−45
x−3
48
let x=−5
y=−45
(−5)−3
¿4−3 ¿1 let x=0
y=−45
(0)−3
¿0−3 ¿−3 let x=5
y=−45
(5)−3
¿−4−3 ¿−7
x y −51 0−3 5−7
13) x+5 y=−15−x −x
5 y5
=−x5
−155
y=−15
x−3
let x=−5
y=−15
(−5)−3
¿1−3 ¿−2 let x=0
y=−15
(0)−3
¿0−3 ¿−3 let x=5
y=−15
(5)−3
¿−1−3 −4
x y −5−2 0−3 5−4
15) 4 x+ y=5 −4 x−4 x
49
y=−4 x+5
let x=−1 y=−4 (−1)+5 ¿4+5 ¿9 let x=0 y=−4 (0)+5 ¿0+5 ¿5 let x=1 y=−4 (1)+5 ¿−4+5
x y −19 05 11
¿1
17) 2 x− y=2 −2 x−2x
− y−1
=−2x−1
+ 2−1
y=2x−2
let x=−1 y=2(−1)−2 ¿−2−2 ¿−4 let x=0 y=2(0)−2 ¿0−2 ¿−2 let x=1 y=2(1)−2 ¿2−2 ¿0
x y −1−4 0−2 10
50
19) x+ y=−1 −x −x y=−x−1
let x=−1 y=−(−1)−1 ¿1−1 ¿0 let x=0 y=−(0)−1 ¿0−1 ¿−1 let x=1 y=−(1)−1 ¿−1−1 ¿−2
x y −1−0 0−1 1−2
21) x− y=−3
−x−x
− y−1
=−x−1
− 3−1
y=x+3
let x=−1 y=(−1)+3 ¿−1+3 ¿2 let x=0 y=(0)+3 ¿3 let x=1 y=(1)+3 ¿4
x y −12 03 14
2.2
1) m=6
4=32
3)
m=10=undefined
5)
51
6
4
1
6
m=56
7)
m=−22
=−1
9)
m=−11
=−1
11) (−2 ,10 ) ,(−2 ,−15)
m= −15−10−2− (−2 )
=−250
=¿
undefined
13) (−15 ,10 ) ,16 ,−7 ¿
m= −7−1016−(−15 )
=−1731
15) (10 ,18 ) ,(−11 ,−10)
m=−10−18−11−10
=−28−21
=4 /3
17) (−16 ,−14 ) ,(11 ,−14)
m=−14−(−14 )11−(−16)
= 027
=0
19) (−4 ,14 ) , (−16 ,8 )
m= 8−14−16−(−4 )
= −6−12
= 12
21) (12 ,−19 ) , (6 ,14 )
m=14−(−19 )6−12
= 33−6
= 11−2
23) (−5 ,−10 ) ,(−5 ,20)
m=20—10−5—5
=300
=¿
undefined
25) (−17 ,19 ) ,(10 ,−7)
m= −7−1910− (−17 )
=−2627
27) (7 ,−14 ) ,(−8 ,−9)
m=−9−(−14 )−8−7
= 5−15
=−13
29) (−5 ,7 ) ,(−18 ,14)
m= 14−7−18−(−5 )
= 7−13
31) (2 ,6 )∧( x ,2 ); slope 47
47=2−6x−2
( x−2 )1
47= −4x−2
( x−2 )1
47
( x−2 )=−4
(7)(47 x−87 )=−4 (7)
4 x−8=−28 +8+8
4 x4
=−204
x=−5
33) (−3 ,−2 )∧( x ,6 ); slope=−85
−85
=6−(−2 )x−(−3 )
52
5
22
11
( x+3 ) −85
= 8x+3
(x+3 )
−85
( x+3 )=8
5(−85 x−245 )=8(5)
−8 x−24=40 +24+24
−8 x−8
= 64−8
x=−8
35) (−8 , y )∧ (−1,1 ) ; slope=67
67= 1− y
−1−(−8 )
(7 ) 67=1− y
7(7 )
6=1− y −1−1
5
−1=− y
−1 −5= y
37) ( x ,−7 )∧(−9 ,−9 ) ;slope=25
25=
−9−(−7 )−9−x
(−9−x ) 25= −2
−9−x(−9−x )
25
(−9−x )=−2
5(−185 −25x )=(−2 ) 5
−18−2x=−10 +18+18
−2x−2
= 8−2
x=−4
39) ( x ,5 )∧(8 ,0 ) ; slope=−56
−56
=0−58−x
(8−x )−56= −58−x
(8−x )
−56
(8−x )=−5
6¿ −40+5x=−30 +40+40
5x5
=105
x=2
2.3
1) slope=2 y−intercept=5 y=mx+b y=2x+5
3) slope=1 y−intercept=−4 y=mx+b y=x−4
5) slope=−34
y−intercept=−1
y=mx+b
y=−34
x−1
7) slope=13y−intercept=1
y=13x+1
53
9)
y=mx+b y=−x+5
11)
y=mx+b y=x−1
13)
y=mx+b y=−4 x
15) x+10 y=−37 −x−x
10 y10
=−x10
−3710
y=−110
x−3710
17) 2 x+ y=−1
−2 x−2x y=−2x−1
19) 7 x−3 y=24 −7 x−7 x
−3 y−3
=−7 x−3
+ 24−3
y=73x−8
21) x=−8
23) y−4=−( x+5 ) y−4=−x−5 +4+4 y=−x−1
25) y−4=4 ( x−1 ) y−4=4 x−4 +4+4 y=4 x
27) y+5=−4 (x−2) y+5=−4 x+8 −5−5 y=−4 x+3
54
11
m=11=1
b=−1
m=−41
=−4
b=0
4
1
b=5
m=−55
=−1
5
−5
29) y+1=−12
(x−4 )
y+1=−12
x+2
−1−1
y=−12
x+1
31) y=13x+4
m=13,b=4
33) y=65x−5
m=65, b=−5
35) y=32x
m=32,b=0
37) x− y+3=0 −x−3−x−3
− y−1
=−x−1
− 3−1
y=x+3 m=1 , b=3
39) − y−4+3x=0 +4−3x−3x+4
− y−1
=−3 x−1
+ 4−1
y=3 x−4 m=3 , b=−4
55
5
6
b=−5
b=0
2
3
b=3
11
b=41
3
41) −3 y−3
=−5 x−3
+ 9−3
y=53x−3
m=53,b=−3
2.4
1) Through (2 ,3 ) , slope=undefined x=2
3) Through (2 ,2 ) , slope=12
y− y1=m(x−x1)
y−2=12(x−2)
5) Through (−1 ,−5 ) , slope=9 y− y1=m(x−x1) y+5=9(x+1)
7) Through (−4 ,1 ) , slope=34
y− y1=m(x−x1)
y−1=34(x+4 )
9) Through (0 ,−2 ) , slope=−3 y− y1=m(x−x1) y+2=−3(x−0) y+2=−3x
11) Through (0 ,−5 ) , slope=−14
y− y1=m(x−x1)
y+5=−14
(x−0)
y+5=−14
x
13) Through (−5 ,−3 ) , slope=15
y− y1=m(x−x1)
y+3=15(x+5)
56
1
3
b=−4
b=−3 3
5
15) Through (−1,4 ) , slope=−54
y− y1=m(x−x1)
y−4=−54
( x+1)
17) Through (−1 ,−5 ) , slope=2 y− y1=m(x−x1) y+5=2(x+1) y+5=2x+2 −5−5 y=2x−3
19) Through (5 ,−1 ) , slope=−35
y− y1=m(x−x1)
y+1=−35
(x−5)
y+1=−35
x+3
−1−1
y=−35
x+2
21) Through (−4,1 ) , slope=12
y− y1=m(x−x1)
y−1=12(x+4 )
y−1=12x+2
+1+1
y=12x+3
23) Through (4 ,−2 ) , slope=−32
y− y1=m(x−x1)
y+2=−32
(x−4)
y+2=−32
x+6
−2−2
y=−32
x+4
25) Through (−5 ,−3 ) , slope=−25
y− y1=m(x−x1)
y+3=−25
(x+5)
y+3=−25
x−2
−3−3
y=−25
x−5
27) Through (2 ,−2 ) , slope=1 y− y1=m(x−x1) y+2=1( x−2) y+2=x−2 −2−2 y=x−4
29) Through (−3 ,4 ) , slope=undefined x=−3
31) Through (−4 ,2 ) , slope=−12
y− y1=m(x−x1)
y−2=−12
(x+4)
57
y−2=−12
x−2
+2+2
y=−12
x
33) Through (−4 ,3 )∧ (−3 ,1 )
m= 1−3−3−(−4 )
=−21
=−2
y− y1=m(x−x1) y−3=−2(x+4)
35) Through (5 ,1 )∧(−3 ,0 )
m= 0−1−3−5
=−1−8
=18
y− y1=m(x−x1)
y−1=18(x−5)
37) Through (−4 ,−2 )∧(0 ,4 )
m= 4− (−2 )0−(−4 )
=64=32
y− y1=m(x−x1)
y+2=32(x+4)
39) Through (3 ,5 )∧ (−5 ,3 )
m= 3−5−5−3
=−2−8
=14
y− y1=m(x−x1)
y−5=12(x−3)
41) Through (3 ,−3 )∧(−4 ,5 )
m=5− (−3 )−4−3
= 8−7
y− y1=m(x−x1)
y+3=−87
(x−3)
43) Through (−5 ,1 )∧(−1 ,−2 )
m= −2−1−1− (−5 )
=−34
y− y1=m(x−x1)
y−1=−34
(x+5)
y−1=−34
x−154
y− 44=−34
x−154
+44
+ 44
y=−34
x−114
45) Through (−5 ,5 )∧ (2,−3 )
m= −3−52−(−5 )
=−87
y− y1=m(x−x1)
y−5=−87
(x+5)
y−5=−87
x− 407
y−357=−87
x−407
+357
+ 357
y=−87
x−57
47) Through (4 ,1 )∧(1 ,4 )
m=4−11−4
= 3−3
=−1
y− y1=m(x−x1) y−1=−1(x−4) y−1=−x+4 +1+1 y=−x+4
58
49) Through (0 ,2 )∧ (5 ,−3 )
m=−3−25−0
=−55
=−1
y− y1=m(x−x1) y−2=−1(x−0) y−2=−x +2+2 y=−x+2
51) Through (0 ,3 )∧(−1 ,−1 )
m=−1−3−1−0
=−4−1
=4
y− y1=m(x−x1) y−3=4(x−0) y−3=4 x +3+3 y=4 x+3
2.5
1) y=2x+4 m=2 ∥m=2
3) y=4 x−5 m=4 ∥m=4
5) x− y=4 −x−x
(− y )−1
=−x−1
+ 4−1
y=x−4 m=1 ∥m=1
7) 7 x+ y=−2 −7 x−7 x y=−7x−2 m=−7 ∥m=−7
9) x=3 m=undefined
⊥m=0
11) y=−13
x
m=−13
⊥m=3
13) x−3 y=−6 −x −x
−3 y−3
=−x−3
− 6−3
y=13x+2
m=13
⊥m=3
15) x+2 y=8
59
−x −x
2 y2
=−x2
+ 82
y=−12
x+4
m=−12
⊥m=2
18) Through (2 ,5 ) , par ¿ x=4 m=undefined ∥m=undefined x=2
19) Through (3 ,4 ) , par ¿ y=92x−5
m=92
∥m=92
y−4=92(x−3)
21) Through (2,3 ) , par¿ x=0
m=75
∥m=75
y−3=75( x−2 )
23) Through (4 ,2 ) , par ¿x=0 m=undefined ∥m=underfined x=4
25) Through (1 ,−5 ) , perp¿ y=x+1 m=1 ⊥m=−1 y+5=−1(x−1)
27) Through (5 ,2 ) perp¿ y=−5 x−3 m=−5
⊥m=15
y−2=15(x−5)
29) Through (4 ,2 ) perp¿ y=4 x m=4
⊥m=−14
y−2=−14
(x−4)
31) Through (2 ,−2 ) , perp¿ y=13x
m=13
⊥m=−3 y+2=−3(x−2)
33) Through (4 ,−3 ) , par ¿ y=−2 x m=−2 ∥m=−2 y+3=−2(x−4) y+3=−2 x+8 −3−3 y=−2x+5
35) Through (−3 ,1 ) , par ¿ y=−43
x−1
m=−43
∥m=−43
60
y−1=−43
(x+3)
y−1=−43
x−4
+1+1
y=−43
x−3
37) Through (−4 ,−1 ) par y=−12
x+1
m=−12
∥m=−12
y+1=−12
(x+4)
y+1=−12
x−2
−1−1
y=−12
x−3
39) Through (−2 ,−1 ) par y=−12
x−2
m=−12
∥m=−12
y+1=−12
(x+2)
y+1=−12
x−1
−1−1
y=−12
x−2
41) Through (4 ,3 ) , perp¿ y=−x−1 m=−1 ⊥m=1 y−3=1 ( x−4 ) y−3=x−4 +3+3 y=x−1
43) Through (5 ,2 ) , perp¿ x=0 m=undefined ⊥m=0 y=2
45) Through (−2 ,5 ) , perp¿ y=x−2 m=1 ⊥m=−1 y−5=−1(x+2) y−5=−x−2 +5+5 y=−x+3
47) Through (4 ,−3 ) , perp¿ y=12x−3
m=12
⊥m=−2 y+3=−2(x−4) y+3=−2 x+8 −3−3 y=−2x+5
61
62
Chapter 3: Inequalities
63
3.1
1) n>−5
−5 0 (−5 ,0)
3) −2≥k
−2 0 ¿
5) 5≥x
05 ¿
7) −20 x←2
9) 05 x≥5
11) −20 x>−2
13) (11) x11
≥10 (11 )
x≥110
0110 ¿
15) 2+r<3 −2−2 r<1
01 (−∞,1)
17) 8+ n3≥6
−8−8
(3 ) n3≥−2 (3)
n≥−6
−60 ¿
19) (5 )2> a−25
(5 )
10>a−2 +2+2 12>a
012 (−∞,12)
21) −47≥8−5 x −8−8
−55−5
≥− 5 x−5
11≤ x
0 11 ¿
23) −2 (3+k )←44 −6−2k←44 +6+6
−2k−2
← 38−2
k>19
0 19 (19 ,∞)
25) 18←2(−8+p) 18<16−2 p −16 −16
2
−2<−2 p
−2 −1> p
−1 0 (−∞,−1)
27) 24≥−6(m−6) 24≥−6m+36 −36 −36
−12−6
≥−6m−6
2≤m
02 ¿
29) −r−5 (r−6 )←18 −r−5 r+30←18 −6 r+30←18 −30−30
−6 r−6
← 48−6
r>8
0 8 (8 ,∞ )
31) 24+4b<4 (1+6b) 24+4b<4+24b −4b −4b
64
( l
] l
l ]
) l
l [
( l
l [
l )
[ l
l )
l [
l (
) l
l [
l (
)
24<4+20b −4 −4
2020
< 20b20
1<b
0 1
(1 ,∞)
33) −5v−5←5(4v+1) −5v−5←20v−5 +20 v+20v 15v−5←5 +5+5 15v<0 v<0
0 (−∞,0)
35) 4+2 (a+5 )←2(−a−4) 4+2a+10<2a+8 14+2a<2a+8 −2a−2a 14<8 false No solution∅
37) −(k−2 )>−k−20 −k+2>−k−20 +k+k 2>−20 true Allreal numbers R
3.2
1) (3 ) n3≤−3 (3 )∨−5n
−5≤−10
−5 n≤−9∨n≥2
−9 02 (−∞,−9 ]∪¿
3) x+7≥12∨9 x9
←459
−7−7∨x←5 x≥5 −5 05 (−∞,−5 )∪¿
5) x−6←13∨6 x6
≤−606
+6+6 x←7∨x ≤−10
−10−7 0
(−∞,−7)
7) (8 ) v8>−1 (8 )∧v−2<1
+2+2 v>−8∧v<3
−8 03 (−8 ,3)
9) −8+b← 3∧4 b4
< 204
+8+8 b<5∧b<5
65
l (
)
] l [
) l [
l ) l [
] )
(( l )
))
l )
05 (−∞,5)
11) a+10≥ 3∧8a8≤ 488
−10−10 a≥−7∧a≤6
−7 0 6 [−7 ,6]
13) 3≤9+x≤7 −9−9−9 −6≤ x≤−2
−6−20 [−6 ,−2]
15) 11<8+k ≤12 −8−8−8 3<k≤4
034 ¿
17) −3<x−1<1 +1+1+1 −2<x<2
−202 (−2 ,2)
19) −4<8−3m≤11 −8−8−8
−12−3
<−3m−3
≤ 3−3
4>m≥−1 −1≤m<4
−104 ¿
21) −16≤2n−10≤−22 −16≤−22
No solution∅
23) −5b+10≤30∧7b+2≤−40 −10−10 −2−2
−5b−5
≤
20−5
∧7b
7≤−427
b≥−4∧b≤−6
−6−4 0 NoSolution∅
25) 3 x−9<2 x+10∧5+7 x ≤10 x−10 −2 x−2x−7 x−7 x x−9<10∧5≤3x−10 +9+9 +10+10
x<19 153≤ 3 x3
5≤x
0519 ¿
27) −8−6 v ≤8−8 v∧7 v+9≤6+10 v +8 v+8v −7 v −7 v −8+2 v≤8∧¿ 9≤6+3v +8+8 −6−6
2v2
≤ 162 ¿
33≤ 3 v3
v≤8 ¿ 1≤v
018
66
[
] [ l ]
[ ] l
l ( ]
( l )
l l l
[ l )
] [
l [ ) [
)
l [ ] [
]
[1,8]
29) 1+5k ≤7 k−3∨k−10>2k+10 −5 k−5k −k −k 1≤2k−3∨−10>k+10 +3+3 −10−10
42≤ 2k2
∨−20>k
2≤k
−2002(−∞,−20 )∪¿
31) 2 x+9≥10 x+1∧3 x−2<7 x+2 −2 x−2x −3 x−3 x
9≥8 x+1∧−2<4 x+2
−1−1 −2−2 88≥ 8x8
¿ −44
< 4 x4
1≥x ¿ −1<x
−101 ¿
3.3
1) |n|≤−11 false NoSolution∅
3) |b|≤−10 false NoSolution∅
5) |x|>5 x>5∨x←5 −505 (−∞,−5)∪(5 ,∞ )
7)10|n|10
>3010
|n|>3 n>3∨n←3
−303 (−∞,−3)∪(3 ,∞)
9)−3|x|−3
< 36−3
x>−12 false
No Solution∅
11) |n|+4>−5 −4−4 |n|>−9 true All Real Numbers R
13) 10−8|p|≥18 −10−10
−8|p|−8
≥ 8−8
|p|≥−1false
No solutoin∅
15) 9|n|−3≥42 +3+3
9∨n∨ ¿9≥ 459
¿
|n|≥5 n≥5∨n≤−5
−505 (−∞,−5)∪(5 ,∞ )
67
) l [ ( l ]
( ]
) l (
) l (
] l [
17) |m9 |≥−5 true All Real Numbers R
19) |9+ x|>−2 true All Real Numbers R
21) |x+73 |≥5 (3 ) v+7
3≥5 (3 )∨ (3 ) v+7
3≤−5(3)
v+7≥15∨v+7≤−15 −7−7 −7−7 v≥8∨v ≤−22
−22 0 8(−∞,−22 ]∪¿
23)7|−7 x|7
≥ 987
|−7 x|≥14
−7 x−7
≥ 14−7
∨−7 x−7
≤−14−7
x≤−2∨x ≥2 −202 (−∞,−2)∪(2 ,∞ )
25) −5+|−8k|≥51 +5+5 |−8 k|≥56
|−8 k|−8
≥ 56−8
∨|−8 k|−8
≤−56−8
k ≤−7∨k≥7 −707 (−∞,−7)∪(7 ,∞)
27) 8−4|x9|>12 −8−8
−4|x9|−4
> 4−4
|x9|←1 false
No Solution∅
29) 7|−9+m|+3≥66 −3−3
7|−9+m|
7≥ 637
|−9+m|≥9 −9+m≥9∨−9+m≤−9 +9+9 +9+9 m≥18∨m≤0 018 (−∞,0)∪(18 ,∞)
31) |3n+10|≤−26 false NoSolution∅
33) |10b+10|>70 10b+10>70∨10b+10←70 −10−10 −10−10
10b10
> 6010
∨10b10
<−8010
b>6∨b←8 −806 (−∞,−8)∪ (6 ,∞)
68
] l [
] l [
] l [
] [
) l (
35) |−10+x|≥8 −10+x ≥8∨−10+x ≤−8 +10+10+10+10 x≥18∨x≤2 0218 (−∞,2)∪(18 ,∞ )
37) |−10+a|−3≥7 +3+3 |−10+a|≥10 −10+a≥10∨−10+a≤−10 +10+10 +10 +10 a≥20∨a≥0 020 (−∞,0)∪ (20 ,∞)
39) |3 x−1|−9≤−8 +9+9
|3 x−1|≤1 −1≤3 x−1≤1 +1 +1 +1
03≤ 3 x3
≤ 23
0≤ x≤ 23
0 231
[0 , 23]
41) −8|8n−1|+4≥−116 −4−4
−8|8n−1|
−8≥−120
−8 |8n−1|≥15 −15≤8n−1≤15 +1+1+1
−148
≤ 8n8
≤ 168
−74
≤n≤2
−7402
[−74
,2]
43) −10+9|3 p−9|←37 +10+10
9|3 p−9|
9<−279
|3 p−9|←3 false
NoSolution∅
45) 9∨2−10n∨−8>100 +8+8
9|2−10n|
9> 1089
|2−10n|>12
69
l ] [
] [
[ ] l
[ l ]
2−10n>12∨2−10n←12 −2 −2 −2 −2
−10n−10
> 10−10
∨¿ −10n−10
<−14−10
n←1∨n>75
−10 75
(−∞,−1)∪( 75,∞)
70
) l (
Chapter 4: Systems of Equations
71
4.1
1) y=−x+1 y=−5x−3
3) y=−3 y=−x−4
5) y=−34
x+1
y=−34
x+2 No Solution
7) y=13x+2
y=−53
x−4
9) y=53x+4
y=−23
x−3
11) x+3 y=−95 x+3 y=3 −x−x−5 x−5 x
3 y3
=−x3
−933 y3
=−5 x3
+ 33
y=−13
x−3 y=−53
x+1
72
(−1 ,2)
(−3 ,1)
(−1 ,−3)
(−3 ,−1)
13) x− y=42 x+ y=−1 −x−x−2 x−2 x
− y−1
=−x−1
+ 4−1 y=−2x−1
y=x−4
15) 2 x+3 y=−62 x+ y=2 −2 x−2x−2x−2 x
3 y3
=−2 x3
−63y=−2 x+2
y=−23
x−2
17) 2 x+ y=2x− y=4 −2 x−2x−x−x
y=−2x+2− y−1
=−x−1
+ 4−1
y=x−4
19) 2 x+ y=2 x+3 y=9 −2 x−2x−x−x
y=−2x+2 3 y3
=−x3
+ 93
y=−13
x+3
21) 0=−6 x−9 y+3612=6 x−3 y +9 y+9 y−6 x−6 x
9 y9
=−6 x9
+369
−6 x−3
+ 12−3
=−3 y−3
y=−23
x+42 x−4= y
73
(3 ,−4)
(1 ,−3)
(−3,4)
(3 ,−4) (3 ,2)
(−2 ,−2)
23) 2 x− y=−10=−2x− y−3 −2 x−2x+ y+ y
− y−1
=−2x−1
− 1−1
y=−2 x−3
y=2x+1
25) 3+ y=−x− 4−1
−6 x−1
=− y−1
−3−34+6 x= y y=−x−3
27) − y+7 x=4− y−3+7 x=0 −7 x−7 x+ y+ y
− y−1
=−7 x−1
+ 4−1 −3+7 x= y
y=7 x+4 No Solution
29)−124
+ x4=4 y4124
−5 x4
=4 y4
−3+ 14x= y 3−5
4x= y
4.2
1) y=−3x y=6x−9 −3 x=6 x−9 −6 x−6 x
−9 x−9
=−9−9
x=1 y=−3 (1 )=−3
(1 ,−3)
74
(−1 ,−1)
(−1 ,−2)
(4 ,−2)
3) y=−2x−9 y=2x−1 −2 x−9=2x−1 +2 x+2x −9=4 x−1 +1+1
−84
=4 x4
−2=x y=−2 (−2 )−9 y=4−9 y=−5 (−2 ,−5)
5) y=6x+4 y=−3x−5 6 x+4=−3 x−5 +3 x+3 x 9 x+4=−5 −4−4
9 x9
=−99
x=−1 y=6 (−1 )+4 y=−6+4 y=−2 (−1 ,−2)
7) y=3 x+2 y=−3x+8 3 x+2=−3 x+8 +3 x+3 x 6 x+2=8 −2−2
6 x6
=66
x=1 y=3 (1 )+2 y=3+2 y=5 (1 ,5)
9) y=2x−3
y=−2x+9 2 x−3=−2x+9 +2 x+2x 4 x−3=9 +3+3
4 x4
=124
x=3 y=2 (3 )−3 y=6−3 y=3 (3 ,3)
11) y=6x−6 −3 x−3 y=−24 −3 x−3 (6 x−6 )=−24 −3 x−18 x+18=−24 −21 x+18=−24 −18−18
−21x−21
=−42−21
x=2 y=6 (2 )−6 y=12−6 y=6 (2 ,6)
13) y=−6 3 x−6 y=30 3 x−6 (−6 )=30 3 x+36=30 −36−36
3x3
=−63
x=−2 (−2 ,−6)
15) y=−5 3 x+4 y=−17
3 x+4 (−5 )=−17 3 x−20=−17 +20+20
3x3
=33
x=1 (1 ,−5 )
17) −2 x+2 y=18 y=7 x+15 −2 x+2 (7 x+15 )=18 −2 x+14 x+30=18 12 x+30=18 −30−30
12x12
=−1212
x=−1 y=7 (−1 )+15 y=−7+15 y=8 (−1 ,8)
19) y=−8 x+19 −x+6 y=16 −x+6 (−8 x+19 )=16 −x−48x+114=16 −49 x+114=16
−114−114
−49 x−49
=−98−49
75
x=2 y=−8 (2 )+19 y=−16+19 y=3 (2 ,3)
21) 7 x−2 y=−7 y=7 7 x−2 (7 )=−7 7 x−14=−7 +14+14
7 x7
=77
x=1 (1 ,7)
23) x−5 y=7 2 x+7 y=−20 x−5 y=7 +5 y+5 y x=5 y+7 2 (5 y+7 )+7 y=−20 10 y+14+7 y=−20 17 y+14=−20 −14−14
17 y17
=−3417
y=−2 x−5 (−2 )=7 x+10=7 −10−10 x=−3 (−3 ,−2)
25) −2 x− y=−5 x−8 y=−23 +8 y+8 y x=8 y−23
−2 (8 y−23 )− y=−5 −16 y+46− y=−5 −17 y+46=−5 −46−46
−17 y−17
=−51−17
y=3 x−8 (3 )=−23 x−24=−23 +24+24 x=1 (1 ,3)
27) −6 x+ y=20 −3 x−3 y=−18 −6 x+ y=20 +6 x+6 x y=6x+20
−3 x−3 (6 x+20 )=−18 −3 x−18 x−60=−18 −21 x−60=−18 +60+60
−21x−21
= 42−21
x=−2 y=20+6(−2) y=20−12 y=8 (−2 ,8)
29) 3 x+ y=9
2 x+8 y=−16 3 x+ y=9 −3 x−3 x y=−3x+9 2 x+8 (−3x+9 )=−16 2 x−24 x+72=−16 −22 x+72=−16 −72−72
−22x−22
=−88−22
x=4 y=−3 (4 )+9 y=−12+9 y=−3 (4 ,−3 )
32) 2 x+ y=2 3 x+7 y=14 2 x+ y=2 −2 x−2x y=2−2x 3 x+7 (2−2 x )=14 3 x+14−14 x=14 −11 x+14=14 −14−14
−11 x−11
= 0−11
x=0 y=2−2(0) y=2−0 y=2 (0 ,2)
76
34) x+5 y=15 −3 x+2 y=6 x+5 y=15 −5 y−5 y x=15−5 y −3 (15−5 y )+2 y=6 −45+15 y+2 y=6 −45+17 y=6 +45+45
17 y17
=5117
y=3 x=15−5(3) x=15−15 x=0
(0 ,3)
36) −2 x+4 y=−16 y=−2 −2 x+4 (−2 )=−16 −2 x−8=−16 +8+8
−2x−2
=−8−2
x=4 (4 ,−2)
38) −6 x+6 y=−12 8 x−3 y=16 −6 x+6 y=−12
+6 x+6 x
6 y6
=6 x6
−126
y=x−2 8 x−3 ( x−2 )=16 8 x−3 x+6=16 5 x+6=16 −6−6
5x5
=105
x=2 y= (2 )−2 y=0 (2 ,0)
39) 2 x+3 y=16 −7 x− y=20 +7 x+7 x
− y−1
=7 x−1
+20/−1
y=−7x−20 2 x+3 (−7 x−20 )=16 2 x−21 x−60=16 −19 x−60=16 +60+60
−19x−19
= 76−19
x=−4 y=−7 (−4 )−20 y=28−20 y=8 (−4 ,8)
4.3
1) 4 x+2 y=0 −4 x−9 y=−28
−7 y−7
=−28−7
y=4 4 x+2 (4 )=0 4 x+8=0 −8−8
4 x4
=−84
x=−2 (−2 ,4)
77
3) −9 x+5 y=−22 9 x−5 y=13
0=−9 false NoSolution∅
5) −6 x+9 y=3 6 x−9 y=−90=−6
false NoSolution∅
7) −1 (4 x−6 y )=(−10 )−1 4 x−6 y=−14 −4 x+6 y=10
0=10 false NoSolution∅
9) −1 (−x−5 y )=28(−1) −x+4 y=−17 x+5 y=−28
9 y9
=−459
y=−5 −x−5 (−5 )=28 −x+25=28 −25−25
−x−1
= 3−1
x=−3 (−3 ,−5)
11) 2 (2x− y )=(5 )2 5 x+2 y=−28 4 x−2 y=10
9 x9
=−189
x=−2 2 (−2 )− y=5 −4− y=5 +4+4
− y−1
= 9−1
y=−9 (−2 ,−9)
13) 10 x+6 y=24 −6 (−6 x+ y )= (4 )(−6) 10 x+6 y=24 36 x−6 y=−24
46 x46
= 046
x=0 10 (0 )+6 y=24
6 y6
=246
y=4 (0 ,4 )15) 3 (2x+4 y )=(24 )3
4 x−12 y=8 6 x+12 y=72
10x10
=8010
x=8 2 (8 )+4 y=24 16+4 y=24 −16−16
4 y4
=84
y=2 (8 ,2)
17) 2 (−7 x+4 y )=(−4 )2 10 x−8 y=−8 −14 x+8 y=8
(−4 x4 )=−16−4
x=4 −7 (4 )+4 y=−4
−28+4 y=−4 +28+28
4 y4
=244
y=6 (4 ,6)
19) 5 x+10 y=20 2 (−6 x−5 y )=(−3 )2 5 x+10 y=20 −12 x−10 y=−6
(−7 x7 )=14 /−7 x=−2
5 (−2 )+10 y=20 −10+10 y=20 +10+10
10 y10
=3010
y=3 (−2 ,3)
21) 5 (−7 x−3 y )=12(5) −3 (−6 x−5 y )=20 (−3) −35 x−15 y=60 18 x+15 y=−60
−17 x−17
= 0−17
x=0 −7 (0 )−3 y=12
−3 y−3
= 12−3
y=−4 (0 ,−4)
23) 7 (9 x−2 y )=(−18 )7 −2 (5x−7 y )=(−10 )(−2)
78
63 x−14 y=−126 −10 x+14 y=20
53x53
=−10653
x=−2 9 (−2 )−2 y=−18 −18−2 y=−18 +18+18
−2 y−2
= 0−2
y=0 (−2 ,0)
25) 3 (9 x+6 y )=(−21 )3 2 (−10 x−9 y )=28(2) 27 x+18 y=−63 −20 x−18 y=56
7 x7
=−77
x=−1 9 (−1 )+6 y=−21 −9+6 y=−21 +9+9
6 y6
=−126
y=−2 (−1 ,−2)
27) 3 (−7 x+5 y )=(−8 )3 5 (−3 x−3 y )=12(5) −21 x+15=−24 −15 x−15=60
−36 x36
= 36−36
x=−1 −7 (−1 )+5 y=−8 7+5 y=−8 −7−7
5 y5
=−155
y=−3 (−1 ,−3)
29) 5 (−8 x−8 y )= (−8 )5 4 (10x+9 y )=(1 )4 −40x−40 y=−40 40 x+36 y=4
−4 y−4
=−36−4
y=9 −8 x−8 (9 )=−8 −8 x−72=−8 +72+72
−8 x8
=648
x=8 (8 ,9)
31) 9 y=7−x −18 y+4 x=−26 9 y=7−x +x+x 2 (9 y+x )=(7 )2 −18 y+4 x=−26 18 y+2x=14
6 x6
=−126
x=−2 9 y=7−(−2)
9 y9
=99
y=1 (−2 ,1)
33) 0=9 x+5 y
(7 ) y=27x (7)
7 y=2 x −7 y−7 y (−9 )0=(2 x−7 y )(−9) 2 (0 )= (9x+5 y )2 0=−18 x+63 y 0=18 x+10 y
073
=73 y73
0= y 0=9 x+5(0)
79
09=9 x9
0=x (0,0 )
4.4
1) ( I )a−2b+c=5 ( I )a−2b+c=5 ( I )2 (a−2b+c )=(5 )2 ( II )2a+b−c=−1 ( II )2a+b−c=−1 ( III )3a+3b−2c=−4 ( III ) 3a+3b−2c=−4 A :3a−b=4 2a−4b+2c=10
B:5 a−b=6 A :−1 (3a−b )=4 (−1) B:5 a−b=6 A :3 (1 )−b=4 ( I ) (1 )−2 (−1 )+c=5 −3a+b=−4 3−b=4 1+2+c=5
2a2
=22 −3−3 3+c=5
a=1 −b−1
= 1−1 −3−3
(1 ,−1 ,2) b=−1 c=2
3) ( I )3 x+ y−z=11 ( I )−1 (3x+ y−z )=11(−1) ( I )−3 (3 x+ y−z )=11(−3) ( II ) x+3 y=z+13 ( II ) x+3 y−z=13 ( III ) x+ y−3 z=11 ( III ) x+ y−3 z=11 −3 x− y+z=−11 −9 x−3 y+3 z=−33
A : −2 x+2 y=2 B:−8 x−2 y=−22 ( II ) x+3 y=z+13 −z−z ( II ) x+3 y−z=13 A :−2x+2 y=2 A :−2 (2 )+2 y=2 ( I )3 (2 )+3−z=11 B:−8 x−2 y=−22 −4+2 y=2 6+3−z=11
−10x−10
=−20−10 +4+4 9−z=11
x=2 2 y2
=62 −9−9
(2 ,3 ,−2) y=3 −z−1
= 2−1
z=−2
5) ( I ) x+6 y+3 z=4 ( I ) x+6 y+3 z=4 ( II )2 (2 x+ y+2 z )= (3 )2 ( II )2x+ y+2 z=3 ( III )3 (3x−2 y+z )= (0 )3 ( III )3 x−2 y+z=0 ( III )3 x−2 y+z=0 x+6 y+3 z=4 4 x+2 y+4 z=6
9 x−6 y+3 z=0 B:7 x+5 z=6
80
A :10 x+6 z=4
A :−5 (10 x+6 z )=4(−5) A :10 (−2 )+6 z=4 ( I ) (−2 )+6 y+3 (4 )=4 B: 6 (7x+5 z )=6 (6) −20+6 z=4 −2+6 y+12=4 −50 x−30 z=−20 +20+20 10+6 y=4
42 x+30 z=36 6 z6
=246 −10−10
−8 x−8
= 16−8 z=4
6 y6
=−66
x=−2 (−2 ,−1,4 ) y=−1
7) ( I ) x+ y+z=6 ( I ) x+ y+z=6 ( II )2 (1 )− y−z=−3 ( II )2x− y−z=−3 ( II )2x− y−z=−3 2− y−z=−3
( III ) x−2 y+3 z=6 3x3
=33 −2 −2
x=1 A : − y−z=−5 A :3 (− y−z )= (−5 )3 B:−2 y+3 z=5 A :−(2 )−z=−5 ( III )1−2 y+3 z=6 −3 y−3 z=−15 +2 +2 −1 −1
−5 y−5
=−10−5
−z−1
=−3−1 B:−2 y+3 z=5
y=2 z=3 (1 ,2,3)
9) ( I ) x+ y−z=0 ( I ) x+ y−z=0 ( II ) x− y−z=0 ( II ) x− y−z=0 ( II ) x− y−z=0 ( III ) x+ y+2 z=0 ( III ) x+ y+2x=0 A :2 x−2 z=0 B: 2x+z=0
A : (−1 ) (2x−2 z )=0 (−1) A :2 x−2 (0 )=0 ( I )0+ y−0=0
B: 2x+z=0 2x2
=02 y=0
−2 x+2 z=0 x=0
3 z0
=0
x=0 (0 ,0 ,0)
81
11) ( I )−2 x+ y−3 z=1 ( I )−2 x+ y−3 z=1 ( I )2 (−2 x+ y−3 z )=(1 )2 ( II ) x−4 y+z=6 ( II )2 ( x−4 y+z )=6 (2 ) ( III )4 x+16 y+4 z=24 ( III )4 x+16 y+4 z=24 −2 x+ y−3 z=1 −4 x+2 y−6 z=2 2 x−8 y+2 z=12 B:18 y−2 z=26
A :−7 y−z=13
A :−2 (−7 y−z )=13(−2) A :−7 (0 )−z=13 ( I )−2 x+0−3 (−13 )=1
B:18 y−2 z=26 −z−1
= 13−1 −2 x+39=1
14 y+2 z=−26 z=−13 −39−39
32 y32
= 032
−2x−2
=−38−2
y=0 (19 ,0 ,−13 ) x=19
13) ( I )2 x+ y−3 z=0 ( I )4(2x+ y−3 z )=(0 )4 ( II )4 (x−4 y+ z)= (0 )4 ( II ) x−4 y+z=0 ( II ) x−4 y+z=0 ( III )4 x+16 y+4 z=0 ( III )4 x+16 y+4 z=0 −8 x+4 y−12 z=0 4 x−16 y+4 z=0
A :9 x−11 z=0 B: 8x+8 z=2 A :8 (9 x−11 z )=0 (8) B:−9 (8 x+8 z )=2(−9) A :9 x−11 (0 )=0 ( I )2 (0 )+ y−3 (0 )=0
72 x−88 z=0 9 x9
=09 y=0
−72 x−72 z=0 x=0
−160 z−160
=0 (0 ,0 ,0)
z=0
15) ( I )3 x+2 y+2x=3 ( II ) x+2 y−z=5 ( I )3 x+2 y+2x=3 ( II ) x+2 y−z=5 ( III )2x−4 y+z=0 ( II )2 ( x+2 y−z )=(5 )2 ( III )2x−4 y+z=0 A :3 x−2 y=5 3 x+2 y+2 z=3
2 x+4 y−2 z=10 A :3(3 x−2 y )=(5 )5 B:5 x+6 x=13 B:5 x+6 y=13
9 x−6 y=15 A :3 (2 )−2 y=5 ( I )3 (2 )+2( 12 )+2 z=314 x14
=2814 6−2 y=5 6+1+2 z=3
x=2 −6−6 7+2 z=3
−2 y−2
=−1−2 −7−7
82
y=12 2 z2
=−42
(2 , 12,−2) z=−2
17) ( I ) x−2 y+3 z=4 ( II )2x− y+z=−1 ( I ) x−2 y+3 z=4 ( II )2x− y+z=−1 ( III )4 x+ y+z=1 ( III )2 (4 x+ y+z )=(1 )2 ( III )4 x+ y+z=1 A :6 x+2 z=0 x−2 y+3 z=4
8 x+2 y+2 z=2 A :3(6 x+2 y)=(0 )3 B: 9x+5 z=6 B :−2 (9 x+5 z )=6(−2) 18 x+6 z=0 A :6 x+2 (3 )=0 ( I ) (−1 )−2 y+3 (3 )=4 −18 x−10 z=−12 6 x+6=0 8−2 y=4
−4 z−4
=−12−4 −6−6 −8−8
z=3 6 x6
=−66
−2 y−2
=−4−2
x=−1 y=2(−1 ,2 ,3)
19) ( I ) x− y+2 z=0 ( I ) (−1 ) ( x− y+2 z )=0(−1) ( I ) (−2 ) ( x− y+2 z )=0(−2) ( II ) x−2 y+3 z=−1 ( II ) x−2 y+3 z=−1 ( III )2x−2 y+z=−3 ( III )2x−2 y+z=−3 −x+ y−2 z=0 −2 x+2 y−4 z=0
A :− y+z=−1 −3 z−3
=−3−3
− y+(1 )=−1 z=1 −1−1
( I ) x− (2 )+2 (1 )=0 − y−1
=−2−1
x−2+2=0 y=2 x=0 (0 ,2 ,1)
21) ( I )4 x−3 y+2 z=40 ( I )3 (4 x−3 y+2 z)=(40 )3 ( I )8 (4 x−3 y+2 z )=(40 )8 ( II )5x+9 y−7 z=47 ( II )5x+9 y−7 z=47 ( III )3 (9 x+8 y−3 z )=(97 )3 ( III )9 x+8 y−3 z=97 12 x−9 y+6 z=120 32 x−24 y+16 z=320
A :17 x−z=167 27 x+24 y−9 z=291 B:59 x+7 z=611
A :7 (17 x−z )=(167 )7 A :17 (10 )−z=167 ( I )4 (10 )−3 y+2 (3 )=40
83
B:59 x+7 z=611 170− z=167 46−3 y=40 119 x−7 z=1169 −170−170 −46−46
178 x178
=1780178
−z−1
=−3−1
−3 y−3
=−6−3
x=10 z=3 y=2 (10 ,2 ,3)
23) ( I )3 x+3 y−2 z=13 ( II )6 x+2 y−5 z=13 ( I )2 (3 x+3 y−2 z )=(13 )2 ( II )6 x+2 y−5 z=13 ( III )5 x−2 y−5 z=−1( III )3 (5x−2 y−5 z )=(−1 )3 ( III ) 5 x−2 y−5 z=−1 A :11 x−10 z=12 6 x+6 y−4 z=26
15 x−6 y−15 z=−3 A :19 (11 x−10 z )=(12 )19 B: 21x−19 z=23 B: (−10 ) (21 x−19 z )=23(−10) 209 x−190 z=228 A :11 (2 )−10 z=12 ( I )3 (2 )+3 y−2 (1 )=13 −210+190 z=−230 22−10 z=12 4+3 y=13
−x−1
=−2−1 −22−22 −4−4
x=2 −10 z−10
=−10−10
3 y3
=93
(2 ,3 ,1) z=1 y=3
25) ( I )3 x−4 y+2 z=1 ( I )3 x−4 y+2 z=1 ( II )2x+3 y−3 z=−1 ( II )2x+3 y−3 z=−1 ( III )(−3)(x+10 y−8 z )=7 (−3) ( III )(−2)(x+10 y−8 z )=7(−2) ( III ) x+10 y−8 z=7 3 x−4 y+2 z=1 2 x+3 y−3 z=−1
−3 x−30 y+24 z=−21 −2 x−20 y+16 z=−14 A :−34 y+26 z=−20 B:−17 y+13 z=−15 A :−34 y+26 z=−20 B:−2 (−17 y+13 z )=−15(−2) −34 y+26 z=−20 34 y−26 z=30 false
0=10 No solutionθ
27) ( I )m+6n+3 p=8 ( II ) (−5 ) (3m+4n )=(−3)(−5) ( II )3m+4 (18 )=−3
84
( II )3m+4n=−3 ( III )3 (5m+7 n )=(1)3 3m+72=−3 ( III )5m+7n=1 −15m−20n=15 −72−72
15m+21n=3 3m3
=−753
n=18 m=−25 ( I ) (−25 )+6 (18 )+3 p=8
−25+108+3 p=883+3 p=8
−83−83 (−25 ,18 ,−25)
3 p3
=−753
p=−25
29) (I ) −2w+2 x+2 y−2 z=−10 ( II )w+x+ y+z=−5 ( III )3w+2 x+2 y+4 z=−11 ( IV )w+3x−2 y+2 z=−6
(I ) −2w+2 x+2 y−2 z=−10 ( I) (−1)(−2w+2 x+2 y−2 z )=(−10)(−1) ( II )(−2)(w+x+ y+z )=(−5)(−2) ( III )3w+2 x+2 y+4 z=−11 −2w+2 x+2 y−2 z=−10 2w−2x−2 y+2 z=10 −2w−2x−2 y−2 z=10 B:5w+6 z=−1
A :−4w−4 z=0
A :3 (−4w−4 z )=0(3) A :−4 (1 )−4 z=0B: 2 (5w+6 z )=(−1 )2 −4−4 z=0 −12w−12 z=0 +4+4
10w+12 z=−2 −4 z−4
= 4−4
−2w−2
=−2−2 z=−1
w=1
( III )3(1)+2x+2 y+4 (−1)=−11 ( IV )(1)+3 x−2 y+2(−1)=−6 3+2 x+2 y−4=−11 1+3 x−2 y−2=−6
85
2 x+2 y−1=−11 3 x−2 y−1=−6 +1+1 +1+1
C :2 x+2 y=−10 D :3x−2 y=−5
C :2 x+2 y=−10 C :2 (−3 )+2 y=−10D :3x−2 y=−5 −6+2 y=−10
5x5
=−155 +6+6
x=−3 2 y2
=−42
y=−2 (1 ,−3 ,−2 ,−1)
31) ( I )w+x+ y+z=2 ( II )w+2 x+2 y+4 z=1 ( III ) −w+x− y−z=−2 ( IV )−w+3 x+ y−z=−2 ( I )w+x+ y+z=2 ( III ) (−1 ) (−w+x− y−z )=(−6)(−1) ( III ) −w+x− y−z=−6 ( IV )−w+3 x+ y−z=−2
2x2
=−42 w−x+ y+z=6
x=−2 2 x+2 y=42 (−2 )+2 y=4−4+2 y=4
( II )w+2 x+2 y+4 z=1 +4+4
( IV )−2+3 x+ y−z=−2 2 y2
=82
5 x+3 y+3 z=−1 y=45 (−2 )+3 (4 )+3 z=−1−10+12+3 z=−1 ( I )w+(−2 )+( 4 )+(−1 )=22+3 z=−1 w+1=2
86
−2−2 −1−1
3 z3
=−33 w=1
z=−1 (1 ,−2 ,4 ,−1 )
4.5
1) A collection of dimes and quarters is worth S15.25. There are 103 coins in all. How many of each is there?
(−10 ) (D+Q )=(103)(−10)10D+25Q=1525
−10D−10Q=−1030
15Q15
=49515
−33−33 D=70 70dimes
33Quarters
3) The attendance at a school concert was 578. Admission was $2.00 for adults and $1.50 for children. The total receipts were S985.00. How many adults and how many children attended?
−2 (A+C )=(578 ) (−2 )2 A+1.5C=985
−2 A−2C=−1156 236 Adults
−0.5C−0.5
=−1156−0.5 342Children
A+342=578 C=342 −342−342
A=236
5) A boy has $2.25 in nickels and dimes. If there are twice as many dimes as nickels, how many of each kind has he?
5N+20N=225
25N25
=22525 9Nickels
N=9 18Dimes
D=2 (9 )=18
7) A collection of 27 coins consisting of nickels and dimes amounts to $2.25. How many coins of each kind are there?
87
N V TD 10 10DQ 25 25Q
103 1525
N V TA 2 2AC 1.5 1.5C
578 985
N V TN 5 5N
D=2N
10 20N
225
(−10 ) (N+D )=(27)(−10) 5N+10D=225
−10N−10D=−270
−5N−5
=−45−5
N=9 9+D=27 −9−9 18Dimes D=18 9Nickels
9) There were 429 people at a play. Admission was $1 each for adults and 75 cents each for children. The receipts were $372.50. How many children and how many adults attended?
(−1 ) (A+C )=(429)(−1) A+.75C=372.5
−A−C=−429
−.25C−.25
=−56.5−.25
A+226=429 C=226 −226−226
A=203 203 Adults226Children
11) There were 203 tickets sold for a volleyball game. For activity-card holders, the price was $1.25 each and for non-card holders the price was $2 each. The total amount of money collected was $310. How many of each type of ticket was sold?
−2 (A+N )=(203)(−2) 1.25 A+2N=310
−2 A−2N=−406
−.75 A−.75
=−96−.75
128+N=203 A=128 −128−128 N=75 75NonCard
128 Activity Card
13) At a recent Vikings game $445 in admission tickets was taken in. The cost of a student ticket was $1.50 and the cost of a non-student ticket was $2.50. A total of 232 tickets were sold. How many students and how many nonstudents attended the game?
−1.5 (5+N )=(232)(−1.5) 1.55+2.5N=445
88
N V TN S SND 10 10D27 225
N V TA 1 AC .75 .75C
429 372.50
N V TA 1.25 1.25AN 2 2N
203 310
N V T5 1.5 1.55N 2.5 2.5N
232 445
−1.55−1.5N=348 N=97
S+97=232 −97−97 97Non−Students S=135 135Students
15) A coin purse contains 18 coins in nickels and dimes. The coins have a total value of $1.15. Find the number of nickels and dimes in the coin purse.
−5 (N+D )=(18)¿ 5N+10D=115
−5N−5D=−90
5D5
=255
N+5=18 D=5 13Nickels −5−5 5Dimes N=13
17) ) A postal clerk sold some 15¢ stamps and some 25¢ stamps. Altogether, 15 stamps were sold for a total cost of $3.15. How many of each type of stamps were sold?
−15 (F+T )= (15 ) (−15 ) F+9=15 15 F+25T=315 −9−9
−15F−15T=−225 F=6
10T10
=9010
T=9 6 Fifteencents ,9 twenty−five cents
19) The total value of dimes and quarters in a bank is $6.05. There are six more quarters than dimes. Find the number of each type of coin in the bank.
10D+25D+150D=605 Q=13+6 35D+150=605 Q=19
−150−150
35D35
= 45535 13Dimes
D=13 19Quarters
89
N V TN 5 5ND 10 10018 115
N V TF 15 15FT 25 25T
15 315
N V TD 10 10D
Q=D+6 25 25D+150605
21) A coin bank contains nickels and dimes. The number of dimes is 10 less than twice the number of nickels. The total value of all the coins is $2.75. Find the number of each type of coin in the bank.
5N+20N−100=275 D=2 (15 )−10 25N−100=275 D=30−20
+100+100 D=10
25N25
=37525 20Dimes
N=15 15Nickels
23) A bank teller cashed a check for $200 using twenty dollar bills and ten dollar bills. In all, twelve bills were handed to the customer. Find the number of twenty dollar bills and the number of ten dollar bills.
−10 (W+T )=(12)(−10) 8+T=12 20W+10T=200 −8−8
−10W−10T=−120 T=4
10W10
=8010
W=8 4 Tens8Twenties
25) A total of $27000 is invested, part of it at 12% and the rest at 13%. The total interest after one year is $3385. How much was invested at each rate?
−.12 ( x+ y )=(27000)(−.12) x+14500=27000 .12 x+.13 y=3385 −14500−14500
−.12 x−.12 y=−3240 x=12500
−.01 y−.01
=−145−.01
y=14500 $12,500@12%$14,500@13%
27) A total of $9000 is invested, part of it at 10% and the rest at 12%. The total interest after one year is $1030. How much was invested at each rate?
−.1 ( x+ y )=(9000)(−.1) x+6500=9000 .1+.12 y=1030 −6500−6500
−.1 x−.1 y=−900 x=2500
.02 y.02
=130.02
y=6500 $2500@10%
90
N V TN 5 5N
D=2N−¿10
10 20N−¿100275
N V TW 20 20WT 10 10T
12 200
N V Tx .12 .12xy .13 .13y
27000 3385
N V Tx .10 .1xy .12 .12y
9000 1030
$6500@12%
29) An inheritance of $10000 is invested in 2 ways, part at 9.5% and the remainder at 11%. The combined annual interest was $1038.50. How much was invested at each rate?
−.095 ( x+ y )=(10000)(−.095) x+5900=10000 .095 x+.11 y=1038.50 −5900−5900
−.095 x−.095 y=−950 x=4100
0.015 y.015
=88.5.015
y=5900 [email protected]%$5900@11%
31) Jason earned $256 interest last year on his investments. If $1600 was invested at a certain rate of return and $2400 was invested in a fund with a rate that was double the rate of the first fund, find the two rates of interest.
1600 x+4800 x=256
6400 x6400
= 2566400
x=0.04 $1600@4% 2 x=0.08 $2400@8%
33) A total of $8500 is invested, part of it at 6% and the rest at 3.5%. The total interest after one year is $385. How much was invested at each rate?
−.035 ( x+ y )=(8500)(−.035) 3500+ y=8500 .06 x+ .035 y=385 −3500−3500
−.035 x−.035 y=−297.5 y=5000
.025 x.025
=87.5.025
x=3500 $3500@6%[email protected]%
35) A total of $15000 is invested, part of it at 8% and the rest at 11%. The total interest after one year is $1455. How much was invested at each rate?
−.08 ( x+ y )=(15000)(−.08) x+8500=15000 .08 x+.11 y=1455 −8500−8500
91
N V Tx .095 .095xy .11 .11y
10000 1038.50
N V T1600 x 1600x2400 2x 4800x
256
N V Tx .06 .06xy .035 .035y
8500 385
N V Tx .08 .08xy .11 .11y
15000 1455
−.08 x−.08 y=−1200 x=6500
.03 y.03
=255.03
y=8500 $6500@8%$8500@11%
37) A total of $6000 is invested, part of it at 4.25% and the rest at 5.75%. The total interest after one year is $300. How much was invested at each rate?
−.0425 ( x+ y )=(6000)(− .0425) x+3000=6000 .0425 x+.0575 y=300 −3000−3000
−.0425 x−.0425 y=−255 x=3000
.015 y.015
= 45.014
y=3000 [email protected]%[email protected]%
39) A total of $11000 is invested, part of it at 6.8% and the rest at 8.2%. The total interest after one year is $797. How much was invested at each rate?
−.068 ( x+ y )=(11000)(−.068) x+3500=11000 .068 x+.082 y=797 −3500−3500
−.068 x−.068 y=−748 x=7500
.014 y.014
= 49.014
y=3500 [email protected]%[email protected]%
42) Samantha earned $1480 in interest last year on her investments. If $5000 was invested at a certain rate of return and $11000 was invested in a fund with a rate that was two-thirds the rate of the first fund, find the two rates of interest.
3(5000 x+ 220003 x)=(1480 )3
15000 x+22000 x=4440 $5000@12%
92
N V Tx .0425 .0425xy .0575 .0575y
6000 300
N V Tx .068 .068xy .082 .082y
11000
797
N V T5000 x 5000x1100
023x
220003x
1480
37000 x37000
= 444037000 $11000@8%
x=.1223
( .12 )=.08
44) 30 coins having a value of $3.30 consists of nickels, dimes and quarters. If there are twice as many quarters as dimes, how many coins of each kind were there?
N+D+2D=30 (−5 ) (N+3D )=(30)(−5) 5N+10D+50D=330 5N+60D=330
−5N−15D=−150
45D45
=18045
N+3 (4 )=30 D=4 N+12=30
−12−12 Q=2 (4 )=8 N=18
18Nickels4 Dimes8Quarters
4.6
1) A tank contains 8000 liters of a solution that is 40% acid. How much water should be added to make a solution that is 30% acid?
3200=2400+.3w −2400−2400
800.3
= .3w.3
3) Of 12 pounds of salt water 10% is salt; of another mixture 3% is salt. How many pounds of the second should be added to the first in order to get a mixture of 5% salt?
1.2+.03 x=.6+.05 x −.03 x−.03 x
1.2=.6+ .02x
−.6−.6
.6.02
= .02 x.02
93
N V TN 5 5ND 10 10D
Q=2D 25 25D30 330
A P T8000 .4 3200
w 0 08000+
w.3 2400+.3w
A P T12 .1 1.2x .03 .03x
12+x .05 .6+.05x
x=30 lbs
5) How many pounds of a 4% solution of borax must be added to 24 pounds of a 12% solution of borax to obtain a 10% solution of borax?
.04 x+2.88=.1 x+2.4 −.04 x −.04 x
2.88=.06x+2.4
−2.4−2.4
x=8 lbs
7) A 100 LB bag of animal feed is 40% oats. How many pounds of oats must be added to this feed to produce a mixture which is 50% oats?
40+ x=50+.5 x −.5 x−.5 x
40+ .5x=50
−40−40.5x.5
=10.5
x=20 lbs
9) How many pounds of tea that cost $4.20 per pound must be mixed with 12 lb of tea that cost $2.25 per pound to make a mixture that costs $3.40 per pound?
4.2 x+27=3.4 x+40.8 −3.4 x−3.4 x
0.8 x+27=40.8
−27−27
0.8 x0.8
=13.8.8
x=12.25 lbs
94
A P Tx .04 .04x
24 .12 2.88x+24 .10 .1x+.24
A P T100 .4 40
x 1 X100+
x.5 50+.5x
A P Tx 4.2 4.2x
12 2.25 27x+1
23.40 3.4x+40.8
11) How many kilograms of hard candy that cost $7.50 per kilogram must be mixed with 24 kg of jelly beans that cost $3.25 per kilogram to make a mixture that sells for $4.50 per kilogram?
7.5 x+78=4.5x+108 −4.5x−4.5 x
3 x+78=108
−78−783x3
=303
x=10kg
13) How many pounds of lima beans that cost 90¢ per pound must be mixed with 16 lb of corn that cost 50¢ per pound to make a mixture of vegetables that costs 65¢ per pound?
.9 x+8=.65 x+10.4 −.65 x−.65 x
.25 x+8=10.4
−8−8
.25x.25
=2.4.25
x=9.6 lbs
15) Solution A is 50% acid and solution B is 80% acid. How much of each should be used to make 100cc. of a solution that is 68% acid?
−.5 (A+B )=(100)(−.5) A+60=100 .5 A+.8B=68 −60−60
−.5A−.5 B=−50 A=40
.3B.3
=18.3 60cc of 80%
B=60 40 ccof 50%
17) A farmer has some cream which is 21% butterfat and some which is 15% butter fat. How many gallons of each must be mixed to produce 60 gallons of cream which is 19% butterfat?
−.15 (A+B )=(60)(−.15) 40+B=60 .21 A+.15B=11.4 −40−40
95
A P Tx 7.5 7.5x
24 3.25 78x+24 4.5 4.5x+108
A P Tx .9 .9x
16 .5 8X+16 .65 .65x+10.4
A P TA .5 .5AB .8 .8B
100 .68 68
A P TA .21 .21A B .15 .15B60 .19 11.4
−.15 A−.15 B=−9 B=20
.06 A.06
=2.4.06
A=40 40 gal 21%20 gal15%
19) A chemist wants to make 50ml of a 16% acid solution by mixing a 13% acid solution and an 18% acid solution. How many milliliters of each solution should the chemist use?
−.13 ( x+ y )=(50)(− .13) x+30=50 .13 x+.18 y=8 −30−30
−.13 x−.13 y=−6.5 x=20
.05 y.05
=1.5.05
y=30 20mL13%30mL18%
21) A paint that contains 21% green dye is mixed with a paint that contains 15% green dye. How many gallons of each must be used to make 60 gal of paint that is 19% green dye?
−.15 ( x+ y )=(60)(−.15) 40+ y=60 .21 x+.15 y=11.4 −40−40
−.15 x−.15 y=−9 y=20
.06x.06
=2.5.06
x=40 40 gal 21%20 gal15%
23) To make a weed and feed mixture, the Green Thumb Garden Shop mixes fertilizer worth $4.00/lb. with a weed killer worth $8.00/lb. The mixture will cost $6.00/lb. How much of each should be used to prepare 500 lb. of the mixture?
−4 ( x+ y )=(500)(−4) x+250=500 4 x+8 y=3000 −250−250
−4 x−4 y=−2000 x=250
4 y4
=10004
y=250 250 lbs@$ 4250 lbs@$ 8
96
A P Tx .13 .13xy .18 .18y
50 .16 8
A P Tx .21 .21xy .15 .15y
60 .19 11.4
A P Tx 4 4xy 8 8y
500
6 3000
25) A grocer wishes to mix sugar at 9 cents per pound with sugar at 6 cents per pound to make 60 pounds at 7 cents per pound. What quantity of each must he take?
−6 ( x+ y )=(60)(−6) 20+ y=60 9 x+6 y=420 −20−20
−6 x−6 y=−360 y=40
3x3
=603
x=20 20 lbs@9¢40 lbs@6¢
27) A goldsmith combined an alloy that costs S4.30 per ounce with an alloy that costs $1.80 per ounce. How many ounces of each were used to make a mixture of 200 oz costing $2.50 per ounce?
−1.8 ( x+ y )=(200)(−1.8) 56+ y=200 4.3 x+1.8 y=500 −56−56
−1.8 x−1.8 y=−360 y=144
2.5x2.5
=1402.5
x=56 56oz .@$ 4.30144oz .@$1.80
29) The manager of a garden shop mixes grass seed that is 60% rye grass with 70 lb of grass seed that is 80% rye grass to make a mixture that is 74% rye grass. How much of the 60% mixture is used?
.6 x+56=.74 x+51.8 −.6 x−.6 x
56=.45x+51.8
−51.8−51.8
4.2.14
= .14 x.14
30 lbs=x
31) A caterer made an ice cream punch by combining fruit juice that cost $2.25 per gallon with ice cream that costs $3.25 per gallon. How many gallons of each were used to make 100 gal of punch costing $2.50 per pound?
−2.25 ( x+ y )=(100)(−2.25) x+25=100 2.25 x+3.25 y=250 −25−25
97
A P Tx 9 9xy 6 6y
60 7 420
A P Tx 4.30 4.3xy 1.80 1.80y
200 2.50 500
A P Tx .6 .6x
70 .8 56x+7
0.74 .74x+51.8
A P Tx 2.25 2.25xy 3.25 3.25y
100 2.5 250
−2.25 x−2.25 y=−225 x=75y=25 75 gal@$2.25
25 gal@$3.25
33) A carpet manufacturer blends two fibers, one 20% wool and the second 50% wool. How many pounds of each fiber should be woven together to produce 600 lb of a fabric that is 28% wool?
−.2 ( x+ y )=(600)(−.2) x+160=600 .2 x+.5 y=168 −160−160
−2. x−.2 y=−120 x=440
.3 y.3
=48.3
y=160 440 lbs@20%160 lbs@50%
35) The manager of a specialty food store combined almonds that cost $4.50 per pound with walnuts that cost S2.50 per pound. How many pounds of each were used to make a 100 lb mixture that cost $3.24 per pound?
−2.5 ( x+ y )=(100)(−2.5) 37+ y=100 4.5 x+2.5 y=324 −37−37
−2.5 x−2.5 y=−250 y=63
2x2
=742
x=37 37 lbs@$ 4.5063 lbs@$ 2.50
37) How many ounces of dried apricots must be added to 18 oz of a snack mix that contains 20% dried apricots to make a mixture that is 25% dried apricots?
x+3.6=.25 x+4.5 −.25 x−.25 x
.75 x+3.6=4.5
−3.6−3.6
.75x.75
=0.9.75
x=1.2oz
98
A P Tx .2 .2xy .5 .5y
600 .28 168
A P Tx 4.50 4.5xy 2.50 2.5y
100 3.24 324
A P Tx 1 x
18 .2 3.6x+18 .25 .25x+4.5
39) How many ounces of pure bran flakes must be added to 50 oz. of cereal that is 40% bran flakes to produce a mixture that is 50% bran flakes?
x+20=.5 x+25 −.5 x−.5 x
.5 x+20=25
−20−20.5x.5
= 5.5
x=10oz
41) How many grams of pure water must be added to 50 g of pure acid to make a solution that is 40% acid?
50=.4w+20 −20−20
30.4
= .4w.4
75 g=w
43) How many ounces of pure water must be added to 50 oz of a 15% saline solution to make a saline solution that is 10% salt?
7.5= .1x+5 −5−5
2.5.1
= .1 x.1
25oz=x
99
A P Tx 1 x
50 .4 20x+50 .5 .5x+25
A P Tw 0 050 1 50
w+50
.4 .4w+20
A P Tx 0 0
50 .15 7.5x+5
0.10 .1x+5
100
Chapter 5: Polynomials
101
5.1
1) 4 ∙44 ∙44=49
3) 4 ∙22=22 ∙22=24
5) 3m∙4mn=12m2n
7) 2m4n2 ∙4 nm2=8m6n3
9) (33 )4=312
11) (44 )2=48
13) (2u3 v2 )2=4u6 v4
15) (2a4 )4=24a16=16 a16
17)45
43=42
19) 32
3=3
21) 3nm2
3n=m2
23)4 x3 y4
3 x y3=4 x
2 y3
25) (x3 y4 ∙2 x2 y3 )2 (2 x5 y7 )2
22 x10 y14
4 x10 y14
27) 2 x (x4 y4 )4
2 x( x16 y16) 2 x17 y16
29)2 x7 y5
3x3 y ∙4 x2 y3= 2x7 y5
12 x5 y4= x2 y6
31) ( (2x )3
x3 )2
=( 23 x3
x3 )2
=( 8 x3
x3 )2
=82=64
33)
( 2 y17
(2 x2 y4 )4 )3
=( 2 y17
24 x8 y16 )3
=( 2 y17
16x8 y16 )3
=( y8x8 )
3
= y3
83 x24= y3
512 x24
35)
( 2mn4∙2m4n4
mn4 )3
=( 4m5n8mn4 )3
=(4m4n4 )3=43m12n12=64m12n12
37) 2x y5 ∙2x2 y3
2x y4 ∙ y3= 4 x
3 y8
2 x y7=2 x2 y
39)
rr q3 r2 ∙ (2 p2q2 r3 )2
2 p3=q3 r2 (22 p4q4 r2 )
2 p3= q3 r2 ∙4 p4 p4 r6
2 p3=4 p
4q7 r8
2 p3=2 pq7r 8
41)
( z y3 ∙ z3 x4 y4x3 y3 z3 )4
=( z4 y7 x4x3 y3 z3 )4
=(x y 4 z )4=x4 y16 z4
43)2x2 y2 z6 ∙2 z x2 y2
(x2 z3 )2=4 x
4 y4 z7
x4 z6=4 y4 z
102
5.2
1)
2 x4 y−2 (2 x y3 )4=2 x4 y−2 (24 x4 y12 )=25 x8 y10=32x8 y10
3)
(a4b−3 )32a3b−2=a12 b9 ∙2a3b−2=2a15b−11=2a15
b11
5) (2 x2 y2 )4 x−4=24 x8 y8 x−4=16 x4 y8 7) (x3 y4 )∧3 x−4 y 4=x9 y12 x04 y4=x5 y16
9)
2 x−3 y2
3x−3 y3 ∙3 x0= 2 y2 x3
x3 ∙3 y3 ∙3 x0=2 y
3 x3
9x3 y3= 29 y
11)4 x y−3∙ x−4 y0
4 y−1 =4 x y0 y4 y3 x4
= 4 xy4 y3 x4
= 1x3 y2
13)
u2v−1
2u0 v4 ∙2uv= u2
v ∙2u0v4 ∙2uv= u2
4u v6= u4 v6
15)u2
4u0v3∙3v2= u2
12v5
17)2 y
(x0 y2 )4= 2 yx0 y8
= 2y7
19)
( 2a2b3a−1 )4
=(2a2ab3 )4=(2a3b3 )4=24a12b12=16a12b12
21)2nm4
(2m2n2 )4= 2nm4
24m8n8= 123m4n7
= 18m4n7
23)(2mn )4
m0n−2=24m4n4
m0n−2 =24m4n4n2=16m4n6
25)
y3 ∙ x−3 y2
( x4 y2 )3= y3 x−3 y2
x12 y6= y3 y2
x3 x12 y6= y5
x15 y6= 1
x15 y
27)
2u−2v3 (2uv4 )−1
2u−4 v0=2u
−2 v3 ∙2−1u−1 v−4
2u−4 v0= 2v3 ∙u4
u22uv 4 ∙2v0=2 v
3u4
4u3v4= u2v
29) ( 2x0 y4y 4 )3
=(2 )3=8
31) y (2 x4 y2 )2
2 x4 y0= y (22 x8 y4 )
2x4 y0=4 x
8 y5
2 x4 y0=2 x4 y5
33)
2 yz x2
2x4 y4 z−2 ( z y2 )4= 2 yz x2
2 x4 y4 z−2 z4 y8= 2 y z3x2
2x4 y12 z4= 1
x2 y11 z
35)
2k h0 ∙2h−3k0
(2k j3 )2=2k h
0 ∙2h−3k0
22k2 j6= 2k ∙2h3 ∙4k2 j6
= 4k4 k2h3 j6
= 1k h3 j6
37)
(c b3 )2 ∙2a−3b2
(a3b−2c3 )3= c2b6 ∙2a−3b2
a9b−6 c9= c2b62b2b6
a3a9 c9=2b
14 c2
a12c9= 2b
14
a12c7
39)
( y x−4 z2 )−1
z3 x2 y3 z−1= y−1 x4 z−2
x3 x2 y3 z−1= x4 z
y z2 x3 x2 y3= x4 zx2 y4 z5
= x2
y4 z4
5.3
103
1) 885 8.85 x102
3) 0.081 8.1 x10−2
5) 0.039 3.9 x10−2
7) 8.7 x105 870,000
9) 9 x10−4 0.0009
11) 2 x100 2
13) (7 x10−1 ) (2x 10−3 ) 14 x10−4
1.4 x101 x10−4
1.4 x10−3
15) (5.26 x10−5)(3.16 x 10−2) 16.6216 x10−7
1.66216 x101 x10−7
1.66216 x10−6
17) (2.6 x10−2 ) (6 x10−2) 15.6 x10−4
1.56 x101 x10−4
1.56 x10−3
19) 4.9x 101
2.7 x10−3=1.81 x104
21)
5.33x 10−6
9.62x 10−2=0.554 x 10−4=5.54 x10−1 x 10−4=5.54 x10−5
23) (5.5 x10−5 )2
30.25 x10−10 3.025 x101 x10−10
3.025 x10−9
25) (7.8 x10−2 )5 28.872 x10−10
2.8872 x101x 10−10
2.8872 x10−9
27) (8.03 x104 )−4
0.000241 x10−16
2.41 x10−4 x 10−16 2.41 x10−20
29) 6.1x 10−6
5.1x10−4=1.196 x10−2
31) (3.6 x100 ) (6.1x 10−3 )
21.96 x10−3
2.196 x101 x10−3
2.196 x10−2
33) (1.8 x10−5 )−3 0.1715 x1015
1.715 x10−1 x1015
1.715 x1014
35)9 x104
7.83x 10−2=1.149 x106
104
37)
3.22 x10−3
7 x 10−6=0.46 x103=4.6 x10−1x 103=4.6x 102
39)
2.4 x106−66.5 x100
=0.3692 x10−6=3.692 x10−1 x10−6=3.692 x 10−7
41) 6 x103
5.8x 10−3=1.034 x106
5.4
1) −a3a2+6a−21at a=−4
−(−4 )3−(−4 )2+6 (−4 )−21 −(−64 )−(16 )+6 (−4 )−21 64−16−24−21 3
3) n3−7n2+15n−20whenn=2
(2 )3−7 (2 )2+15 (2 )−20 8−7 (4 )+15 (2 )−20 8−28+30−20 −10
5) −5n4−11n3−9n2−n−5whenn=−1
−5 (−1 )4−11 (−1 )3−9 (−1 )2− (−1 )−5 −5 (1 )−11 (−1 )−9 (1 )−1 (−1 )−5 −5+11−9+1−5
−7
7) x2+9 x+23when x=−3 (−3 )2+9 (−3 )+23 9+9 (−3 )+23 9−27+23 5
9) x4−6 x3+x2−24whenx=6
(6 )4−6 (6 )3+ (6 )2−24 1296−6¿ 1296−1296+36−24 12
11) (5 p−5 p4 )−(8 p−8 p4)
5 p−5 p5−8 p+8 p4
3 p4−3 p
13) (3n2+n3 )−(2n3−7n2) 3n2+n3−2n3−7n2
−n3+10n2
15) (8n+n4 )−(3n−4 n4) 8n+n4−3n+4n4
5n4+5n
17) (1+5 p3 )−(1−8 p3) 1+5 p3−1+8 p3
13 p3
19) (5n4+6n3 )+(8−3n3−5n4) 3n3+8
21) (3+b4 )+(7+2b+b4) 2b4+2b+10
23) (8 x3+1 )−(5 x4−6 x3+2) 8 x3+1−5 x4+6 x3−2 −5 x4+14 x3−1
25) (2a+2a4 )−(3a2−5a4+4a) 2a+2a4−3a2+5a4−4a
105
7a4−3a2−2a
27) (4 p2−3−2 p )−(3 p2−6 p+3) 4 p2−3−2 p−3 p2+6 p−3 p2+4 p−6
29) (4b3+7b2−3 )+(8+5b2+b3) 5b3+12b2+5
31) (3+2n2+4n4 )+(n3−7n2−4n4)
n3−5n2+3
33) (n−5n4+7 )+(n2−7n4−n) −12n4+n2+7
35) (8 r 4−5 r3+5 r2 )+(2r 2+2 r3−7 r4+1) r 4−3 r3+7 r2+1
37) (2n2+7n4−2 )+(2+2n3+4 n2+2n4) 9n4+2n3+6n2
39) (8−b+7 b3 )−(3b4+7b−8−7b2 )+(3−3b+6b3 ) 8−b+7b3−3b4−7b+8+7b2+3−3b+6b3
−3b4+13b3−7b2−11b+19
41) (8 x4+2 x3+2x )+(2x+2−2x3−x4)−(x3+5 x4+8 x ) 8 x4+2 x3+2 x+2x+2−2x3−x4−x3−5 x4−8 x 2 x4−x3−4 x+2
5.5
1) 6( p−7) 6 p−42
3) 2(6 x+3) 12 x+6
5) 5m4(4m+4) 20m5+20m4
7) (4n+6)(8n+8) 32n2+32n+48n+48 32n2+80n+48
9) (8b+3)(7 b−5) 56b2−40b+21b−15 56b2−19b−15
11) (4 x+5)(2 x+3) 8 x2+12x+10 x+15 8 x2+22x+15
13) (3v−4)(5v−2) 15v2−6v−20 v+8
106
15v2−26v+815) (6 x−7)(4 x+1)
24 x2+6 x−28 x−7 24 x2−22 x−7
17) (5 x+ y)(6 x−4 y ) 30 x2−20 xy+6 xy−4 y2
30 x2−14 xy−4 y2
19) (x+3 y)(3 x+4 y) 3 x2−4 xy+9xy+12 y2
3 x2+13 xy+12 y2
21) (7 x+5 y)(8 x+3 y ) 56 x2+21xy+40xy+15 y2
56 x2+61xy+15 y2
23) (r−7)(6 r2−4+5) 6 r3−r2+5 r−42r2+7 r−35 6 r3−43 r2+12 r−35
25) (6n−4 )(2n2−2n+5) 12n3−12n2+30n−8n2+8n−20 12n3−20n2+38n−20
27) (6 x+3 y)(6x2−7 xy+4 y2)
36 x3−42x2 y+24 x y2+18x2 y−21 x y2+12 y3
36 x3−24 x2 y+3 x y2+12 y3
29) (8n2+4n+6)(6 n2−6n+6) 48n4−40n3+48n2+24 n3−20n2+24 n+36n2−30n+36 48n4−16 n3+64n2−6 n+36
31) (5k2+3k+3)(3k 2+3 k+6) 15k4+15k3+30k2+9 k3+9k 2+18 k+9k2+9k+18 15k4+24k3+48 k2+27k+18
33) 3(3 x−4)(2 x+1) 3(6 x2+3 x−8 x−4) 3(6 x2−5 x−5) 18 x2−15x−12
35) 3(2 x+1)(4 x−5) 3(8x2−10 x+4 x−5) 3(8x2−6 x−5) 24 x2−18 x−15
37) 7(x−5)( x−2) 7(x2−2x−5 x+10) 7(x2−7 x+10) 7 x2−49 x+70
39) 6(4 x−1)(4 x+1) 6(16x2+4 x−4 x−1) 6(16x2−1) 96 x2−6
5.6
107
1) (x+8)(x−8) x2−64
3) (1+3 p)(1−3 p) 1−9 p2
5) (1−7 n)(1+7 n) 1−49n2
7) (5n−8)(5n+8) 25n2−64
9) (4 x+8)(4 x−8) 16 x2−64
11) (4 y−x )(4 y+x ) 16 y2−x2
13) (4m−8n)(4m+8n) 16n2−64n2
15) (6 x−2 y )(6 x+2 y ) 36 x2−4 y2
17) (a+5 )2
2 (5a )=10a a2+10a+25
19) ( x−8 )2 2 (−8 x )=−16 x x2−16 x+64
21) ( p+7 )2 2 (7 p )=14 p p2+14 p+49
23) (7−5n )2 2 (−35n )=−70n 49−70n+25n2
25) (5m−8 )2 2 (−40m )=−80m 25m2−80m+64
27) (5 x+7 y )2 2 (35xy )=70 xy
25 x2+70 xy+49 y2
29) (2 x+2 y )2
2 (4 xy )=8 xy 4 x2+8 xy+4 x= y2
31) (5+2 r )2 2 (10 r )=20 r 25+20 r+4 r2
33) (2+5 x )2 2 (10x )=20x 4+20 x+25x2
35) (4 v−7)(4 v+7)
16v2−49
37) (n−5)(n+5) n2−25
39) (4 k+2 )2
2 (8k )=16 k 16k2+16k+4
5.7
1)20x4+ x3+2 x2
4 x3=20 x
4
4 x3+ x3
4 x3+ 2 x
2
4 x3=5 x+ 1
4+ 12 x
3) 20n4+n3+40n2
10n=20n
4
10n+ n3
10n+ 40n
2
10n=2n3+ n2
10+4 n
5) 12x4+24 x3+3 x2
6 x=12 x
4
6 x+ 24 x
3
6x+ 3 x
2
6 x=2 x3+4 x2+ x
2
7)10n4+50n3+2n2
10n2=10n
4
10n2+ 50n
3
10n2+ 2n
2
10n2=n2+5n+ 1
5
108
9) x2−2x−71x+8
x−10+ 9x+8
x+8 x2−2 x−71 −x2+(−8 x )
−10 x−71 +10 x+80
9
11) n2+13n+32n+5
n+8− 8n+5
n+5n2+13n+32 −n2−5n
8n+32 −8n−40
−8
13) v2−2v−89v−10
v+8− 9v−10
v−10 v2−2 v−89 −v2+10 v
8 v−89 −8 v+80
9
15) a2−4a−38a−8
a+4− 6a−8
a−8a2−4 a−38 −a2+8a
4 a−38 −4a+32
−6
17) 45 p2+56 p+199 p+4
5 p+4+ 39 p+4
9 p+445 p2+56 p+19 −45 p2−20 p
36 p+19 −36 p−16
3
109
19) 10x2−32 x+910 x−2
x−3+ 310 x−2
10 x−210x2−32 x+9−10 x2+2 x
−30 x+9+30 x−6
3
21) 4 r2−r−14 r+3
r−1+ 24 r+3
4 r+34 r2−r−1 −4 r2−3 r
−4 r−1 +4 r+3
2
23) n2−4n−2
n+2
n−2n2−0n−4 −n2−2n
−2n−4 +2n+4
0
25) 27b2+87b+353b+8
9b+5− 53b+8
3b+827b2+87b+35 −27b2−72b
15b+35 −15b−40
−5
27) 4 x2−33 x+284 x−5
x−7− 74 x−5
4 x−54 x2−33 x+28 −4 x2+5 x
28 x+28 −28 x−35
110
−7
29) a3+15a2+49a−55a+7
a2+8a−7− 6a+7
a+7a3+15a2+49 a−55 −a3−7a2
8a2+49a −8a2−56a
−7a−55 +7a+55
0
31) x3−26 x−41x+4
x2−4 x−10− 1x+4
x+4 x3−0x2−26 x−41 −x3−4 x2
−4 x2−26 x +4 x2+16 x
−10 x−41 +10 x+40
1
33) 3n3+9n2−64 n−68
n+63n2−9n−10− 8
n+6 n+63n3+9n2−64n−68 −3n3−18n2
−9n2−64n +9n2+54 n
−10n−68 +10n+60
−8
35) x3−46 x+22x+7
x2−7 x+3+ 1x+7
x+7 x3+0 x2−46 x+22
111
−x3−7 x −7 x−46 x +7 x+49 x
3 x+22 −3 x−21
1
37) 9 p3+45 p2+27 p−5
9 p+9 p2+4 p−1+ 4
9 p+9 9 p+99 p3+45 p2+27 p−5 −9 p3−9 p
36 p2+27 p −36 p2−36 p
−9 p−5 +9 p+9 4
39) r3−r2−16 r+8r−4
r2+3 r−4− 8r−4
r−4 r 3−r2−16 r+8 −r3+4 r2
3 r2−16 r −3 r2+12 r
−4 r+8+4 r−16
−8
41) 12n3+12n2−15n−42n+3
6n2−3n−3+ 52n+3
2n+3 12n3+12n2−15n−4 −12n3−18n2
−6n2−15n+6n2+9n
−6n−4+6n+9 5
112
43) 4 v3−21v2+6 v+194 v+3
v2−6 v+6+ 14v+3
4 v+34 v3−21 v2+6 v+19 −4v3−3 v2
−24 v2+6v +24 v2+18 v
24 v+19 −24 v−18
1
113
Chapter 6: Factoring
114
6.1
1) 9+8x 1(9+8x )
3) 45 x2−25 5(9x2−5)
5) 56−35 p 7(8−5 p)
7) 7ab−36a2b 7ab (1−5a)
9) −3a2b+6 a3b2 −3a2b(1−2ab)
11) −5 x2−5 x3−15 x4 −5 x2(1+x+3 x2)
13) 20 x4−30 x+30 10(2 x4−3x+3)
15) 28m4+40m3+8 4 (7m4+10m3+2)
17) 30b9+5ab−15a2 5(6 b9+ab−3a2)
19) −48a2b2−56a3b−56a5b
−8a2b (6b+7a+7a3 )
21) 20 x8 y2 z2+15 x5 y2 z+35x3 y3 z 5 x3 y2 z(4 x5 z+3 x2+7 y)
23) 50 x2 y+10 y2+70x z2
10(5 x2 y+ y2+7 x z2)
25) 30qpr−5qp+5q 5q (6 pr−p+1)
27) −18n5+3n3−21n+3 −3(6n5−n3+7n−1)
29) −40x11−20x12+50 x13−50 x14 −10 x11(4+22 x−5 x2+5 x3)
31) −32mn8+4m6n+12mn4+16mn −4mn(8n7−m5−3n3−4)
6.2
1) 40 r 3−8 r2−25 r+5 8 r2 (5 r−1 )−5 (5 r−1) (5 r−1)(8 r2−5)
3) 3n2−2n2−9n+6 n2 (3n−2 )−3(3n−2) (3n−2)(n2−3)
5) 15b3+21b2−35b−49 3b2 (5b+7 )−7(5b+7)
(5b+7)(3b2−7)
7) 3 x3+15 x2+2x+10 3 x2 ( x+5 )+2(x+5) (x+5)(3 x2+2)
9) 35 x3−28 x2−20 x+16 7 x2 (5 x−4 )−4 (5 x−4)
115
728 9 17
8-8 -1 -9
-10-10 1 -9
324 8 12
-70 10 -7 3
15-3 -5 -8
(5 x−4)(7 x2−4)
11) 7 xy−49 x+5 y−35 7 x ( y−7 )+5( y−7) ( y−7)(7x+5)
13) 32 xy+40 x2+12 y+15 x 8 x (4 y+5x )+3(4 y+5x ) (4 y+5 x)(8 x+3)
15) 16 xy−56 x+2 y−7 8 x (2 y−7 )+1(2 y−7) (2 y−7)(8 x+1)
17) 2 xy−8 x2+7 y3−28 y2 x 2 x ( y−4 x )+7 y2( y−4 x) ( y−4 x )(2x+7 y2)
19) 40 xy+35 x−8 y2−7 y 5 x (8 y+7 )− y (8 y+7) (8 y+7)(5 x− y )
21) 32uv−20u+24 v−15 4 u (8 v−5 )+3(8v−5) (8 v−5)(4u+3)
23) 10 xy+30+25 x+12 y 10 ( xy+3 )+1(25 x+12 y ) No! 10 xy+25 x+12 y+30 5 x (2 y+5 )+6(2 y+5) (2 y+5)(5x+6)
25) 3uv+14u−6u2−7v u (3v+14 )−1(6u2+7 v) No! 3uv−6u2−7v+14u 3u (v−2u )−7(v−2u) (v−2u)(3u−7)
27) 16 xy−3 x−6 x2+8 y x (16 y−3 )−1(6x2−8 y) No! 16 xy−6 x2+8 y−3 x 2 x (8 y−3 x )+1(8 y−3x ) (8 y−3x )(2x+1)
6.3
1) p2+17 p+72 ( p+8)( p+9)
3) n2−9n+8 (n−8)(n−1)
5)x2−9 x−10
(x−10)(x+1)
7)b2+12b+32
9) x2+3 x−70 (x+10)(x−7)
11) n2−8n+15 (n−3)(n−5)
116
549 6 15
56-7 -8 -15
15-5 -3 -8
-84 -2 2
18-9 -2 -11
-124 -3 1
-126 -2 4
2010 2 12
-328 -4 3
21 2 3
639 7 16
252-6 -42 -48
141 14 15
-1407 -20 -13
4-4 -1 -5
7014 5 19
13) p2+15 p+54 ( p+9 )(p+6)
15) n2−15n+56 (n−7)(n−8)
17) u2−8uv+15 v2
(u−5v )(u−3 v)
19) m2−2mn−8n2
(m+4 n)(m−2n)
21) x2−11 xy−12 y2 ( x−9 y )(x−2 y )
24) x2+ xy−12 y2 (x+4 y)(x−3 y )
26) x2+4 xy−12 y2
(x+6 y )( x−2 y )
27) 5a2+60a+100 5(a2+12a+20) 5(a+10)(a+2)
29) 6a2+24 a−192 6(a2+4 a−32) 6(a+8)(a−4)
31) 6 x2+18xy+12 y2
6( x2+3 xy+2 y2) 6( x+ y )(x+2 y)
35) 6 x2+96 xy+378 y2
6( x2+16 xy+63 y2) 6( x+9 y )(x+7 y)
6.4
1) 7 x2−48 x+36 7 x2−6 x−42x+36 x (7 x−6 )−6 (7 x−6) (7 x−6)(x−6)
3) 7b2+15b+2 7b2+b+14b+2 b (7b+1 )+2(7b+1) (7b+1)(b+2)
5)5a2−13a−28
5a2+7a−20a−28
a (5a+7 )−4(5 a+7) (5a+7)(a−4)
7)
2 x2−5x+2
2 x ( x−2 )−1(x−2) (x−2)(2 x−1)
9) 2 x2+19 x+35 2 x2+14 x+5 x+35 2 x ( x+7 )+5(x+7) (x+7)(2x+5)
117
-32 -3 -1
3010 3 13
60-12 -5 -17
3015 2 17
-24535 -7 28
-14-14 1 -13
-2106 -35 -29
56-2 -28 -30
1205 24 29
16-16 -1 -17
88 1 9
-36 -12 3 -9
1212 1 13
21021 10 31
11) 2b2−b−3 2b2+2b−3b−3 2b (b+1 )−3(b+1) (b+1)(2b−3)
13) 5k2+13k+6 5k2+10k+3k+6 5k (k+2 )+3 (k+2) (k+2)(5k+3)
15) 3 x2−17 x+20 3 x2−12x−5 x+20 3 x ( x−4 )−5(x−4) (x−4)(3 x−5)
17) 3 x2+17 xy+10 y2
3 x2+15 xy+2 xy+10 y2
3 x ( x+5 y )+2 y ( x+5 y ) (x+5 y)(3 x+2 y)
19) 5 x2+28 xy−49 y2
5 x2+35 xy−7xy−49 y2
5 x ( x+7 y )−7 y (x+7 y ) (x+7 y)(5x−7 y)
21) 6 x2−39 x−21 3(2 x2−13 x−7) 3(2 x2−14 x+x−7) 3(2 x (x−7 )+1 (x−7 )) 3(x−7)(2x+1)
23) 21k2−87 k−90 3(7 k2−29k−30) 3(7 k2+6k−35 k−30)
3(k (7 k+6 )−5 (7 k+6 )) 3(7 k+6)(k−5)
25) 14 x2−60 x+16 2(7 x2−30 x+8) 2(7 x2−2 x−28 x+8) 2 (x (7 x−2 )−4 (7 x−2 ) ) 2 (7 x−2 ) (x−4 )
28) 6 x2+29 x+20 6 x2+5x+24 x+20 x (6 x+5 )+4(6x+5) (6 x+5)(x+4)
30) 4 k2−17 k+4 4 k2−16 k−k+4 4 k (k−4 )−1(k−4 ) (k−4 )(4 k−1)
33) 4 x2+9 xy+2 y2
4 x2+8 xy+xy+2 y2
4 x ( x+2 y )+ y (x+2 y ) (x+2 y)(4+ y )
33) 4m2−9mn−9n2
4m2−12mn+3mn−9n2
4m (m−3n )+3n(m−3n) (m−3n)(4m+3n)
37) 4 x2+13 xy+3 y2
4 x2+12 xy+xy+3 y2
4 x ( x+3 y )+ y (x+3 y ) (x+3 y)(4 x+ y)
39) 12 x2+62 xy+70 y2
2(6 x2+31xy+35 y2) 2(6 x2+21xy+10 xy+35 y2) 2 (3 x (2 x+7 y )+5 y (2x+7 y ) )
118
12-12 -1 -13
11 1 2
93 3 6
9-3 -3 -3
25-5 -5 -10
22515 15 30
100-10 -10 -20
36-6 -6 -12
2(2 x+7 y)(3 x+5 y)
40) 24 x2−52 xy+8 y2
4 (6 x2−13 xy+2 y2 )
4 (6 x2−12 xy−xy+2 y2) 4 (6 x ( x−2 y )− y ( x−2 y ) ) 4 (x−2 y )(6 x− y )
6.5
1) r2−16 (r )(4) (r+4)(r−4)
3) v2−25 ( v )(5) (v+5)(v−5)
5) p2−4
( p+2)( p−2)
7) 9k 2−4
(3k+2)(3k−2)
9) 3 x2−27 3(x2−9) ( x )(3) 3(x+3)(x−3)
11) 16 x2−36 4 (4 x2−9) (2 x )(3) 4 (2 x+3)(2x−3)
13) 18a2−50b2
2 (9a2−25b2 ) (3a ) (5b ) 2(3a+5b)(3a−5b)
15) a2−2a+1
(a−1 )2
17) x2+6 x+9 ( x+3 )2
19) x2−6 x+9 ( x+3 )2
21) 25 p2−10 p+1
(5 p−1 )2
23) 25a2+30ab+9b2 (5a+3b )2
25) 4 a2−20 ab+25b2
(2a−5b )2
27) 8 x2−24 xy+18 y2
2 (4 x2−12 xy+9 y2 ) 2 (2x−3 y )2
29) 8−m3
(2 )(m) (2−m)(4+2m+m2)
31) x3−64 ( x )(4 ) (x−4)(x2+4 x+16)
33) 216−u3
(6 )(u) (6−u)¿
119
20-4 -5 -9
-3010 -3 7
35) 125a3−64 (5a )(4) (5a−4 )(25a2+20a+16)
37) 64 x3+27 y3
(4 x )(3 y) (4 x+3 y ) (16 x2−12xy+9 y2)
39) 54 x3+250 y3
2 (27 x3+125 y3 ) (3 x )(5 y) 2(3 x+5 y)¿
41) a4−81 (a2 )(9) (a2+9)(a2−9) (a )(3) (a2+9)(a+3)(a−3)
43) 16−z4
(4 ) ( z2 ) (4+z2)(4−z2)
(2 )(z ) (4+z2)(2+z )(2−z )
45) x4− y4
(x2 ) ( y2 ) (x2+ y2 )( x2− y2)
( x ) ( y ) (x2+ y2)(x+ y )(x− y )
47) m4−81b4
(m2 ) (9b2 ) (m2+9b2 ) (m2−9b2 )
(m ) (3b ) (m2+9b2)(m+3b)(m−3b)
6.6
1) 24 az−18ah+60 yz−45 yh 3(8az−6ah+20 yz−15 yh) 3¿ 3(4 z−3h)(2a+5 y )
3) 5u2−9uv+4v2
5u2−4uv−5uv+4 v2
u (5u−4v )−v (5u−4 v ) (5u−4 v)(u−v)
5) −2 x3+128 y3
−2 (x3−64 y3 ) ( x ) (4 y )
−2 ( x−4 y ) (x2+4 xy+16 y2 )
7)
5n3+7n2−6n
n(5 n2+7 n−6) n(5 n2+10n−3n−6) n (5n (n+2 )−3 (n+2 ) ) n (n+2 ) (5n−3 )
9) 54u3−16 2(27u3−8)
120
3-1 -3 -4
-105 -2 3
105 2 7
20-4 -5 -9
16-4 -4 -8
-6012 -5 7
-105 -2 3
(3u )(2) 2 (3u−2 ) (9u2+6u+4 )
11) n2−n n(n−1)
13) x2−4 xy+3 y2
x2−xy−3 xy+3 y2
x (x− y )−3 y (x− y ) (x− y )(x−3 y )
15) 9 x2−25 y2
(3 x )(5 y) (3 x+5 y)(3 x−5 y )
17) m2−4n2
(m )(2n) (m+2n)(m−2n)
19) 36b2 c−16 xd−24b2d+24 xc 4 (9b2c−4 xd−6b2d+6 xc) 4 ¿ 4 (9b2c−6b2d+6 xc−4 xd ) 4 ¿ 4 (3c−sd )(3b2+2 x)
21) 128+54 x3 2(64+27 x3) (4 )(3 x) 2 (4+3x ) (16−12x+9 x2 )
23) 2 x3+6 x2 y−20 y2 x
2 x (x2+3 xy−10 y2 ) 2 x ( x+5 y ) ( x−2 y )
25) n3+7n2+10n n(n2+7n+10) n(n+5)(n+2)
27) 27 x3−64 (3 x )(4) (3 x−4)(9 x2+12 x+16)
29) 5 x2+2 x x (5 x+2)
31) 3k3−27k2+60k 3k (k2−9k+20) 3k (k−4)(k−5)
33) mn−12 x+3m−4 xn 1 (mn−12x )+1(3m−4 xn) mn+3m−4 xn−12 x m (n+3 )−4 x (n+3) (n+3 ) (m−4 x )
35) 16 x2−8 xy+ y2
(4 x− y )2
37) 27m2−48n2
3 (9m2−16n2 ) (3m )(4 n) 3(3m+4n)(3m−4 n)
39) 9 x3+21x2 y−60 y2x 3 x (3 x2+7 xy−20 y2 ) 3 x (3 x2+12 xy−5 xy−20 y2) 3 x (3 x ( x+4 y )−5 y ( x+4 y ) ) 3 x ( x+4 y ) (3 x−5 y )
41) 2m2+6mn−20n2
2(m2+3mn−10n2) 2 (m+5n ) (m−2n )
121
-14 7 -2 5
1055 21 26
-10-10 1 -9
4-4 -1 -5
168-3 56 53
6.7
1) (k−7 ) (k+2 )=0 k−7=0k+2=0 +7+7−2−2 k=7k=−2
3) ( x−1 ) ( x+4 )=0 x−1=0x+4=0 +1+1−4−4 x=1 x=−4
5) 6 x2−150=0
6 ( x+5 ) ( x−5 )=0 x+5=0 x−5=0 −5−5+5+5 x=−5 x=5
7)2n2+10n−28=0
2 (n+7 ) (n−2 )=0 n+7=0n−2=0 −7−7+2+2 n=−7n=2
9) 7 x2+26x+15=0 7 x2+5x+21 x+15=0 x (7 x+5 )+3 (7 x+5 )=0 (7 x+5 ) (x+3 )=0 7 x+5=0 x+3=0
−5−5−3−3
7 x7
=−57
x=−3
x=−57
11) 5n2−9n−2=0 5n2−10n+n−2=0 5n (n−2 )+1 (n−2 )=0 (n−2 ) (5n+1 )=0 n−2=05n+1=0 +2+2−1−1
n=2 5n5=−15
n=−15
13) x2−4 x−8=−8 +8+8
x2−4 x=0 x (x−4 )=0 x=0 x−4=0 +4+4
x=4
15) x2−4 x−1=−5 +5+5
x2−4 x+4=0 ( x−4 ) ( x−1 )=0 x−4=0 x−1=0 +4+4+1+1 x=4 x=1
17) 49 p2+371 p−241=0 7 (7 p2+53 p−24 )=0
122
-84-4 21 17
124 3 7
-16-8 2 -6
-12015 -8 7
633 21 24
-44 -1 3
7 (7 p2−3 p+56 p−24 )=0 7¿ 7 (7 p−3 ) ( p+8 )=0 7 p−3=0 p+8=0 +3+3−8−8
7 p7
=37p=−8
p=37
19) 7 x2+17x−20=−8 +8+8 7 x2+17x−12=0 7 x2−4 x+21 x−12=0 x (7 x−4 )+3 (7 x−4 )=0 (7 x−4 ) ( x+3 )=0 7 x−4=0x+3=0 +4+4−3−3
7 x7
=47x=−3
x=47
21) 7 r2+84=−49 r +49 r+49 r 7 r2+49 r+84=0 7 (r2+7 r+12 )=0 7 (r2+7 r+12 )=0 7 (r+4 ) (r+3 )=0 r+4=0 r+3=0 −4−4−3−3 r=−4 r=−3
23) x2−6 x=16 −16−16 x2−6 x−16=0 ( x−8 ) ( x+2 )=0 x−8=0 x+2=0
+8+8−2−2 x=8 x=−2
25) 3v2+7 v=40 −40−40 3v2+7 v−40=0 3v2+15v−8v−40=0 3v (v+5 )−8 (v+5 )=0 ( v+5 ) (3 v−8 )=0 v+5=03 v−8=0 −5−5+8+8
v=−5 3 v3
=83
v=83
27) 35 x2+120 x=−45 +45+45 35 x2+120 x+45=0 5 (7 x2+24 x+9 )=0 5 (7 x2+3 x+21 x+9 )=0 5 (x (7 x+3 )+3 (7 x+3 ) )=0 5 (7 x+3 ) ( x+3 )=0 7 x+3=0 x+3=0 −3−3−3−3
7 x7
=−37
x=−3
x=−37
29) 4 k2+18k−23=6 k−7 −6k+7−6 k+7 4 k2+12k−16=0 4 (k2+3k−4 )=0 4 (k+4 ) (k−1 )=0 k+4=0k−1=0 −4−4+1+1 k=−4 k=1
123
805 16 21
-120-4 30 26
31) 9 x2−46+7 x=7 x+8x2+3 −8 x2−3−7 x−7 x−8 x2−3 x2−49=0 ( x+7 ) ( x−7 )=0 x+7=0x−7=0 −7−7+7+7 x=−7 x=7
33) 2m2+19m+40=−2m +2m+2m 2m2+21m+40=0 2m2+4m+16m+40=0 m ¿ (2m+5 ) (m+8 )=0 2m+5=0m+8=0 −5−5−8−8
2m2
=−52
m=−8
m=−52
35) 40 p2+183 p−168=p+5 p2
−5 p2−p−p−5 p2
35 p2+182 p−168=0 7 (5 p2+26 p−24 )=0 7 (5 p2−4 p+30 p−24 )=0 7 ( p (5 p−4 )+6 (5 p−4 ) )=0 7 (5 p−4 ) ( p+6 )=0 5 p−4=0 p+6=0 +4+4−6−6
5 p5
=45 p=−6
p= 45
124
Chapter 7: Rational Expressions
125
7.1
1) 3k2+30 kk+10
k+10≠0 −10−10 k ≠−10
3) 15n2
10n+25 10n+25≠0 −25−25
10n10
≠−2510
n≠−52
5) 10m2+8m10m
10m10
≠ 010
m≠0
7) r2+3 r+125 r+10
5 r+10≠0 −10−10
5 r5≠−10
5 r ≠−2
9)b2+12b+32b2+4b−32
b2+4b−32≠0 (b+8)(b−4)≠0 b+8≠0b−4 ≠0 −8−8+4+4 b≠−8b≠4
11) 21x2
18 x=7 x6
13)24 a40a2
= 35a
15)32x3
8 x4=4x
17) 18m−2460
=6 (3m−4 )60
= 3m−410
19)204+2 p
= 202 (2+ p )
= 102+ p
21)x+1
x2+8 x+7= x+1
( x+7 ) (x+1 )= 1
x+7
23)32x2
28x2+28 x= 32x2
28 x (x+1 )= 8 x7 ( x+1 )
25)n2+4 n−12n2−7 n+10
=(n+6 ) (n−2 )(n−5 ) (n−2 )
= n+6n−5
27)9v+54
v2−4v−60=
9 (v+6 )(v−10 ) ( v+6 )
= 9v−10
29)12x2−42x30x2−42x
=6 x (2 x−7 )6 x (5 x−7 )
=2 x−75 x−7
31)6a−1010 a+4
=2 (3a−5 )2 (5a+2 )
=3 a−55a+2
33) 2n2+19n−109n+90
=(2n−1 ) (n+10 )9 (n+10 )
=2n−19
35)8m+1620m−12
= 8 (m+2 )4 (5m−3 )
=2 (m+2 )5m−3
126
37) 2x2−10 x+8
3 x2−7 x+4=2 (x−4 ) ( x−1 )(3 x−4 ) ( x−1 )
=2 ( x−4 )3 x−4
39) 7n2−32n+164n−16
=(7n−4 ) (n−4 )4 (n−4 )
=7n−44
41) n2−2n+16n+6
= (n−1 )2
6 (n+1 )
43)
7a2−26 a−456a2−34a+20
=(7a+9 ) (a−5 )2 (3a−2 ) (a−5 )
= 7a+92 (3a−2 )
7.2
1) 8 x2
9∙ 92=4 x2
3)9n2n
∙ 75n
= 6310n
5) 5x2
4∙ 65=3 x
2
2
7)7 (m−6 )m−6
∙ 5m (7m−5 )7 (7m−5 )
=5m
9)7 r
7 r (r+10 )÷ r−6
(r−6 )2= 7 r7 r (r+10 )
∙ (r−6 )2
r−6= r−6r+10
11)25n+255
∙ 430n+30
=25 (n+1 )5
∙ 430 (n+1 )
=23
13)x−1035x+21
÷ 73x+21
= x−1035 x+21
∙ 35 x+217
= x−107 (5 x+3 )
∙ 7 (5 x+3 )7
= x−107
15) x2−6 x−7x+5
∙ x+5x−7
=(x−7 ) ( x+1 )
x+5∙ x+5x−7
=x+1
17)8 k
24k2−40k÷ 115k−25
= 8 k24 k2−40k
∙ 15k−251
= 8 k8k (3k−5 )
∙ 5 (3k−5 )1
=5
127
4
2
3
2
3
3
19) (n−8 ) ∙ 610n−80
=n−81
∙ 610 (n−8 )
=35
21)4m+36m+9
∙ m−55m2 =
4 (m+9 )m+9
∙ m−55m2
=4 (m−5 )5m2
23)3x−612x−24
∙ ( x+3 )= 3 ( x−2 )12 ( x−2 )
∙ x+31
= x+34
25)b+2
40b2−24b∙ (5b−3 )= b+2
8b (5b−3 )∙ 5b−31
=b+28b
27)n−76n−12
∙ 12−6nn2+13n+42
= n−76 (n−2 )
∙ −6 (n−2 )(n−7 ) (n−6 )
= −1n−6
29)27a+369a+63
÷ 6a+82
=27 a+369a+63
∙ 26a+8
=9 (3a+4 )9 (a+7 )
∙ 22 (3a+4 )
= 1a+7
31)x2−12 x+32x2−6 x−16
∙ 7 x2+14 x
7 x2+21 x=
( x−8 ) ( x−4 )( x−8 ) ( x+2 )
∙ 7 x( x+2 )
7 x ( x+3 )= x−4
x+3
33) (10m2+100m) ∙ 18m3−36m2
20m2−40m=10m (m+10 )
1∙ 18m
2 (m−2 )20m (m−2 )
=9m2 (m+10 )
35) 7 p2+25 p+126 p+48
∙ 3 p−821 p2−44 p−32
=(7 p+4 ) (p+3 )6 ( p+8 )
∙ 3 p−8(7 p+4 ) (3 p−8 )
= p+36 ( p+8 )
37)10b2
30b+20∙ 30b+202b2+10b
= 10b2
10 (3b+2 )∙ 10
(3b+2 )2b (b+5 )
= 5bb+5
39) 7 r2−53 r−247 r+2
÷ 49 r+2149 r+14
=7 r2−53r−247 r+2
∙ 49 r+1449 r+21
=(7 r+3 ) (r−8 )7 r+2
∙ 7(7 r+2 )7 (7 r+3 )
=r−8
41)
x2−12x−4
∙ x2−4x2−x−2
÷ x2+x−23x−6
= x2−12 x−4
∙ x2−4x2−x−2
∙ 3 x−6x2+x−2
=( x+1 ) ( x−1 )2 ( x−2 )
∙( x+2 ) ( x−2 )( x−2 ) (x+1 )
∙ 3 ( x−2 )( x+2 ) ( x−1 )
=32
128
5
4
9
5
43)
x2+3 x+9x2+x−12
∙ x2+2 x−8x3−27
÷ x2−4x2−6 x+9
= x2+3 x+9x2+x−12
∙ x2+2 x−8x3−27
∙ x2−6 x+9x2−4
= x2+3x+9(x+4 )(x−3)
∙ ( x+4 ) ( x−2 )( x−3 ) (x2+3 x+9 )
∙ ( x−3 )2
( x−2 ) ( x+2 )=¿¿
1x+2
7.3
1)(6 )(6 )38= ?48
1848
3)( y )( y )
ax= ?xy
ayxy
5)(3a2 c3 )(3a2 c3 )
23 a3b2 c
= ?9a3b2c4
6 a2c3
9a3b2 c4
7)( x−4 )( x−4 )
2x+4
= ?x2−16
2 x−8
( x+4 ) ( x−4 )
9)( x+3 )( x+3 )
( x−4 )( x+2 )
= ?x2+5x+6
x2−4 x+3 x−12
( x+2 ) ( x+3 )= x2−x−12
(x+2 ) ( x+3 )
11) 2a3 ,6a4b2 ,4 a3b5 12a4b5
13) x3−3 x , x−3 , x x (x2−3) x (x−3)
15) x+2 , x−4 (x+2)( x−4)
17) x2−25 , x+5 (x+5)(x−5) (x+5)(x−5)
19) x2+3 x+2 , x2+5 x+6 ( x+1 ) ( x+2 )(x+2)(x+3) (x+1)(x+2)(x+3)
21)(2a3 )(2a3 )
3a5b2
, 210a3b
(b )(b )
LCD=10a3b2
6a4
10a3b2, 2b10a3b2
23)( x+2 )( x+2 )
x+2x−3
, ( x−3 )(x+2 )
( x−3 )( x−3 )
LCD=(x−3)(x+2)
x2+4 x+4( x−3 ) ( x+2 )
, x2−6 x+9( x−3 ) ( x+2 )
25)( x−4 )( x−4 )
xx2−16
, 3 xx2−8x+16
( x+4 )( x+4 )
( x−4 ) ( x+4 )(x−4 )(x−4 )
129
( x+4 ) (x−4 )
( x+2 ) (x+3 )
LCD=( x−4 )2(x+4 )
27)4 x
x2−x−6, x+2x−3
( x+2 )( x+2 )
(x−3)(x+2)
LCD : ( x−3 ) ( x+2 )
4 x( x−3 ) ( x+2 )
, x2+4 x+4( x−3 ) ( x+2 )
7.4
1)2
a+3+ 4a+3
= 6a+3
3)t2+4 tt−1
+ 2t−7t−1
= t 2+6 t−7t−1
=(t+7 ) (t−1 )
t−1=t+7
5)2x2+3
x2−6 x+5+−x2+5 x+(−9 )
x2−6 x+5=
x2+5x−6x2−6 x+5
=(x+6 ) ( x−1 )( x+5 ) ( x−1 )
=x+6x+5
7)(4 )(4 )
56 r
+−58 r
(3 )(3 )
LCD :24 r
2024 r
+−1524 r
= 524 r
9)(2 )(2 )
89 t3
+ 56 t 2
(3 t )(3 t )
LCD :18 t 3
1618t 3
+ 15t18 t 3
=16+15 t18 t 3
11)(2 )(2 )
a+22
+−a+44
LCD : 4
2a+44
+−a+44
=a+84
13)x−14 x
+−2 x−3x
(4 )(4 )
LCD : 4 x
x−14 x
+−8 x−124 x
=−7 x−134 x
15)( y )( y )5 x+3 y2x2 y
+−3 x−4 yx y2
(2x )(2x )
LCD :2 x2 y2
5xy+3 y2
2x2 y2+−6 x2−8xy
2x2 y2=−6 x2−3 xy+3 y2
2 x2 y2
17)( z+1 )( z+1 )
2 zz−1
+−3 zz+1
( z−1 )( z−1 )
LCD :(z−1)(z+1)
2 z2+2 z
( z−1 ) (z+1 )+ −3 z2+3 z
( z−1 ) ( z+1 )= −a2+5 z
(z−1 ) ( z+1 )
19)8
x2−4+ −3x+2
( x−2 )( x−2 )
LCD : ( x+2 ) ( x−2 )
8( x+2 ) ( x−2 )
+ −3 x+6( x+2 ) ( x−2 )
= −3 x+14(x+2 ) ( x−2 )
21)(4 )(4 )
tt−3
+ −54 t−12
LCD : 4(t−3)
4 t
4 (t−3)+ −54 (t−3)
= 4 t−54 (t−3)
130
23)(3 )(3 )
25x2+5 x
+ −43x+3
(5 x )(5 x )
5 x ( x+1 )3 (x+1) LCD :15 x ( x+1 )
6
15x ( x+1 )+ −20 x15x ( x+1 )
= 6−20x15 x ( x+1 )
25)( y+ t )( y+ t )
ty− t
+ − yy+t
( y−t )( y−t )
LCD :( y+t)( y−t)
yt+t 2
( y+ t ) ( y−t )+ − y2+ yt
( y+t ) ( y−t )= t 2+2 yt− y2
( y+ t ) ( y−t )
27)( x+1 )( x+1 )
xx2+5x+6
+ −2x2+3 x+2
( x+3 )( x+3 )
( x+2 ) (x+3 )(x+1)(x+2) LCD :(x+1)(x+2)(x+3)
x2+x( x+1 ) ( x+2 ) ( x+3 )
+ −2x−6(x+1 ) ( x+2 ) ( x+3 )
= x2−x−6( x+1 ) ( x+2 ) ( x+3 )
=( x−3 ) ( x+2 )
( x+1 ) ( x+2 ) (x+3 )= x−3
( x+1 ) ( x+3 )
29)( x+6 )( x+6 )
xx2+15 x+56
+ −7x2+13x+42
( x+8 )( x+8 )
( x+7 ) ( x+8 )(x+7)(x+6) LCD :(x+6)(x+7)(x+8)
x2+6 x( x+6 ) ( x+7 ) ( x+8 )
+ −7 x−56( x+6 ) ( x+7 ) ( x+8 )
= x2−x−56(x+6 ) ( x+7 ) ( x+8 )
=( x−8 ) ( x+7 )
( x+6 ) ( x+7 ) ( x+8 )= x−8
( x+6 ) (x+8 )
31)( x+3 )( x+3 )
5 xx2−x−6
+ −18x2−9
( x+2 )( x+2 )
( x−3 ) ( x+2 )(x+3)(x−3) LCD :(x+2)(x−3)(x+3)
5 x2+15 x( x+2 ) ( x−3 ) ( x+3 )
+ −18x−36( x+2 ) (x−3 ) (x+3 )
= 5 x2−3 x−36( x+2 ) (x−3 ) ( x+3 )
=(5 x+12 ) ( x−3 )
( x+2 ) ( x−3 ) ( x+3 )= 5x+12
( x+2 ) ( x+3 )
33)( x+3 )( x+3 )
2xx2−1
+ −4x2+2 x−3
( x+1 )( x+1 )
( x+1 ) ( x−1 ) ( x+3 )(x−1) LCD :(x+3)(x+1)(x−1)
2x2+6 x( x+3 ) ( x+1 ) (x−1 )
+ −4 x−4( x+3 ) ( x+1 ) ( x−1 )
= 2 x2+2 x−4( x+3 ) ( x+1 ) ( x−1 )
=2 (x+2 ) ( x−1 )
( x+3 ) (x+1 ) ( x−1 )=
2 ( x+2 )(x+3)(x+1)
131
35)( x+2 )( x+2 )
x+1x2−2 x−35
+ x+6x2+7 x+10
(x−7 )(x−7 )
( x−7 ) ( x+5 )(x+5)(x+2) LCD :(x+2)(x+5)(x−7)
x2+x+2x+2( x+2 ) ( x+5 ) (x−7 )
+ x2−7 x+6 x−42( x+2 ) ( x+5 ) ( x−7 )
= 2 x2+2x−40( x+2 ) (x+5 ) ( x−7 )
=2 ( x+5 ) ( x−4 )
( x+2 ) (x+5 ) ( x−7 )=
2 ( x−4 )( x−7 ) ( x+2 )
37)(−1 )(−1 )
4−a2
a2−9+−a+23−a
(a+3 )(a+3 )
(a+3 ) (a−3 )(−1)(a−3) LCD :(−1)(a+3)(a−3)
a2−4(−1 ) ( a+3 ) (a−3 )
+−a2−3a+2a+6(−1 ) (a+3 ) ( a−3 )
= −a+2(−1 ) (a+3 ) ( a−3 )
39)(2 z+1 )(2 z+1 )
2 z1−2 z
+(−1 ) (2 z−1 )(−1 ) (2 z−1 )
3 z2 z+1
+ −34 z2−1
(−1 )(−1 )
(−1 ) (2 z−1 )(2 z+1)(2 z−1) LCD :(−1)(2 z−1)(2 z+1)
4 z2+2 z(−1 ) (2 z−1 ) (2 z+1 )
+ −6 z2+3 z(−1 ) (2 z−1 ) (2 z+1 )
+ 3(−1 ) (2 z−1 ) (2 z+1 )
= −2 z2+5 z+3(−1 ) (2 z−1 ) (2 z+1 )
=(−1 ) (2 z+1 ) (z−3 )
(−1 ) (2 z−1 ) (2 z+1 )= z−32 z−1
41)( x+3 )x+3
2 x−3x2+3 x+2
+ 3 x−1x2+5 x+6
( x+1 )( x+1 )
( x+1 ) ( x+2 )(x+3)(x+2)LCD :(x+1)(x+2)(x+3)
2x2−3 x+6 x−9
( x+1 ) ( x+2 ) ( x+3 )+ 3 x2+3 x−x−1
(x+1 ) ( x+2 ) ( x+3 )=
(5x2+5 x−10 )( x+1 ) ( x+2 ) ( x+3 )
=5 ( x+2 ) ( x−1 )
( x+1 ) ( x+2 ) (x+3 )=
5 ( x−1 )( x+1 ) ( x+3 )
43)( x+5 )(x+5)
(2 x+7 )( x2−2x−3 )
+ −3 x+2x2+6x+5
( x−3 )( x−3 )
( x−3 ) ( x+1 )(x+5)(x+1) LCD :(x+1)(x−3)(x+5)
2x2+7 x+10 x+35( x+1 ) ( x−3 ) ( x+5 )
+(−3x2+9 x+2x−6 )
( x+1 ) (x−3 ) (x+5 )= −x2+28 x+29
( x+1 ) ( x−3 ) ( x+5 )=
−1 ( x−29 ) ( x+1 )(x+1 ) ( x−3 ) ( x+5 )
=(−1 ) ( x−29 )(x−3 ) ( x+5 )
132
7.5
1)
(x2 )1+ 1x
(x2 )
(x2)1− 1x2
(x2 )= x2+xx2−1
=x ( x+1 )
( x+1 ) ( x−1 )= xx−1
3)(a )a−2 (a )
(a ) 4a−a (a )
=a2−2a4−a2
=a (a−2 ) (−1 )(2+a ) (2−a )
= −aa+2
5)(a2 ) 1
a2−1a
(a2 )
(a2) 1a2
+ 1a
(a2 )=1−a1+a
7)( x+2 )2− 4
x+2( x+2 )
( x+2 )5− 10x+2
( x+2 )= 2x+4−45 x+10−10
=2 x5 x
=25
9)(2a−3 ) 3
(2a−3 )+2 (2a−3 )
(2a−3 ) −6(2a−3 )
−4 (2a−3 )= 3+4a−6
−6−8a+12= 4 a−3
−8a+6= 4 a−3
−2 (4 a−3 )=−12
11)x ( x+1 ) x
x+1−1
xx (x+1 )
x ( x+1 ) xx+1
+ 1xx ( x+1 )
= x2−x−1x2+ x+1
13)(x2 ) 3
x
(x2) 9x2
=3x9
= x3
133
15)(16 a2b2 ) a
2−b2
4 a2b
(16a2b2 ) a+b16 ab2
=4 b (a2−b2 )a (a+b )
=4b (a+b ) (a−b )
a (a+b )=4 b (a−b )
a
17)( x2 )1−3
x(x2 )−10
x2(x2 )
(x2)1+ 11x
(x2 )+ 18x2
(x2 )= x2−3x−10x2+11 x+18
=(x−5 ) (x+2 )( x+9 ) (x+2 )
= x−5x+9
19)(3 x−4 )1− 2x
(3x−4 )(3 x−4 )
(3x−4 ) x− 32(3 x−4 )
(3 x−4 )= 3 x−4−2x3 x2−4 x−32
= x−4(3 x+8 ) ( x−4 )
= 13 x+8
21)
( x−4 ) x−( x−4 )1+ 2( x−4 )
( x−4 )
( x−4 ) x+ ( x−4 )3+ 6( x−4 )
( x−4 )= x2−4 x−x+4+2x2−4 x+3 x−12+6
= x2−5 x+6x2−x−6
=( x−2 ) ( x+3 )( x−3 ) ( x+2 )
=( x−2 )x+2
23)(2x+3 ) x−(2 x+3 )4+ 9
(2x+3 )(2x+3 )
(2 x+3 ) x+(2 x+3 )3− 5(2 x+3 )
(2 x+3 )=2x
2+3 x−8 x−12+92x2+3 x+6 x+9−5
=(2 x+1 ) ( x−3 )(2x+1 ) ( x+4 )
= x−3x+4
25)b (b+3 ) 2
b− 5b+3
b (b+3 )
b (b+3 ) 3b+ 3b+3
b (b+3 )=2b+6−5b3b+9+3b
=−3b+66b+9
=(−3 ) (b−2 )3 (2b+3 )
=(−1 ) (b−2 )2b+3
27)(a2b2 ) 2
b2−(a2b2 ) 5
ab− 3a2
(a2b2 )
(a2b2 ) 2b2
−(a2b2) 7ab
+ 3a2
(a2b2 )=2a
2−5ab−3b2
2a2+7ab+3b2=
(2a+b ) (a−3b )(2a+b ) (a+3b )
=a−3ba+3b
29)( y+2 ) ( y−2 ) y
y+2− y
y−2( y+2 ) ( y−2 )
( y+2 ) ( y−2 ) yy+2
+ yy−2
( y+2 ) ( y−2 )= y2−2 y− y2−2 y
y2−2 y+ y2+2 y=−4 y2 y2
=−2y
134
31) x−2− y−2
x−1+ y−1=(x2 y2 ) 1
x2+ 1y2
(x2 y2 )
( x2 y2 ) 1x+ 1y
(x2 y2)= y2−x2
x y2+x2 y=
( y+x ) ( y−x )xy ( y+ x )
= y− xxy
33) x−3 y−x y−3
x−2− y−2 =(x3 y3 ) y
x3− x
y3(x3 y3 )
(x3 y3 ) 1x2
− 1y2
(x3 y3 )= y4−x4
x y3−x3 y=
( y2+ x2 ) ( y2−x2 )xy ( y2−x2 )
= y2+ x2
xy
35) x−2−6 x−1+9x2−9
=(x2) 1
x2−(x2 ) 6
x+9 (x2 )
(x2 ) x2−9=1−6 x+9x
2
x2 ( x2−9 )=
(1−3 x )2
(x2) ( x+3 ) ( x−3 )
7.6
1)10a
=68
806
=6a6
13.3=a
3)76=2k
7k7
=127
k=1.71
5)6x=82
128
=8 x8
1.5= x
7)m−15
=82
2 (m−1 )=40 2m−2=40 +2+2
2m2
=422
m=21
9)29= 10
p−4
2 (p−4 )=90 2 p−8=90 +8+8
2 p2
=982
p=49
11)b−107
=b4
4 (b−10 )=7b 4 b−40=7b −4b−4b
−403
=3b3
−13.3=b
13)x5= x+2
9
135
9 x=5 ( x+2 ) 9 x=5 x+10 −5 x−5 x
4 x4
=104
x=2.5
15)310
= aa+2
3 (a+2 )=10a 3a+6=10a −3a−3a
67=7a7
0.86=a
17)v−5v+6
=49
9 ( v−5 )=4 (v+6) 9 v−45=4 v+24 −4v−4 v 5v−45=24 +45+45
5v5
=695
v=13.8
19)7
x−1= 4x−6
7 ( x−6 )=4 (x−1) 7 x−42=4 x−4 −4 x−4 x 3 x−42=−4 +42+42
3x3
=383
x=12.67
21)x+55
= 6x−2
( x+5 ) ( x−2 )=30 x2+5 x−2 x−10=30 x2+3 x−10=30 −30−30 x2+3 x−40=0 ( x+8 ) ( x−5 )=0 x+8=0x−5=0 −8−8+5+5 x=−8 x=5
23)m+34
= 11m−4
(m+3 ) (m−4 )=44 m2−4m+3m−12=44 m2−m−12=44 −44−44 m2−m−56=0 (m−8 ) (m+7 )=0 m−8=0m+7=0 +8+8−7−7 m=8m=−7
25)2
p+4= p+5
3
6=( p+4)( p+5) 6=p2+5 p+4 p+20 6=p2+9 p+20 −6−6 0=p2+9 p+14 0=( p+7 )( p+2) p+7=0 p+2=0 −7−7−2−2 p=−7 ,−2
27)n+43
= −3n−2
136
(n+4 ) (n−2 )=−9 n2−2n+4n−8=−9 n2+2n−8=−9 +9+9 n2+2n+1=0 (n+1 )2=0 n+1=0 −1−1 n=−1
29)3
x+4= x+2
5
15=(x+4 )(x+2) 15=x2+2 x+4 x+8 15=x2+6 x+8 −15−15
0=x2+6 x−7 0=(x+7)(x−1) x+7=0x−1=0 −7−7+1+1 x=−7 x=1
31) The currency in Western Samoa is the Tala. The exchange rate is approximately S0.70 to 1 Tala. At this rate, how many dollars would you get if you exchanged 13.3 Tala?
T$= 10.70
=13.3x
x=$9.31
39) Kali reduced the size of a painting to a height of 1.3 in. What is the new width if it was originally 5.2 in. tall and 10 in. wide?
hw
=5.210
=1.3x
x=2.5∈¿
41) A bird bath that is 5.3 ft tall casts a shadow that is 25.4 ft long. Find the length of the shadow that a 8.2 ft adult elephant casts.
hs= 5.325.4
=8.2x
x=39.3 ft
43) The Vikings led the Timberwolves by 19 points at the half. If the Vikings scored 3 points for every 2 points the Timberwolves scored, what was the half time score?
VT
=( x+19 )
x= 32
137
2 ( x+19 )=3 x 2 x+38=3 x −2 x−2x 38=x Timberwolves :38 Vikings :57
45) Computer Services Inc. charges S8 more for a repair than Low Cost Computer Repair. If the ratio of the costs is 3 : 6, what will it cost for the repair at Low Cost Computer Repair?
CSILCCR
= x+8x
=63
3 ( x+8 )=6 x 3 x+24=6 x −3 x−3 x
243
=3 x3
$8=x
7.7
1) (2 x )3 x−(2 x ) 12−(2x ) 1
x=0 (2 x)
LCD :2 x
2x2
≠ 02
¿ x≠0∗¿ 6 x2−x−2=0 (2 x+1 ) (3 x−2 )=0 2 x+1=03 x−2=0 −1−1+2+2
138
2x2
=−123 x3
=23
x=−12
x=23
3) x (x−4 )+ 20( x−4 )
( x−4 )= 5 x( x−4 )
( x−4 )−2 (x−4 )
LCD : ( x−4 ) x−4≠0 +4+4 ¿ x≠ 4∗¿ x2−4 x+20=5 x−2 x+8 x2−4 x+20=3 x+8 −3 x−8−3 x−8 x2−7 x+12=0 ( x−4 ) ( x−3 )=0 x−4=0 x−3=0 +4+4+3+3 x=4 x=3
5) x (x−3 )+ 6( x−3 )
( x−3 )= 2 x( x−3 )
( x−3 )
LCD=x−3 x−3≠0 +3+3 ¿ x≠3∗¿ x2−3 x+6=2 x −2 x−2x x2−5 x+6=0 ( x−2 ) ( x−3 )=0 x−2=0 x−3=0 +2+2+3+3 x=2 x=3
7)2 x3x−4
(6 x−1 ) (3x−4 )= 4 x+56 x−1
(6 x−1 ) (3x−4 )− 33x−4
(6 x−1 ) (3x−4 )
LCD : (6 x−1 ) (3 x−4 ) 6 x−1≠03 x−4≠0 +1+1+4+4
6 x6
≠ 163 x3
≠ 43
139
¿ x≠ 16∗¿ x≠ 4
3∗¿
12 x2−2x=12 x2−16 x+15 x−20−18 x+3 12 x2−2x=12 x2−19 x−17 −12 x2−12x2
−2 x=−19 x−17+19 x+19x
17 x17
=−1717
x=−1
9)3m2m−5
(2 ) (2m−5 ) (3m+1 )− 73m+1
(2 ) (2m−5 ) (3m+1 )=32
(2 ) (2m−5 ) (3m+1 )
LCD : (2 ) (2m−5 ) (3m+1 ) 2m−5≠03m+1≠0 +5+5−1−1
2m2
≠ 523m3
≠−13
¿ x≠ 52∗¿ x≠−1
3∗¿
18m2+6m−28m+70=18m2−39m−15 18m2−22m+70=18m2−39m−15 −18m2−18m2
−22m+70=−39m−15+39m+39m 17m+70=−15
−70−70
17m17
=−8517
m=−5
140
11)4−x1−x
(1−x ) (3−x )= 123−x
(1−x ) (3−x )
LCD : (1− x ) (3− x ) 1−x ≠03−x≠0 +x+x+ x+x ¿1≠x∗¿3≠ x∗¿ 12−4 x−3 x+x2=12−12 x x2−7 x+12=12−12 x
+12 x−12−12+12 x x2+5x=0
x (x+5 )=0x=0 x+5=0
−5−5 x=−5
13)7
y−3(2 ) ( y−3 ) ( y−4 )−1
2(2 ) ( y−3 ) ( y−4 )= y−2
y−4(2 ) ( y−3 ) ( y−4 )
LCD : (2 ) ( y−3 ) ( y−4 ) y−3≠0 y−4≠0 +3+3+4+4 *y ≠3∗¿ y ≠4∗¿ 14 y−56− y2+3 y+4 y−12=2 y2−6 y−4 y+12 − y2+21 y−68=2 y2−10 y+12
+ y2−21 y+68+ y2−21 y+680=3 y2−31 y+800=(3 y−16)( y−5)
3 y−16=0 y−5=0 +16+16+5+5
3 y3
=163
y=5
15)1
x+2( x+2 ) ( x−2 )+ 1
2−x( x+2 ) (x−2 )=3x+8
x2−4( x+2 ) ( x−2 )
x−2 (x−2)(x+2) LCD : ( x+2 ) ( x−2 ) x+2≠0 x−2≠0 −2−2+2+2 x≠−2x ≠2 x−2+x+2=3 x+8 2 x=3 x+8
141
−3 x−3 x
−x−1
= 8−1
x=−8
17)( x+1 )x−1
(6 ) (x−1 ) ( x+1 )+−x+1x+1
(6 ) ( x−1 ) ( x+1 )=56
(6 ) ( x−1 ) ( x+1 )
LCD : (6 ) ( x−1 ) ( x+1 ) x−1≠0x+1≠0 +1+1−1−1 ¿ x≠1∗¿x ≠−1∗¿ 6 x2+6 x+6 x+6−6 x2+6 x+6 x−6=5 x2−5 24 x=5 x2−5
−24 x−24 x 0=5 x2−24 x−50=(5 x+1)( x−5)
5 x+1=0 x−5=0 −1−1+5+5
5x5
=−15
x=5
x=−15
19)3
2x+1(2x+1 ) (2x−1 )
1+−2x−12x−1
(2 x+1 ) (2 x−1 )1
=1(2 x+1 ) (2 x−1 )
1− 8x2
4 x2−1(2x+1 ) (2x−1 )
1
LCD : (2 x+1 ) (2 x−1 )2 x+1≠02x−1≠0−1−1+1+1
2x2
≠−122 x2
≠ 12
* x≠−12∗¿x ≠ 1
2∗¿
6 x−3−4 x2−2x−2 x−1=4 x2−1−8x2
−4 x2+2 x−4=−4 x2−1 +4 x2 +4 x2
2 x−4=−1 +4+4
2x2
=32
142
x=32
21)x−2x+3
( x+3 ) ( x−2 )− 1x−2
( x+3 ) ( x−2 )= 1x2+x−6
( x+3 ) ( x−2 )
(x−2)(x+3) LCD : ( x+3 ) ( x−2 ) x+3≠0 x−2≠0 −3−3+2+2 ¿ x≠−3∗¿x ≠2∗¿ x2−4 x+4−x−3=1 x2−5 x+1=1
−1−1 x2−5 x=0 x (x−5 )=0x=0 x−5=0
+5+5 x=5
5(x+4)
23)3
x+2+ x−1x+5
=5 x+206 x+24
6( x+4)
3
x+2(6 ) ( x+2 ) ( x+5 )+ x−1
x+5(6 ) ( x+2 ) ( x+5 )=5
6(6 ) ( x+2 ) ( x+5 )
LCD : (6 ) ( x+2 ) ( x+5 ) x+2≠0 x+5≠0 −2−2−5−5 *x≠−2∗¿ x ≠−5∗¿ 18 x+90+6 x2+12 x−6 x−12=5x2+25 x+10x+50 6 x2+24 x+78=5 x2+35 x+50
143
−5 x2−35 x−50−5 x2−35 x−50 x2−11 x+28=0 ( x−7 ) ( x−4 )=0
x−7=0 x−4=0 +7+7+4+4 x=7 x=4
25)x
x−1( x+1 ) ( x−1 )− 2
x+1( x+1 ) (x−1 )= 4 x2
x2−1( x+1 ) ( x−1 )
( x+1 ) ( x−1 ) LCD : ( x+1 ) (x−1 ) x+1≠0 x−1≠0 −1−1+1+1 ¿ x≠−1∗¿ x ≠1* x2+ x−2x+2=4 x2
x2−x+2=4 x2
−x2+ x−2−x2+x−20=3 x2+x−x0=(3 x−2)(x+1)
3 x−2=0x+1=0 +2+2−1−1
3x3
=23x=−1
x=23
27)2 xx+1
(x+1 ) ( x+5 )− 3x+5
( x+1 ) ( x+5 )= −8x2
x2+6 x+5( x+1 ) ( x+5 )
( x+1 ) ( x+5 ) LCD : ( x+1 ) (x+5 )
144
x+1≠0 x+5≠0 −1−1−5−5 ¿ x≠−1∗¿ x ≠−5∗¿ 2 x2+10 x−3 x−3=−8 x2
2 x2+7 x−3=−8 x2
+8 x2+8 x2
10 x2+7 x−3=0 (10 x−3 ) ( x+1 )=0 10 x−3=0 x+1=0 +3+3−1−1
10x10
= 310
x=−1
x= 310
29)x−5x−9
( x−9 ) ( x−3 )+ x+3x−3
( x−9 ) ( x−3 )= −4 x2
x2−12x+27(x−9 ) ( x−3 )
( x−9 ) ( x−3 ) LCD : ( x−9 ) (x−3 ) x−9≠0 x−3≠0 +9+9+3+3 ¿ x≠9∗¿ x≠3∗¿ x2−3 x−5 x+15+ x2−9 x+3 x−27=−4 x2
2 x2−14 x−12=−4 x2
+4 x2 +4 x2
6 x2−14 x−12=0 2 (3 x2−7 x−6 )=0 2 (3x+2 ) ( x−3 )=0
3 x+2=0 x−3=0 −2−2+3+3
3x3
=−23
x=3
145
x=−23
31)x−3x−6
( x−6 ) ( x+3 )+ x+5x+3
( x−6 ) (x+3 )= −2 x2
x2−3 x−18( x−6 ) ( x+3 )
( x−6 ) ( x+3 ) LCD : ( x−6 ) (x+3 ) x−6≠0 x+3≠0 +6+6−3−3 *x≠6∗¿ x≠−3∗¿ x2−9+x2−6 x+5 x−30=−2x2
2 x2−x−90=−2x2
+2 x2 +2 x2
4 x2−x−90=0(4 x−13 ) ( x+3 )=0
4 x−13=0x+3=0 +13+13−3−3
4 x4
=134
x=−3
x=134
33)4 x+1x+3
( x+3 ) ( x−1 )+5 x−3x−1
( x+3 ) ( x−1 )= 8 x2
x2+2 x−3( x+3 ) ( x−1 )
( x+3 ) ( x−1 ) LCD : ( x+3 ) ( x−1 ) x+3≠0 x−1≠0 −3−3+1+1 ¿ x≠−3∗¿x ≠1∗¿ 4 x2−4 x+x−1+5 x2+15 x−3 x−9=8 x2
9 x2+9 x−10=8 x2
−8 x2 −8 x2
x2+9 x−10=0
146
( x+10 ) (x−1 )=0x+10=0 x−1=0 −10−10+1+1 x=−10 x=1
7.8
1) 7mi ¿ yd
( 7mi1 )( 5280 ft1mi )( 1 yd(3 ft ) )=36960 yd3=12,320 yd
3) 11.2mg ¿g
( 11.2mg1 )( 1 g1000mg )=11.291000
=0.0112 g
5) 9,800,000mm¿mi
( 9,800,000mm1 )( 1m1000mm )(3.29 ft1m )( 1mi
5280 ft )=32,144,000mi5280000=6.088mi
7) 435,000m2 ¿k m2
( 435,000m2
1 )( 1km1,000m )
2
( 435,000m2
1 )( 1k m2
1,000,000m2 )=435,000 k m2
1,000,000=0.435 k m2
9) 0.0065k m3¿m3
( 0.0065 k m3
1 )( 1000mkm )3
( 0.0065 k m3
1 )( (1,000,000,000m3 )km3 )=6,500,000m3
11) 5,500c m3 ¿ y d3
( 5,500 cm3
1 )¿¿
( 5,500 cm3
1 )( 1i n3
16.387064 c m3 )( 1 y d3
46656 in3 )= 5,500 y d3
764554.858=0.00719 yd3
13) 185 yd /min ¿min/hr
147
( 185 ydmin❑ )( 3 ft1 yd )( 1mi5280 ft )( 60min1hr )=33300mi5280hr
=6.307mi /hr
15) 248mi /hr ¿m /sec
( 248mihr )( 1.61km1mi )( 1000m1km )( 1hr3600 sec❑ )= 399.280m
3600 sec❑=110.9m /sec
17) 7.5Ty d2
¿ lbs / i n2
( 7.5Ty d2 )( 2000 lbs1T )¿¿
( 7.5Ty d2 )( 2000 lbs1T )( 1 y d2
1296 in2 )=15000lbs1296 i n2=11.57 lbs /i n2
19) On a recent trip, Jan traveled 260 miles using 8 gallons of gas. How many miles per 1-gallon did she travel? How many yards per 1-ounce?
260mi8 gal
=32.5mi/ gal
( 32.5migal )( 5280 ft1mi )(1 yd3 ft )( 1gal4qt )( 1qt2 pt )( 1 pt2c )( 1c8oz )=171,600 yd384oz=446.875 yd /oz
21) A certain laser printer can print 12 pages per minute. Determine this printer’s output in pages per day, and reams per month. (1 ream = 5000 pages)
( 12 pg1min❑ )( 60min1hr )( 24hr1day )=17280 pg /day
( 17280 pgday )( 30daymon )( 1 ream5000 pg )=5184000reams5000months=103.68 reams /month
23) Blood sugar levels are measured in miligrams of gluclose per deciliter of blood volume. If a person’s blood sugar level measured 128 mg/dL, how much is this in grams per liter?
( 128mgdL )( 1 g100mg )(10 dL1 L )=1280 g1000L
=1.28 g/L
25) A car travels 14 miles in 15 minutes. How fast is it going in miles per hour? in meters per second?
( 14mi15min❑ )( 60min1hr )=840mi15hr
=56mi/hr
( 14mi15min❑ )( 1.61 km1mi )( 1000m1km )( 1min
60 sec❑ )=22540m900 sec=25.04m / sec
148
27) A local zoning ordinance says that a house’s “footprint” (area of its ground floor) cannot occupy
more than 14 of the lot it is built on. Suppose you own a
13acre lot, what is the maximum allowed
footprint for your house in square feet? in square inches? (1 acre = 43560 f t 2)
( 1acre3 )( 43560 f t2
1acre )( 14 )= 43560 f t2
12=3,630 f t 2
( 3630 f t 21 )¿¿
( 3630 f t 21 )(144 i n21 f t 2 )=522,720i n2
29) In April 1996, the Department of the Interior released a “spike flood” from the Glen Canyon Dam on the Colorado River. Its purpose was to restore the river and the habitants along its bank. The release from the dam lasted a week at a rate of 25,800 cubic feet of water per second. About how much water was released during the 1-week flood
( 25,800 f t 31 sec❑ )(3600 sec1hr )( 24hr1day )( 7day1wk )=15,603,840,000 f t 3/week
149
Chapter 8: Radicals
150
8.1
1) r √245 √5 ∙72 7 √5
4) √36 √22 ∙32 2∙3 6
5) √12 √22 ∙3 2√3
7) 3√123√22 ∙3 3 ∙2√3 6√3
9) 6√1286 √276 ∙23√26 ∙8√248√2
11) −8√392 −8√72 ∙23 −8 ∙7 ∙2√2 −112√2
13) √192n √26∙3n 23√3n 8√3n
15) √196 v2 √22∙72v2 2 ∙7 ∙ v
14v 17) √252 x2
√22 ∙32 ∙7 x2 2 ∙3 x √7 6 x √7
19) −√100k4 −√22 ∙52 k4 −2 ∙5 ∙ k2
−10k2
21) −7√64 x4 −7√26 x4 −7 ∙23x2
−7 ∙8 x2
−56 x2
23) −5√36m −5√22 ∙32m −5 ∙2 ∙3√m −30√m
25) √45 x2 y2 √32 ∙5 ∙ x2 y2 3 xy √5
27) √16 x3 y3 √24 x3 y3 22 xy √ xy 4 xy√ xy
29) √320 x4 y 4 √26 ∙5 x4 y4 23 x2 y2√5 8 x2 y2√5
31) 6√80 x y2 6√24 ∙5 x y2
6∙22 y √5 x 6 ∙4 y √5x 24 y √5x
33) 5√245 x2 y3 5√5∙72 x2 y3 5 ∙7 xy √5 y 35 xy √5 y
35) −2√180u3 v −2√22 ∙32∙5u3 v −2 ∙2 ∙3u√5uv −12u√5uv
37) −8√180 x 4 y2 z4
−8√22 ∙32 ∙5 x4 y2 z4 −8 ∙2∙3 x2 y z2√5 −48x2 y z2√5
39) 2√80h j4 k 2√24 ∙5h j4 k 2 ∙22 j2√5hk 2 ∙4 j2√5hk 8 j2√5hk
41) −4√54mn p2 −4√2∙32mn p2 −4 ∙3 p√2∙3mn −12 p√60n
151
152
8.2
1) 3√625 3√54 5 3√5
3) r 3√750 3√2∙3 ∙53 5 3√2∙3 5 3√6
5) 3√875 3√53∙7 5 3√7
7) −4 4√96 −4 4√25 ∙3 −4 ∙2 4√2∙3 −8 4√6
9) 6 4√112 6 4√24 ∙7 6 ∙2 4√7 12 4√7
11) −4√112 −4√24 ∙7 −2 4√7
13) r 4√648a2 4√23 ∙34a2 3 4√23a2 3 4√8a2
15) 5√224n3 5√25∙7n3 2 5√7 n3
17) r 5√224 p5 5√25 ∙7 p5 2 p 5√7
19) −3 7√896 r −3 7√27∙7 r −3 ∙2 7√7 r −6 7√7 r
21) r−2 3√−48v7 −2 3√−13∙24 ∙3 v7
−2 ∙−1 ∙2v2 3√2∙3v 4 v2 3√6 v
23) −7 3√320n6 −7 3√26∙5n6 −7 ∙22n2 3√5 −7 ∙4n2 3√5 −28n2 3√5
25) 3√−135 x5 y3
3√−13 ∙33 ∙5 x5 y3
−1 ∙3 xy 3√5 x2 −3 xy 3√5 x2
27) 3√−32x4 y 4
3√−13 ∙25 x 4 y 4
−1 ∙2 xy 3√22 xy −2 xy 3√4 xy
29) r 3√256 x4 y6 3√28 x4 y6
22 x y2 3√22 x 4 x y2 3√4 x
31) 7 3√−81 x3 y7
7 3√−13 ∙34 x3 y7
7 ∙−1 ∙3x y2 3√3 y −21 x y2 3√3 y
33) 2 3√375u2 v8 2 3√3 ∙53u2 v8 2 ∙5 v2 3√3u2 v2 10 v2 3√3u2v2
35) −3 3√192ab2 −3 3√26 ∙3ab2 −3 ∙22 3√3ab2 −3 ∙4 3√3ab2 −12 3√3ab2
37) 6 3√−54m8n3 p7
6 3√−13 ∙2 ∙33m8n3 p7
6 ∙−1 ∙3m2n p2 3√2m2 p −18m2n p2 3√2m2 p
39) 6 4√648 x5 y7 z2 6 4√23 ∙34 x5 y7 z2 6 ∙3xy 4√23x y3 z2
18 xy 4√8 x y3 z2
41) 7 4√128h6 j8k8 7 4√27h6 j8 k8 7 ∙2h j2k2 4√23h2
153
14h j2 k2 4√8h2
8.3
1) 2√5+2√5+2√5 6√5
3) −3√2+3√5+3√5 −3√2+6√5
6) −2√6−2√6−√6 −5√6
8) 3√6+3√5+2√5 3√6+5√5
10) 2√2−3√18−√2 2√2−3√32∙2−√2 2√2−3 ∙3√2−√2 2√2−9√2−√2 −8√2
12) −3√6−√12+3√3 −3√6−√22 ∙3+3 √3 −3√6−2√3+3√3 −3√6+√3
14) 3√2+2√8−3√18 3√2+2√23−3√2 ∙32 3√2+2∙2√2−3∙3√2 3√2+4 √2−9√2 −2√2
16) 3√18−√2−3√2 3√2∙32−√2−3 √2 3 ∙3 √2−√2−3√2 9√2−2√2−3√2 5√2
19) −3√6−3 √6−√3+3√6 −3√6−√3
20) −2√18−3√8−√20+2√20 −2√2∙32−3√23−√22 ∙5+2√22 ∙5 −2 ∙3√2−3 ∙2√2−2√5+2 ∙2√5 −6√2−6 √2−2√5+4 √5 −12√2−2√5
21) −2√24−2√6+2√6+2√20 −2√23 ∙3−2√6+2√6+2√22 ∙5 −2 ∙2√2∙3−2√6+2√6+2 ∙2√5 −4√6−2√6+2√6+4 √5 −4√6+4 √5
23) 3√24−3√27+2√6+2√8 3√23∙3−3√33+2√6+2√23 3 ∙2√2∙3−3∙3√3+2√6+2∙2√2 6√6−9√3+2√6+4√2 8√6−9√3+4√2
25) −2 3√16+2 3√16+2 3√2 −2 3√24+2 3√24+2 3√2 −2 ∙2 3√2+2∙2 3√2+2 3√2 −4 3√2+4 3√2+2 3√2 2 3√2
27) 2 4√243−2 4√253−4√3 2 4√35−2 4√35−4√3 2 ∙3 4√3−2∙3 4√3−4√3 6 4√3−6 4√3−4√3 −4√3
29) 3 4√2−2 4√2−4√243
154
3 4√2−2 4√2−4√35 3 4√2−2 4√2−3 4√3 4√2−3 4√3
31) −4√324+3 4√324−3 4√4 −4√35+3 4√35−3 4√4 −3 4√3+3 ∙3 4√3−3 4√4 −3 4√3+9 4√3−3 4√4 6 4√3−3 4√4
33) 2 4√2+2 4√3+3 4√64−4√3 2 4√2+2 4√3+3 4√26−4√3 2 4√2+2 4√3+3 ∙2 4√22−4√3 2 4√2+2 4√3+6 4√4−4√32 4√2+ 4√3+6 4√4
35) −3 5√6− 5√64+2 5√192−2 5√64 −3 5√6− 5√26+2 5√26 ∙3−2 5√26 −3 5√6−2 5√2+2∙2 5√2∙3−2 ∙2 5√2 −3 5√6−2 5√2+4 5√6−4 5√2 5√6−6 5√2
37) 2 5√160−2 5√192−5√160−5√−160 2 5√25 ∙5−2 5√26 ∙3− 5√25 ∙5−5√−15∙25 ∙5 2 ∙2 5√5−2 ∙2 5√2∙3−2 5√5−(−1)∙2 5√5 4 5√5−4 5√6−2 5√5+2 5√5 4 5√5−4 5√6
39) −6√256−2 6√4−3 6√320−2 6√128 −6√28−2 6√4−3 6√26∙5−2 6√27 −2 6√22−2 6√4−3∙2 6√5−2 ∙2 6√2 −2 6√4−2 6√4−6 6√5−4 6√2 −4 6√4−6 6√5−4 6√2
155
8.4
2) 3√5∙−4 √16 −12√80 −12√24 ∙5 −12 ∙22√5 −12 ∙4√5 −48√5
5) √12m√15m √180m2 √22 ∙32 ∙5m2
2 ∙3m√5 6m√5
7) 3√4 x3 3√2x4 3√8 x7 3√23 x7 2 x2 3√ x
9) √6(√2+2) √12+2√6 √22 ∙3+2√6 2√3+2√6
11) −5√15(3√3+2) −15√45−10√15 −15√32 ∙5−10√15 −15 ∙3√5−10√15 −45√5−10√15
13) 5√10(5n+√2) 25n√10+5√20 25n√10+5√22+5 25n√10+5∙2√5 25n√10+10√5
15) (2+2√2)(−3+√2) −6+2√2−6 √2+2√4 −6+2√2−6 √2+2√22 −6+2√2−6 √2+2 ∙2 −6+2√2−6 √2+4 −2−4√2
17) (√5−5)(2√5−1) 2√25−√5−10√5+5 2√52−√5−10√5+5 2 ∙5−√5−10√5+5 10−√5−10√5+5 15−11√5
20) (√2a+2√3 a)(3 √2a+√5a) 3√4 a2+√10a2+6√6 a2+2√15a2 3√22a2+√10a2+6 √6a2+2√15a2 3 ∙2a+a√10+6a√6+2a√15 6a+a√10+6a√6+2a√15
21) (−5−4 √3)(−3−4 √3) 15+20√3+12√3+16 √9 15+20√3+12√3+16 √32 15+20√3+12√3+16 ∙3 15+20√3+12√3+48 63+32√3
21)√125√100
= √35√25
= √35√52
= √35 ∙5
=√325
23)√5
4 √125= 14 √25
= 14 √52
= 14 ∙5
= 120
25) √10√6
=√5√3 (√3√3 )=√15
3
156
27)2√43√3 ( √3√3 )=2√123∙3
=2√22 ∙39
=2∙2√39
=4√39
29)5 x2
4 √3 x3 y3= 5 x2
4 xy √3xy= 5 x4 y √3 xy (√3 xy√3 xy )= 5 x√3 xy4 y ∙3 xy
=5 x √3 xy12x y2
31) √2 p2√3 p
=√2 p√3 (√3√3 )=√6 p
3
33)3 3√105 3√27
=33√105 3√32
=33√105 ∙3
=33√1015
=3√105
35)3√54 3√4
=3√54 3√22 (
3√23√2 )=3
3√104 ∙2 =
3 3√108
37)5 4√5 r44√8 r2
=54√5 r24√8
=54√5 r24√23 ( 4√24√2 )=5
4√10 r2
8.5
1)4+2√3
√9=4+2√3
3
3) 4+2√35√4
=4+2√35 ∙2
= 4+2√310
=2(2+√3 )10
=2+√35
5)2−5√54 √13 (√13√13 )=2√13−5√654 ∙13
=2√13−5√6552
7) √2−3√3√3 (√3√3 )=√6−3√9
3=√6−3 ∙3
3=√6−9
3
9)
2 p+3√5 p45√20 p2
=2 p+3√5 p4
5√22 ∙5 p2=2 p+3 p
2√55 ∙2 p√5
=2 p+3 p2√510 p √5 (√5√5 )=2 p√5+3 p2√25
10 p(5)=2 p√5+3 p2 ∙5
50 p=2 p√5+15 p2
50 p=
p(2√5+15 p)50 p
=2√5+15 p50
157
11)
√3m2−4√2m4
5√12m4=√3m2−4 √2m4
5√22∙3m4=m√3−4m2√2
2 ∙5m2√3=m√3−4m2√2
10m2√3 (√3√3 )=m√9−4m2√610m2∙3
=3m−4m2√630m2 =
m(3−4m√6)30m2 = 3−4m√6
30m
13) 5
3√5+√2 ( 3√5+23√5+2 )=15 √5−5√29 ∙5−2
=15√5−5√245−2
=15√5−5√243
15) 2
5+√2 ( 5−√25−√2 )=10−2√225−2
=10−2√223
17) 3
4−3√3 ( 4+3√34+3√3 )=12+9√316−9 ∙3=12+9√316−27
=12+9√3−11
19) 4
3+√5 ( 3−√53−√5 )=12−4√59−5
=12−4 √54
=4 (3−√5)
4 ¿3−√5
21)−4
4−4√2 ( 4+4 √24+4 √2 )=−16−16 √2
16−16 ∙2=−16−16√2
16−32=−16−16√2
−16=
−16(1+√2)−16
=¿ 1+√2
23)5
√n4−5= 5n2−5
25)4 p
3−5√ p4= 4 p3−5 p2
27) 4
5+√5x2= 45+ x√5 (5−x √5
5−x √5 )=20−4 x√525+5x2
29) 5
2+√5 r3= 52+r √5 r ( 2−r √5 r
2−r √5 r )=10−5 r √5 r4−r2 (5 r )=10−5 r √5 r
4−5 r3
31) 5−5v−3√v (−5v+3√v−5v+3√v )=−25v+15√v
25v2−9 v
33) 4√2+33√2+√3 ( 3√2−√3
3√2−√3 )=12√4−4√6+9√2−3√39∙2−3=12∙2−4√6+9√2−3√3
18−3=24−4 √6+9√2−3√3
15
158
35) 2−√5−3+√5 (−3−√5
−3−√5 )=−6−2√5+3√5+√259−5
=−6−2√5+3 √5+54
=−1+√54
37)
5√2+√35+5√2 ( 5−5√25−5√2 )=25√2−25√4+5√3−5√625−25 ∙2
=25√2−25 ∙2+5√3−5√625−50
=25√2−50+5√3−5√6−25
=5(5√2−10+√3−√6)
−25=5√2−10+√3−√6
−5
39) √3+√22√3−√2 ( 2√3+√2
2√3+√2 )=2√9+√6+2√6+√44 ∙3−2
=2∙3+√6+2√6+212−2
=6+√6+2√6+210
=8+3√610
41) √3−√24+√5 ( 4−√5
4−√5 )=4 √3−√15−4√2+√1016−5
=4 √3−√15−4√2+√1011
43)
4+2√2 x25+2√5 x3
= 4+2x √25+2 x√5x ( 5−2 x √5 x
5−2 x √5 x )=20+8 x √5 x+10x √2−4 x2√10 x25−4 x2 (5 x )
=20+8 x√5 x+10 x √2−4 x2√10 x25−20 x3
45)
(2√3m2−√2m4 )5−√3m2 =
(2m√3−m2√2 )5−m√3 ( 5+m√3
5+m√3 )=10m√3+2m2√9−5m2√2−m3√625−3m2 =
10m √3+2m2(3)−5m2√2−m3√625−3m2 =
10m√3+6m2−5m2√2−m3√625−3m2
47)
2b−5√2b−1+√2b4
=2b−5√2b−1+b2√2 (−1−√2
−1−√2 )=−2b−2b3√2+5 √2b−5b2√4 b1−2b4
=−2b−2b3√2+5√2b−2∙5b2√b1−2b4
=−2b−2b3√2+5√2b−10b2√b1−2b4
49)
2−√2 x4 x−5√3 x3
= 2−√2 x4 x−5 x√3 x ( 4 x+5 x√3 x4 x+5 x√3 x )= 8 x+10 x √3 x−4 x √2 x−5 x√6 x2
16x2+25 x2 (3 x )= 8 x+10x √3 x−4 x √2 x−5 x2√6
16 x2+75 x2=
x (8+10√3x−4√2 x−5x √6)x (16 x+75 x)
=8+10 √3 x−4 √2 x−5 x √616 x+75 x
51)
−4 p−√ p−p−√ p3
=−4 p−√ p−p−p √ p (−p+p √ p
−p+p √ p )=4 p2−4 p2√ p+ p√ p−p√ p2
p2−p2 ∙ p=4 p
2−4 p2√ p+p √ p−p2
p2−p3=3 p
2−4 p2√ p+ p√ pp2−p3
=p(3 p−4 p√ p+√ p)
p( p−p2)=3 p−4 p√ p+√ p
p−p2
8.6
1) m35=( 5√m )3
159
3) r (7 x )32=(√7 x )3
5)1
(√6x )3=(6 x )
−32
7)1
( 4√n)7=n
−74
9) 823=( 3√8)2=22=4
11) 432=(√4 )3=23=8
13) y x13 ∙ x y
32
y22 x
13 x
33 y
32
x43 y
52
15) (a12 b12)−1
a−12 b
−12
1
a12 b
12
17)a2b0
3a4= 13a2
19) uv ∙ u (v32)3
uv ∙ uv92
uv22 uv
92
u2 v112
21) (x0 y 13)32x0
160
y12
23) a34 b−1b
74
3b−1 =a34 b
74
3
25)3 y
−54
y−12 y−13
=3 y y
13
2 y54
=3 y
33 y
13
2 y54
=3 y
43
2 y54
=3 y
1612
2 y1512
=3 y
112
2
27) ( m32 n−2
(mn43 )−1 )
74
=( m32 n−2
m−1n−43 )
74
=(m32mn
43
n2 )74
=(m32m
22n
43
n63 )
74
=(m52
n23 )
74
=m358
n76
29) r(m2n
12 )0
n34
= 1
n34
31) r( x−4
3 y−13 y )−1
x13 y−2
=( x−4
3 y−13 y
33 )−1
x13 y−2
=(x−4
3 y23 )−1
x13 y−2
= x43 y
−23
x13 y−2
= x43 y2
x13 y
23
= x43 y
63
x13 y
23
¿ x y43
33)(uv2 )
12
v−14 v2
=u12 v
v−14 v2
=u12 v
14 v
v2=u12 v
14 v
44
v2=u12 v
54
v2=u12 v
54
v84
=u12
v34
8.7
1) 8√16 x4 y6 8√24 x4 y6 4√22 x2 y3 4√4 x2 y3
3) 12√64 x4 y6 z8 12√26 x4 y6 z8
6√23 x2 y3 z4 6√8 x2 y3 z 4
5)
6√ 16 x29 y 4=6√ 24 x232 y4
= 3√ 22 x3 y2 (3√32 y3√32 y )= 3√36 xy
3 y
161
7) 12√x6 y9 4√ x2 y3
9) 8√ x6 y4 z2 4√ x3 y2 z
11) 9√8 x3 y6 9√23x3 y6 3√2x y2
13) 3√5√6 6√52∙63 6√25 ∙216 6√5400
15) √ x 3√7 y 6√ x3 ∙72 y2 6√49 x3 y2
17) √ x 3√ x−2 6√ x3 ( x−2 )2
19) 5√ x2 y √xy 10√x4 y2 ∙ x5 y5 10√x9 y7
21) 4√ x y2 3√x2 y 12√x3 y6 ∙ x8 y4 12√x11 y10
23) 4√a2bc2 5√a2b3c 20√a10b5c10 ∙a8b12c4 20√a18b17c14
25) √a 4√a3 4√a2 ∙ a3 4√a5 a 4√a
27) 5√b2√b3 10√b4 ∙ b15 10√b19 b 10√b9
29) √ x y3 3√x2 y 6√ x3 y9 x4 y2 6√ x7 y11 xy 6√x y5
31) 4√9ab3√3a4b 4√32ab3√3a4b 4√32ab3 ∙32a8b2 4√34a9b5 3a2b 4√ab
33) 3√3 x y2 z 4√9 x3 y z2 3√3 x y2 z 4√32x3 y z2 12√34 x4 y8 z4 ∙36 x9 y3 z6 12√310 x13 y11 z10 x 12√59049 x y11z10
35) √27a5(b+1) 3√81a (b+1 )4
162
√33a5(b+1) 3√34a (b+1 )4
6√39a15 (b+1 )3 ∙38a2 (b+1 )8
6√317 a17 (b+1 )11
32a2(b+1) 6√35a5 (b+1 )5
9a2 (b+1 ) 6√243a5 (b+1 )5
37)3√a24√a
=12√ a8
a3=2√a5
39)4√x2 y33√xy
=12√ x6 y9
x4 y4=12√x2 y5
41) √ab3 c5√a2b3 c−1
=10√ a5b15 c5
a4b6 c−2=10√ab9 c7
43)4√(3 x−1 )35√(3 x−1 )3
=20√ (3 x−1 )15
(3 x−1 )12=20√ (3 x−1 )3
45)3√(2 x+1 )25√(2 x+1 )2
=15√ (2 x+1 )10
(2x+1 )6=15√ (2 x+1 )4
163
8.8
1) 3−(−8+4 i) 3+8−4 i 11−4 i
3) r 7 i−(3−2 i) 7 i−3+2 i −3+9i
5) −6 i−(3+7 i) −6 i−3−7 i −3−13i
7) (3−3i )+(−7−8 i) 3−3i−7−8i −4−11 i
9) i−(2+3 i )−6 i−2−3i−6 −8−2i
11) (6 i)(−8 i) −48i2
−48(−1) 48
13) (−5 i)(8i) −40i2
−40(−1) 40
15) (−7 i )2
49 i2
49 (−1) −49
17) (6+5 i )2
36+60 i+25 i2
36+60 i+25(−1) 36+60 i−25 11+60 i
19) (−7−4 i)(−8+6 i) 56−42 i+32i−24 i2
56−42 i+32i−24 (−1) 56−42 i+32i+24 80−10 i
21) (−4+5 i)(2−7 i) −8+28 i+10 i−35 i2
−8+28 i+10 i−35(−1) −8+28 i+10 i+35 27+38 i
23) (−8−6 i)(−4+2i) 32−16 i+24 i−12 i2
32−16 i+24 i−12(−1) 32−16 i+24 i+12 44+8 i
25) (1+5 i)(2+i) 2+i+10 i+5 i2
2+i+10 i+5 (−1) 2+i+10 i−5 −3+11 i
27)(−9+5 i )
i(i )(i )
=−9 i+5 i2
i2=
−9 i+5(−1)−1
=−9i−5−1
=9 i+5
164
29)(−10−9i )
6i(i )(i )
=−10 i−9i2
6 i2=
−10i−9 (−1 )6 (−1 )
=−10 i+9−6
31)(−3−6 i )4 i
( i )( i )
=−3 i−6i2
4 i2=
−3 i−6 (−1 )4 (−1 )
=−3i+6−4
33)(10−i )−i
(i )(i )
=10 i−i2
−i2=10 i−(−1 )−(−1 )
=10 i+11
=10 i+1
35) 4 i−10+i
(−10−i )(−10−i )
=−40 i−4 i2
100−i2=
−40i−4 (−1)100−(−1 )
=−40i+4100+1
=−40 i+4101
37)8
7−6 i(7+6 i )(7+6 i )
= 56+48 i49−36 i2
= 56+48 i49−36 (−1 )
=56+48 i49+36
=56+48 i85
39)7
10−7 i(10+7 i )(10+7 i )
= 70+49 i100−49 i2
= 70+49 i100−49(−1)
=70+49 i100+49
=70+49 i149
41)5i
−6−i(−6+ i )(−6+ i )
=−30 i+5i2
36−i2=
−30 i+5 (−1 )36−1 (−1 )
=−30 i+5 (−1 )36−1 (−1 )
=−30 i−536+1
=−30 i−537
43) √−81 √−1∙32
32 i 9 i
45) √−10√−2 √−1∙10√−1∙2 i√10 ∙i √2 i2√20 −1√22 ∙5 −1 ∙2√5 −2√5
47) 3+√−276
=3+√−1∙33
6=3+3 i√3
6=3 (1+i√3 )
6=1+√3
2
49) 8−√−164
=8−√−1∙24
4=8−2
2i4
=8−4 i4
=4 (2−i )4
=2−i
165
51) i73=i1=i
53) i48=i0=1
55) i62=i2=−1
57) i154=i2=−1
Chapter 9: Quadratics
166
9.1
1) √2x+3−3=0 +3+3 (√2x+3 )2=32
2 x+3=9 −3−3
2x2
=62
x=3Check: √2 (3 )+3−3=0
√6+3−3=0√9−3=03−3=00=0√
x=3
3) √6 x−5−x=0 +x+x (√6 x−5 )2= x2
6 x−5=x2
−6 x+5−6 x+5 0=x2−6 x+5 0=(x−1)(x−5)
x−1=0x−5=0 +1+1+5+5 x=1 x=5Check: √6 (5 )−5−5=0
√30−5−5=0√25−5=05−5=00=0√
Check: √6 (1 )−5−1=0√6−5−1=0√1−1=01−1=00=0√
x=5 ,1
5) (3+x )2=(√6 x+13 )2
9+6x+x2=6 x+13 −13−6 x−6 x−13 x2−4=0 ( x+2 ) (x−2 )=0 x+2=0 x−2=0 −2−2+2+2 x=−2 x=2Check: 3+(−2)=√6 (−2 )+13
1=√−12+131=√11=1 √
Check: 3+(2 )=√6 (2 )+135=√12+135=√255=5 √
x=−2 ,2
7) √3−3 x−1=2x +1+1 (√3−3 x )2= (2 x+1 )2
3−3x=4 x2+4 x+1 −3+3 x+3 x−3
0=4 x2+7 x−2 0=(4 x−1)(x+2)4 x−1=0 x+2=0 +1+1−2−2
4 x4
=14x=−2
x=14
Check: √3−3 ( 14 )−1=2( 14 )√3−34−1=1
2
167
√ 94−1=1232−1=1
212=12 √
Check: √3−3 (−2)−1=2(−2)√3+6−1=−4√9−1=−43−1=−42=−4No !
x=14
9) √4 x+5−√x+4=2 +√ x+4+√x+4
(√4 x+5 )2=(2+√ x+4 )2
4 x+5=4+4 √x+4+x+4 4 x+5=8+x+4√x+4 −x−8−8−x (3 x−3 )2= (4√ x+4 )2
9 x2−18 x+9=16 (x+4) 9 x2−18 x+9=16 x+64 −16 x−64−16 x−64 9 x2−34 x−55=0 (9 x+11) ( x−5 )=0 9 x+11=0x−5=0 −11−11 +5+5
9 x9
=(−11 )9
x=5
x=−119
Check: √4 (−119 )+5−√−119
+4=2
√−449
+5−√ 259 =2
√ 19−53=2
13−53=2
−2=2No !Check: √4 (5 )+5−√ (5 )+4=2
√20+5−√9=2√25−3=25−3=22=2 √
x=5
11) √2x+4−√x+3=1 +√ x+3+√ x+3 (√2x+4 )2=(1+√x+3 )2
2 x+4=1+2√x+3+x+3 2 x+4=4+x+2√ x+3 −x−4−4−x ( x )2=(2√x+3 )2
x2=4 (x+3)x2=4 x+12
−4 x−12−4 x−12 x2−4 x−12=0 ( x−6 ) ( x+2 )=0 x−6=0 x+2=0 +6+6−2−2 x=6 x=−2Check: √2 (6 )+4−√(6 )+3=1
√12+4−√9=1√16−3=14−3=11=1 √
Check: √2 (−2 )+4−√ (−2 )+3=1√−4+4−√1=1√0−1=10−1=1−1=1No!
x=6
168
13) √2x+6−√ x+4=1 +√ x+4+√x+4
(√2x+6 )2=(1+√ x+4 )2
2 x+6=1+2√x+4+x+4 2 x+6=5+x+2√x+4 −x−5−5−x ( x+1 )2=(2√x+4 )2
x2+2x+1=4(x+4) x2+2x+1=4 x+16 −4 x−16−4 x−16 x2−2 x−15=0 ( x−5 ) ( x+3 )=0 x−5=0 x+3=0 +5+5−3−3 x=5 x=−3Check: √2 (5 )+6−√(5 )+4=1
√10+6−√9=1√16−3=14−3=11=1 √
Check: √2 (−3 )+6−√(−3 )+4=1√−6+6−√1=1√0−1=10−1=1−1=1No!
x=5
15) √6−2x−√2 x+3=3 +√2 x+3+√2 x+3
(√6−2x )2=(3+√2 x+3 )2
6−2x=9+6√2x+3+2 x+36−2x=2x+12+6√2x+3
−12−2 x−2 x−12 (−6−4 x )2=(6√2x+3 )2
36+48 x+16 x2=36 (2x+3) 16 x2+48 x+36=72 x+108 −72 x−108−72 x−108 16 x2−24 x−72=0 8 (2 x2−3 x−9 )=0 8 (2 x+3 ) ( x−3 )=0 2 x+3=0 x−3=0 −3−3+3+3
2x2
=−32
x=3
x=−32
Check: √6−2(−32 )−√2(−32 )+3=3√6+3−√−3+3=3√9−√0=33−0=33=3 √
Check: √6−2(3)−√2 (3 )+3=3√6−6−√6+3=3√0−√9=30−3=3−3=3No !
x=−32
169
9.2
1) √ x2=√75 x=±√75 x=±√52 ∙3 x=±5√3
3) x2+5=13 −5−5 √ x2=√8 x=±√23 x=±2√2
5) 3 x2+1=73 −1−1
3x2
3=723
√ x2=√24 x=±√24 x=±√23 ∙3 x=±2√2 ∙3 x=±2√6
7) 5√ ( x+2 )5=5√−243 x+2=−3
−2−2 x=−5
9) (2 x+5 )3−6=21 +6+6 3√ (2x+5 )3=3√27 2 x+5=3
−5−5
2x2
= (−2 )2
x=−1
11) ( x−1 )23=16 `
❑√ ( 3√ x−1 )2=❑√16 ( 3√ x−1 )3=(±4 )3
x−1=±64 +1+1
x=1±64x=65.−63
13) (2−x )32=27
3√ (√2−x )3=¿ 3√27¿ (√2−x )2=32
2−x=9 −2−2
−x−1
= 7−1
x=−7
Check: (2−(−7 ) )32=27
932=27
(√9 )3=27 33=27 27=27 √
15) (2 x−3 )23=4
√ ( 3√2x−3)2=√4 ( 3√2 x−3 )3=(±2)3
2 x−3=±8 +3+3
2x2
=3±82
x=112,−52
17) (x+ 12 )−23 =4
√( 3√ 1
x+ 12 )
2
=√4
( 3√ 1
x+ 12 )
3
=(±2 )3
1
x+ 12
=±8
(x+ 12 ) 1
x+12
=±8(x+ 12 )
1±8
=±8(x+ 12 )
±8
± 18=x+ 1
2
−12
−12
−12
± 18=x
x=−38
,−58
19) r ( x−1 )−52 =32
170
( 1x−1 )
52=32
5√(√ 1x−1 )
5
= 5√32
(√ 1x−1 )
2
=(2 )2
( x−1 ) 1x−1
=4 (x−1)
1=4 x−4 +4+4
54=4 x4
54=x
Check: ( 54−1)−52 =32
( 14 )−52 =32
452=32
(√4 )5=32 25=32
32=32 √
x=54
21) (3 x−2 )45=16
4√ ( 5√3 x−2 )4=4√16
( 5√3 x−2 )5=+25
3 x−2=±32 +2+2
3x3
=2±323
x=343,−10
23) (4 x+2 )35=−8
3√ ( 5√4 x+2)3=3√−8 ( 5√4 x+2 )5=(−2 )5
4 x+2=−32 −2−2
4 x4
=−344
x=−172
9.3
1) x2−30 x+¿¿
(−30 ∙ 12 )2
(−15 )2=225 x2−30 x+225 ( x−15 )2
3) m2−36m+¿¿
(−36 ∙ 12 )2
(−18 )2=324 m2−36m+324 (m−18 )2
5) x2−15 x+¿¿
(−15 ∙ 12 )2
(−152 )2
= 2254
x2−15 x+ 2254
(x−152 )2
7) y2− y +¿¿
(−1 ∙ 12 )2
171
(−12 )2
=14
y2− y+ 14
( y−12 )2
9) x2−16 x+55=0 −55−55 x2−16 x=−55
(−16 ∙ 12 )2
(−8 )2=64 x2−16 x+64=−55+64 f √ ( x−8 )2=√9
x−8=±3 +8+8 x=8±3x=11 ,5
11) v2−8v+45=0 −45−45 v2−8v=−45
(−8∙ 12 )2
(−4 )2=16 v2−8v+16=−45+16 √ (v−4 )2=√−29
v−4=± i √29 +4+4 v=4 ±i√29
13) 6 x2+12x+63=0 −63−63
6 x2
6+12 x6
=−636
x2+2x=−212
(2 ∙ 12 )2
(1 )2=1
x2+2x+1=−212
+1
√ ( x+1 )2=√−192 (√2√2 )
x+1=± i √382
−1−1
x=−2±i √382
15) 5k2−10 k+48=0 −48−48
5k2
5−10k5
=−485
k 2−2 k=−485
(−2 ∙ 12 )2
(−1 )2=1
k 2−2 k+1=−485
+1
√ (k−1 )2=√−485 (√5√5 )
k−1=± i √2155
+1+1
k=5± i √2155
17) x2+10 x−57=4 +57+57
172
x2+10x=61
(10 ∙ 12 )2
(5 )2=25 x2+10x+25=61+25 √ ( x+5 )2=√86 x+5=±√86 −5−5 x=−5±√86
19) n2−16n+67=4 −67−67 n2−16n=−63
(−16 ∙ 12 )2
=(−8 )2=64
n2−16n+64=63+64 √ (n−8 )2=√1 n−8=±1 +8+8 n=9 ,7
21) 2 x2+4 x+38=−6 −38−38
2x2
2+ 4 x2
=−442
x2+2x=−22
(2 ∙ 12 )2
=11=1
x2+2x+1=−22+1 √ ( x+1 )2=√−21 x+1=± i√21 −1−1 x=−1± i√21
23) 8b2+16b−37=5 +37+37
8b2
8+ 16b8
= 428
b2+2b=214
(2 ∙ 12 )2
=11=1
b2+2b+1=214+1
√ (b+1 )2=√ 254 b+1=± 5
2 −1−1
b=−1± 52
b=32,−72
25) r x2=−10 x−29 +10 x+10x x2+10x=−29
(10 ∙ 12 )2
=(5 )2=25
x2+10x+25=−29+25 √ ( x+5 )2=√−4 x+5=±2 i −5−5 x=−5±2i
27) n2=−21+10n −10n −10n n2−10n=−21
(−10 ∙ 12 )2
=(−5 )2=25
n2−10n+25=−21+25 √ (n−5 )2=√4 n−5=±2 +5+5 n=5±2 n=7 ,3
173
29) 3k2+9=6k −6 k−9−6k−9
3k2
3−6 k3
=−93
k 2−2k=−3
(−2 ∙ 12 )2
=(−1 )2=1
k 2−2k+1=−3+1 √ (k−1 )2=√−2 k−1=± i√2 +1+1 k=1± i√2
31) 2 x2+63=8 x −8 x−63−8 x−63
2x2
2−8x2
=−632
x2−4 x=−632
(−4 ∙ 12 )2
=(−2 )2=4
x2−4 x+4=−632
+4
√ ( x−2 )2=√−552 (√2√2 )
x−2=± i√1102
+2+2
x=4± i √1102
33) p2−8 p=−55
(−8∙ 12 )2
=(−4 )2=16
p2−8 p+16=−55+16 √ (p−4 )2=√−39 p−4=± i√39 +4+4 p=4± i√39
35) 7n2−n+7=7n+6n2 −6n2−7n−7−6n2−7n−7 n2−8n=−7
(−8∙ 12 )2
=(−4 )2=16
n2−8n+16=−7+16 √ (n−4 )2=√9 n−4=±3 +4+4 n=4±3 n=7 ,1
37) 13b2+15b+44=−5+7b2+3b −7b2−3b−44−44−7b2−3b
6b2
6+ 12b6
=−496
b2+2b=−496
(2 ∙ 12 )2
=12=1
b2+2b+1=−496
+1
√ (b+1 )2=√−436 (√6√6 )
174
b+1=± i √2566
−1−1
b=−6±i√2566
39) 5 x2+5 x=−31−5 x +5 x +5 x
5x2
5+ 10 x5
=−315
x2+2x=−315
(2 ∙ 12 )2
=12=1
x2+2x+1=−315
+1
√ ( x+1 )2=√−265 (√5√5 )
x+1=± i√1305
−1−1
x=−5±i√1305
41) v2+5 v+28=0 −28−28 v2+5 v=−28
(5 ∙ 12 )2
=( 52 )2
=254
v2+5 v+ 254=−28+ 25
4
√(v+52 )2
=√−874
v+ 52=± i √87
2
−52
−52
v=−5± i √872
43) 7 x2−6 x+40=0 −40−40
7 x2
7−6 x7
=−407
x2−67x=−40
7
(−67 ∙ 12 )
2
=(−37 )2
= 949
x2−67x+ 949
=−407
+ 949
√(x−37 )2
=√−27149
x−37=± i√271
7
+37
+ 37
x=3±i √2717
45) k 2−7 k+50=3 −50−50 k 2−7k=−47
(−7 ∙ 12 )2
=(−72 )2
=494
k 2−7 k+ 494=−47+ 49
4
√(k−72 )2
=√−1394
k−72=± i √139
2
+72
+ 72
k=7± i√1392
47) 5 x2+8x−40=8 +40+40
5x2
5+ 8 x5
=485
175
x2+ 85x=48
5
( 85 ∙ 12 )2
=( 45 )2
=1625
x2+ 85x+ 1625
=485
+ 1625
√(x+ 45 )2
=√ 25625 x+ 45
=± 165
−45
−45
x=−4 ±165
49) m2=−15+9m −9m−9m m2−9m=−15
(−9∙ 12 )2
=(−92 )2
=814
m2−9m+ 814
=−15+ 814
√(m−92 )2
=√ 214 m−9
2=± √21
2
+92
+ 92
m=9±√212
51) 8 r2
8+ 10 r8
=−558
r2+ 54r=−55
8
( 54 ∙ 12 )2
=( 58 )2
=2564
r2+ 54r+ 2564
=−558
+ 2564
√(r+ 58 )2
=√−41564
r+ 58=± i√416
8
−58
−58
r=−5± i√4158
53) 5n2−8n+60=−3 n−6+4n2
−4n2+3n−60+3n−60−4 n2
n2+5n=−54
(5 ∙ 12 )2
=( 52 )2
=254
n2+5n+ 254=−54+25
4
√(n+52 )2
=√−1914
n+ 52=± i√191
2
−52
−52
n=−5± i√1912
55) 2 x2+3 x−5=−4 x2
+4 x2+5+4 x2+5
6 x2
6+3 x6
=56
x2+ 12x=56
( 12 ∙ 12 )2
=( 14 )2
= 116
x2+ 12x+ 116
=56+ 116
√(x+ 14 )2
=√ 4348= √434 √3 ( √3√3 )
x+ 14=± √129
12
176
−14
−14
x=(−3±√129 )
12
57) −2 x2+3 x−5=−4 x2
+4 x2+5+4 x2+5
2x2
2+ 3 x2
=52
x2+ 32x=52
( 32 ∙ 12 )2
=( 34 )2
= 916
x2+ 32x+ 916
=52+ 916
√(x+ 34 )2
=√ 4916 x+ 34
=± 74
−34
−34
x=−34
± 74
x=1 ,−52
9.4
1) 4 a2+6=0 a=4 , b=0 ,c=6
−0±√02−4 (4 ) (6 )
2 (4 )=±√−96
8=±√−16 ∙6
8=±4 i√6
8=± i√6
2
3) 2 x2−8 x−2=0 a=2 , b=−8 , c=−2
8±√ (−8 )2−4 (2 ) (−2 )2 (2 )
=8±√64+164
=8±√804
=8±√16 ∙54
=8±4√54
=2±√5
5) 2m2−3=0 a=2 , b=0 ,c=−3
−0±√(0 )2−4 (2 ) (−3 )2 (2 )
=±√244
=±√4 ∙64
=(±2√6 )4
=±√62
7) 3 r2−2 r−1=0 a=3 , b=−2 ,c=−1
177
2±√ (−2 )2−4 (3 ) (−1 )2 (3 )
=2±√4+126
=2±√166
=2± 46
=1 ,−13
9) 4 n2−36=0 a=4 , b=0 ,c=−36
−0±√02−4 (4 ) (−36 )
2 ( 4 )=±√576
8=±248
=±3
11) v2−4 v−5=−8 +8+8 v2−4 v+3=0 a=1 , b=−4 , c=3
4±√ (−4 )2−4 (1 ) (3 )2 (1 )
=4±√16−122
=4±√42
= 4±22
=3,1
13) 2a2+3a+14=6 −14−14 2a2+3a+8=0 a=2 , b=3 , c=8 −3±√¿¿¿
15) 3k2+3k−4=7 −7−7 3k2+3k−11=0 a=3 , b=3 ,c=−11
−3±√32−4 (3 ) (−11 )
2 (3 )=−3±√9+132
6=−3±√141
6
17) 7 x2+3 x−16=−2 +2+2 7 x2+3x−14=0 a=7 , b=3 , c=−14
−3±√32−4 (7 ) (−14 )
2 (7 )=−3±√9+392
14=−3±√401
14
178
19) 2 p2+6 p−16=4 −4−4 2 p2+6 p−20=0 a=2 , b=6 ,c=−20
−6±√62−4 (2 ) (−20 )
2 (2 )=−6±√36+160
4=−6±√196
4=−6±14
4=2 ,−5
21) 3n2+3n=−3
+3+3 3n2+3n+3=0 a=3 , b=3 ,c=3
−3±√ (3 )2−4 (3 ) (3 )2 (3 )
=−3±√9−366
=−3±√−9∙36
=−3±3 i √36
=−1±i√36
23) 2 x2=−7 x+49 +7 x−49+7 x−49 2 x2+7 x−49=0 a=2 , b=7 ,c=−49
−7±√(7 )2−4 (2 ) (−49 )2 (2 )
=−7±√49+3924
=−7±√4414
25) 5 x2=7 x+7 −7 x−7−7 x−7 5 x2−7 x−7=0 a=5 , b=−7 , c=−7
7±√ (−7 )2−4 (5 ) (−7 )2 (5 )
=7±√49+140¿ ¿10
=7±√18910
=7±√9 ∙2110
=7±3√2110
27) 8n2=−3n−8 +3n+8+3n+8 8n2+3n+8=0 a=8 , b=3 , c=8
−3±√32−4 (8 ) (8 )
2 (8 )=−3±√9−256
16=−3±√−247
16=−3±i√247
16
29) 2 x2+5 x=−3
179
+3+3 2 x2+5 x+3=0 a=2 , b=5 , c=3
−5±√52−4 (2 ) (3 )
2 (2 )=−5±√25−24
4=−5±√1
4=−5±1
4=−1 ,−3
2
31) 4 a2−64=0 a=4 , b=0 ,c=−64
−0±√o2−4 (4 ) (−64 )
2 (4 )=±√1024
8=±328
=±4
33) 4 p2+5 p−36=3 p2
−3 p2−3 p2
p2+5 p−36=0 a=1 , b=5 , c=−36
−5±√52−4 (1 ) (−36 )
2 (1 )=−5±√25+144
2=−5±√169
2=−5±13
2=4 ,−9
35) −5n2−3n−52=2−7n2 +7n2−2−2+7n2
2n2−3n−54=0 a=2 , b=−3 ,c=−54
3±√ (−3 )2−4 (2 ) (−54 )2 (2 )
=3±√9+4324
=3±√4414
=3±214
=6 ,−92
37) 7 r2−12=−3 r +3 r+3 r 7 r2+3 r−12=0 a=7 , b=3 , c=−12
−3±√32−4 (7 ) (−12 )
2 (7 )=−3±√9+336
14=−3±√345
14
39) 2n2−9=4 −4−4 2n2−13=0 a=2 , b=0 ,c=−13
180
−0±√02−4 (2 ) (−13 )
2 (2 )=±√104
4=±√4 ∙26
4=±2√26
4=±√26
2
9.5
1) 2 ,5 x=2 , x=5 −2−2−5−5 x−2=0 x−5=0 ( x−2 ) ( x−5 )=0 x2−5 x−2 x+10=0 x2−7 x+10=0
3) 20 ,2 x=20 x=2 −20−20−2−2 x−20=0 x−2=0 ( x−20 ) ( x−2 )=0 x2−20 x−2 x+40=0 x2−22 x+40=0
5) 4 ,4 x=4 x=4 −4−4−4−4 x−4=0 x−4=0 ( x−4 ) ( x−4 )=0 x2−4 x−4 x+16=0 x2−8 x+16=0
7) 0 ,0 x=0 , x=0 xx=0 x2=0
9) −4 ,11
x=−4 x=11 +4+4−11−11 x+4=0 x−11=0 ( x+4 ) (x−11)=0 x2+4 x−11 x−44=0 x2−7 x−44=0
11) 34, 14
4 ( x )=( 34 )4 ,4 ( x )=( 14 )4 4 x=34 x=1 −3−3−1−1 4 x−3=04 x−1=0 (4 x−3 ) (4 x−1 )=0 16 x2−12x−4 x+3=0 16 x2−8 x+3=0
13)12, 13
(2 ) x=12
(2 ) (3 ) x=13(3)
2 x=13 x=1 −1−1−1−1 2 x−1=03 x−1=0 (2 x−1 ) (3 x−1 )=0 6 x2−3 x−2 x+1=0 6 x2−5 x+1=0
15)37,4
(7 ) x=37
( x ) x=4
7 x=3−4−4 −3−3 x−4=0 7 x−3=0 (7 x−3 ) ( x−4 )=0 7 x2−3 x−28 x+12=0 7 x2−31x+12=0
17)−13
, 56
(3 ) x=−13
(3 )6 ( x )=56(6)
3 x=−16 x=5 +1+1−5−5 3 x+1=06 x−5=0 (3 x+1 ) (6 x−5 )=0 18 x2+6 x−15 x−5=0 18 x2−9 x−5=0
19) −6 , 19
x=−6 (9 ) x=19(9)
+6+69 x=1 x+6=0 −1−1
9 x−1=0 ( x+6 ) (9 x−1 )=0 9 x2+54 x−x−6=0 9 x2−53 x−6=0
21) ±5 x2=(±5 )2
181
x2=25 −25−25 x2−25=0
23) ± 15
x2=(± 15 )2
(25)x2= 125
(25)
25 x2=1 −1−1 25 x2−1=0
25) ±√11
x2=(±√11 )2
x2=11 −11−11 x2−11=0
27) ± √34
4 x=± √34
(4)
¿ 16 x2=3 −3−3 16 x2−3=0
29) ± i√13
x2=(± i√13 )2 x2=−13
+13+13 x2+13=0
31) 2±√6 x=2±√6 −2−2 ( x−2 )2= (±√6 )2
x2−4 x+4=6 −6−6 x2−4 x−2=0
33) 1±3 i x=1±3 i −1−1 ( x−1 )2= (±3 i )2
x2−2 x+1=−9 +9+9 x2−2 x+10=0
35) 6± i√3
x=6± i√3 −6−6 ( x−6 )2=(± i√3 )2
x2−12 x+36=−3 +3+3 x2−12 x+39=0
37) −1±√62
(2)x=−1±√62
(2)
2 x=−1±√6 +1+1 (2 x+1 )2=(±√6 )2
4 x2+4 x+1=6 −6−6
4 x2+4 x−5=0
39) 6±i√28
(8 ) x=6±i √28
(8)
8 x=6± i√2 −6−6 (8 x−6 )2=(± i√2 )2
64 x2−96x+36=−2 +2+2 64 x2−96x+38=0
182
9.6
1) x4−5x2+4=0 y=x2 , y2=x4
y2−5 y+4=0 ( y−4 ) ( y−1 )=0 y−4=0 y−1=0 +4+4+1+1 y=4 y=1 √ x2=√4 x2=√1 x=±2 ,±1
3) m4−7m2−8=0 y=m2 y2=m2
y2−7 y−8=0 ( y−8 ) ( y+1 )=0 y−8=0 y+1=0 +8+8−1−1 y=8 y=−1 √m2=√8√m2=√(−1 )2
m=±2√2 , ±i
5) a4−50a2+49=0 y=a2 y2=a4
y2−50 y+49=0 ( y−49 ) ( y−1 )=0 y−49=0 y−1=0 +49+49+1+1 y=49 y=1 √a2=√49√a2=√1 a=±7 ,±1
7) x4−25x2+144=0 y=x2 , y2=x4
y2−25 y+144=0 ( y−9 ) ( y−16 )=0 y−9=0 y−16=0 +9+9+16+16 y=9 y=16 √ x2=√9√x2=√16 x=±3 , ±4
9) m4−20m2+64=0 y=m2 y2=m4
y2−20 y+64=0 ( y−4 ) ( y−16 )=0 y−4=0 y−16=0 +4+4+16+16 y=4 y=16 √m2=√4 √m2=√16 m=±2 , ±4
183
11) z6−216=19 z3 y=z3 y2=z6
y2−216=19 y −19 y−19 y y2−19 y−216=0 ( y−27 ) ( y+8 )=0 y−27=0 y+8=0 +27+27−8−8 y=27 y=−8 z3=27 z3=−8 −27−27+8+8 z3−27=0 z3−8=0 ( z−3 ) ( z2+3 z+9 )=0 z−3=0 z2+3 z+9=0
+3+3−3±√32−4 (1 ) (9 )2 (1 )
=2±√−272
=−3±3i√32
z=3 ( z+2 ) ( z2−2 z+4 )=0 z+2=0 z2−2 x+4=0
−2−2 2±√ (−2 )2−4 (1 ) (4 )2
= 2±√−122
=2±2 i√32
=1±i√3
z=−2
z=3 ,−3±3 i√32
,−2 ,1± i√3
13) 6 z4−z2=12 y=z2 , y2=z4
6 y2− y=12 −12−12 6 y2− y−12=0 (3 y+4 ) (2 y−3 )=0
3 y+4=02 y−3=0 −4−4+3+3
3 y3
=−432 y2
=32
y=−43
, 32
184
√ z2=√−43 (√3√3 )√z2=√ 32 (√2√2 )
z=±2i √33
, ±√62
15) x23−35=2 x
13
y=x13 , y2=x
23
y2−35=2 y −2 y−2 y
y2−2 y−35=0 ( y−7 ) ( y+5 )=0 y−7=0 y+5=0 +7+7−5−5 y=7 , y=−5
x13=−5 x
13=7
( 3√ x )3=(−5 )3 ( 3√x )3=73 x=−125 ,343
17) y−6+7 y−3=8 z= y−3 z2= y−6
z2+7 z=8 −8−8 z2+7 z−8=0 ( z+8 ) ( z−1 )=0 z+8=0 z−1=0 −8−8+1+1 z=−8 , z=1 y−3=−8 , y−3=1
( y3 )( 1y3 )=−8 ( y3 ) ( y3 ) 1y3
=1 ( y3 )
1=−8 y31= y3
+8 y3+8 y3−1−1 8 y3+1=00= y3−1 (2 y+1 ) (4 y2−2 y+1 )=0 0=( y−1)( y2+ y+1) 2 y+1=04 y2−2 y+1=0 y−1=0 y2+ y+1=0
−1−1 2±√ (−2 )2−4 (4 ) (1 )2 (4 )
+1+1−1±√11−4 (1 ) (1 )2 (1 )
2 y2
=−122±√−12
8y=1−1±√−3
2=−1± i√3
2
y=−122±2 i√38
=1± i√34
185
y=−12
, 1± i √34
,1 ,−1± i√32
19) x4−2x2−3=0 y=x2 y2=x4
y2−2 y−3=0 ( y−3 ) ( y+1 )=0 y−3=0 y+1=0 +3+3−1−1 y=3 y=−1 √ x2=√3√x2=√−1 x=±√3 ,± i
21) 2 x4−5 x2+2=0 y=x2 , y2=x4 2 y2−5 y+2=0 (2 y−1 ) ( y−2 )=0 2 y−1=0 y−2=0 +1+1+2+2
2 y2
=12y=2
y=12
√ x2=√ 12 (√2√2 )√ x2=√2
x=± √22
,±√2
23) x4−9 x2+8=0 y=x2 , y2=x4
y2−9 y+8=0
186
( y−8 ) ( y−1 )=0 y−8=0 y−1=0 +8+8+1+1 y=8 y=1 √ x2=√8√x2=√1 x=±2√2 , ±1
25) 8 x6−x3+1=0 y=x3 , y2=x6
8 y2− y+1=0 (8 y−1 ) ( y−1 )=0 8 y−1=0 y−1=0 +1+1+1+1
8 y8
=18y=1
y=18
(8 ) x3=18
(8 ) x3=1
8 x3=1−1−1 −1−1 x3−1=0 8 x3−1=0 ( x−1 ) (x2+x+1 )=0 (2 x−1)¿ x−1=0x2+x+1=0
2 x−1=0 4 x2+2 x+1=0 +1+1−1±√11−4 (1 ) (1 )2 (1 )
+1+1−2±√22−4 (4 ) (1 )2 (4 )
x=1−1±√−32
=−1± i√32
2x2
=12−2±√−12
8
x=12
−2±2 i√38
=−1± i√34
x=12,−1± i√3
4,1 ,−1± i√3
2
27) x8−17 x4+16=0 y=x4 , y2=x8
y2−17 y+16=0 ( y−16 ) ( y−1 )=0
187
y−16=0 y−1=0 +16+16+1+1 y=16 y=1 x4=16x 4=1 −16−16−1−1 x4−16=0 x4−1=0 (x2+4 ) (x2−4 )=0 (x2−1 ) (x2+1 )=0 x2+4=0 x2−4=0 x2−1=0x2+1=0 −4−4+4+4 +1+1−1−1 √ x2=√−4 √x2=√4 √ x2=√1√ x2=√−1 x=±2i x=±2 x=±1 x=± i x=±2i , ±2 , ± i ,±1
29) ( y+b )2−4 ( y+b )=21
z=( y+b ) , z2=( y+b )2 z2−4 z=21 −21−21 z2−4 z−21=0 ( z−7 ) ( z+3 )=0 z−7=0 z+3=0 +7+7−3−3 z=7 z=−3 y+b=7 y+b=−3 −b−b−b−b y=7−b y=−3−b
31) ( y+2 )2−6 ( y+2 )=16
z= y+2 , z2=( y+2 )2
z2−6 z=16 −16−16 z2−6 z−16=0 (z−8)(z+2)=0 z−8=0 z+2=0
188
+8+8−2−2 z=8 z=−2 y+2=8 y+2=−2 −2−2−2−2 y=6 y=−4
33) ( x−3 )2−2 ( x−3 )=35
y= (x−3 ) , y2=( x−3 )2
y2−2 y=35 −35−35 y2−2 y−35=0 ( y−7 ) ( y+5 )=0 y−7=0 y+5=0 +7+7−5−5 y=7 y=−5 x−3=7 x−3=−5 +3+3+3+3 x=10 ,−2
35) (r−1 )2−8 (r−1 )=20 y= (r−1 ) , y2=(r−1 )2
y2−8 y=20 −20−20 y2−8 y−20=0 ( y−10 ) ( y+2 )=0 y−10=0 y+2=0 +10+10−2−2 y=10 y=−2 r−1=10 r−1=−2 +1+1+1+1 r=11 ,−1
37) 3 ( y+1 )2−14 ( y+1 )=5
z=( y+1 ) , z2=( y+1 )2
3 z2−14 z=5 −5−5 3 z2−14 z−5=0 (3 z+1 ) ( z−5 )=0 3 z+1=0 z−5=0 −1−1+5+5
3 z3
=−13
z=5
z=−13
y+1=−13
y+1=5
−1−1−1−1
y=−43
,4
39) (3 x2−2 x )2+5=6(3x2−2 x)
y=(3x2−2 x ) , y2=(3 x2−2x )2 y2+5=6 y
189
−6 y−6 y y2−6 y+5=0 ( y−1 ) ( y−5 )=0 y−1=0 y−5=0 +1+1+5+5 y=1 y=5 3 x2−2 x=13 x2−2 x=5 −1−1−5−5 3 x2−2 x−1=03 x2−2x−5=0 (3 x−5 ) ( x+1 )=0 (3 x+1 ) ( x−1 )=0 3 x−5=0 x+1=03x+1=0x−1=0 +5+5−1−1−1−1+1+1
3x3
=53x=−1 3x
3=−13
x=1
x=53,−1 ,−1
3,1
41) 2 (3x+1 )23−5 (3 x+1 )
13=88
y= (3 x+1 )13, y2=(3 x+1 )
23
2 y2−5 y=88 −88−88 2 y2−5 y−88=0 (2 y+11) ( y−8 )=0 2 y+11=0 y−8=0 −11−11+8+8
2 y2
=−112
y=8
y=−112
190
(3 x+1 )13=
−112
(3 x+1 )13=8
( 3√3 x+1 )3=(−112 )3
( 3√3x+1 )3=83
3 x+1=−13318
3 x+1=512
−1−1 −1−1
3x3
=(−132983 ) 3 x3 =5113
x=132924, 5113
43) (x2+2x )2−2 (x2+2 x)=3
y=( x2+2 x ) , y2= (x2+2 x )2
y2−2 y=3 −3−3 y2−2 y−3=0 ( y−3 ) ( y+1 )=0 y−3=0 y+1=0 +3+3−1−1 y=3 y=−1 x2+2x=3 x2+2x=−1 −3−3+1+1 x2+2x−3=0 x2+2x+1=0 ( x+3 ) ( x−1 )=0¿ x+3=0x−1=0 x+1=0 −3−3+1+1−1−1 x=−3 ,1 ,−1
45) (2 x2−x )2−4 (2x2−x )+3=0
y=(2x2−x ) , y2=(2 x2−x )2
y2−4 y+3=0 ( y−3 ) ( y−1 )=0 y−3=0 y−1=0
191
+3+3+1+1 y=3 y=1 2 x2−x=3 2 x2−x=1 −3−3 −1−1 2 x2−x−3=0 2 x2−x−1=0 (2 x−3 ) ( x+1 )=0 (2 x+1 ) ( x−1 )=0 2 x−3=0x+1=0 2 x+1=0 x−1=0 +3+3−1−1 −1−1+1+1
2x2
=32x=−1
2x2
=−12
x=1
x=32,−1 ,−1
2,1
9.7
1) In a landscape plan, a rectangular flowerbed is designed to be 4 meters longer than it is wide. If 60 square meters are needed for the plants in the bed, what should the dimensions of the rectangular bed be?
x (x+4 )=60 x+4 x2+4 x=60
−60−60 60 x x2+4 x−60=0
( x−6 ) ( x+10 )=0x−6=0 x+10=0 +6+6−10−10 x=6 x=−10
6mby10m
3) A rectangular lot is 20 yards longer than it is wide and its area is 2400 square yards. Find the dimensions of the lot
x (x+20 )=2400 x+20 x2+20x=2400
192
2400 x (20 ∙ 12 )2
=102=100
x2+20x+100=2400+100
√ ( x+10 )2=√2500 x+10=±50 −10−10 x=−10±50
x=40 ,−60 40 yds x60 yds
5) The length of a rectangular lot is 4 rods greater than its width, and its area is 60 square rods. Find the dimensions of the lot.
x (x+4 )=60 x+4 x2+4 x=60
−60−60 60 x x2+4 x−60=0
( x+10 ) (x−6 )=0x+10=0 x−6=0 −10−10+6+6 x=−10 x=6
6 rods x10 rods
7) A rectangular piece of paper is twice as long as a square piece and 3 inches wider. The area of the rectangular piece is 108 i n2. Find the dimensions of the square piece.
2 x ( x+3 )=108 2 x 2 x2+6 x=108
−108−108
108 x+3 2x2
2+ 6 x2
−1082
=02
x2+3x−54=0( x+9 ) ( x−6 )=0 x+9=0x−6=0 −9−9+6+6 x=−9x=6
6∈x6∈¿
9) The area of a rectangle is 48 f t 2 and its perimeter is 32 ft. Find its length and width.
193
2 x+2 y=32 x −2 x−2x
2 y2
=322
−2 x2
48 y y=16−x p=32 xy=48
x (16−x )=48 16 x−x2=48 −16 x+x2−16 x+x2
0=x2−16 x+480=(x−12)(x−4)x−12=0x−4=0 +12+12+4+4 x=12 x=4
12 ft x 4 ft y=16−12=4 y=16−4=12
11) A mirror 14 inches by 15 inches has a frame of uniform width. If the area of the frame equals that of the mirror, what is the width of the frame?
(15+2 x ) (14+2x )=420 210+30 x+28+4 x2=420 14 4 x2+58 x+210=420
15 15+2 x −420−420
4 x2
2+ 58 x2
−−2102
=02
2 x2+29 x−105=0 14+2x (2 x+35 ) ( x−3 )=0 2 x+35=0 x−3=0 A=2 (14 ∙15 )=420 −35−35+3+3
3∈¿ 2x2
=−352
=x=−352
x=3
194
13) A grass plot 9 yards long and 6 yards wide has a path of uniform width around it. If the area of the path is equal to the area of the plot, determine the width of the path.
(6+2 x ) (9+2x )=10854+12x+18 x+4 x2=108
9 4 x2+30 x+54=108 66+2 x −108−108
4 x2
2+ 30 x2
−542
=02
2 x2+15 x−27=09+2 x (2 x−3 ) ( x+9 )=0
2 x−3=0x+9=0 A=2 (9∙6 )=108 +3+3−9−9
2x2
=32x=−9
x=32=1.5
1.5 yds
15) A page is to have a margin of 1 inch, and is to contain 35 i n2of painting. How large must the page be if the length is to exceed the width by 2 inches?
x (x−2 )=35(x−2) x2−2 x=35
−35−35x2−2 x−35=0( x−7 ) ( x+5 )=0
x−7=0 x+5=0 +7+7−5−5 x=7 x=−5
7∈x 9∈¿
17) A rectangular wheat field is 80 rods long by 60 rods wide. A strip of uniform width is cut around the field, so that half the grain is left standing in the form of a rectangular plot. How wide is the strip that is cut?
(80−2x ) (60−2x )=240048000−160 x−120 x+4 x2=2400
80−2x 60−2x 4 x2−280x+4800=2400 −2400−2400
60 4 x2
4−280 x
4+ 24004
=04
195
1 1
35 1 X 1
x2−70 x+600=0 80 ( x−10 ) ( x−60 )=0
A=12
(60∙80 )=2400 x−10=0 x−60=0
+10+10+60+6010 rods x=10 x=60
19) A rectangular field 225 ft by 120 ft has a ring of uniform width cut around the outside edge. The ring leaves 65% of the field uncut in the center. What is the width of the ring? (225−2x ) (120−2 x )=17500
225−2x 120−2x 27000−450 x−240 x+4 x2=17500 120 4 x2−690 x+27000=17500
−17500−17500
4 x2
2−690 x
2+ 94502
=02
225 2 x2−345 x+4725=0A=0.65 (120 ∙225 )17500
345±√3452−4 (2 ) (4725 )2 (2 )
=345±√812254
=345±2854
=3152
∧15
15 ft
21) A frame is 15 in by 25 in and is of uniform width. The inside of the frame leaves 75% of the total area available for the picture. What is the width of the frame?
(15−2x ) (25−2 x )=281.25
375−30x−50x+4 x2=281.2515−2x 25−2 x 4 x2−80 x+375=281.25
−281.25−281.25
25 4 x2
4−80x4
+ 93.754
=04
x2−20 x+23.4375=0 15 −23.4375−23.4375
x2−20 x=−23.4375
A=.75 (25 ∙15 )=281.25 (−20 ∙ 12 )2
=(−10 )2=100
x2−20 x+100=−23.4375+100
√ ( x−10 )2=√(76.5625) x−10=±8.75
+10+10 x=10±8.75x=18.75 ,1.25
196
1.25∈¿
23) The farmer in the previous problem has a neighbor who has a field 325 ft by 420 ft. His neighbor wants to increase the size of his field by 20% by cultivating a band of uniform width around the outside of his lot. How wide a band should his neighbor cultivate?
(420+2x ) (325+2 x )=163800325 420 136500+840 x+650 x+4 x2=163800 420+2x 4 x2+1490 x+136500=163800
−163800−1638004 x2
4+ 1490 x
4−27300
4=04
325+2 x x2+745x−13650=0 A=1.2 (325 ) (420 )=163800 −745±√¿¿¿
−745±815
4=17.5 ,−390
17.5 ft
25) Donna has a garden that is 30 ft by 36 ft. She wants to increase the size of the garden by 40%. How wide a ring around the outside should she cultivate? (36+2 x ) (30+2x )=1512 3036+2 x 1080+72 x+60 x+4 x2=1512
4 x2+132 x+1080=1512 36 −1512−1512
4 x2
4+ 132x
4−4324
=04
30+2 x x2+33x−108=0 A=1.4 (30 ∙36 )=1512−33±√332−4 (1 ) (−108 )
2 (1 )=−33±√1521
2=−33±39
2=3 ,−36
3 ft .
197
9.8
1) Bills father can paint a room in two hours less than Bill can paint it. Working together they can complete the job in two hours and 24 minutes. How much time would each require working alone?
Father : x−2 1x−2
12x ( x−2 )+ 1x12x (x−2 )= 5
1212x (x−2)
Bill : x LCD :12 x (x−2)
Team :2 2460=2 25=125 12 x+12 ( x−2 )=5 x (x−2)
12 x+12 x−24=5 x2−10x 24 x−24=5 x2−10 −24 x+24−24 x+24
0=5 x2−34 x+240=(5 x−4)(x−6)5 x−4=0x−6=0 +4+4+6+6
5x5
=45x=6
x=.8Bill :6hr ,Father : 4
3) Jack can wash and wax the family car in one hour less than Bob can. The two working together can
complete the job in 1 15hours. How much time would each require if they worked alone?
198
Jack : x−1 1x−1 (6 x (x−1 ) )+ 1x (6 x ( x−1 ) )=56 (6 x ( x−1 ) )
Bob : x LCD :6 x (x−1)
Team :1 15=65 6 x+6 ( x−1 )=5 x (x−1)
6 x+6 x−6=5x2−5 x12 x−6=5 x2−5 x −12 x+6−12 x+6
0=5 x2−17 x+6 0=(5 x−2)(x−3)5 x−2=0x−3=0 +2+2+3+3
5x5
=25x=3
x=.4Bob :3hr , Jack :2hr
5) Working alone it takes John 8 hours longer than Carlos to do a job. Working together they can do the job in 3 hours. How long will it take each to do the job working alone
John: x+8 1x+8
3 x ( x+8 )+ 1x3x ( x+8 )=1
33 x (x+8)
Carlos : x LCD :3 x (x+8) Team :3 3 x+3 (x+8 )=x (x+8)
3 x+3x+24=x2+8 x 6 x+24=x2+8 x −6 x−24−6 x−24
0=x2+2 x−24 0=(x+6)(x−4)x+6=0x−4=0 −6−6+4+4 x=−6 x=4
Carlos : 4hr , John :12hr
7) A can do a piece of work in 4 days and B can do it in half the time. How long will it take them to do the work together?
A :4 14
(4 x )+ 12
(4 x )=1x(4 x)
199
B: 2 LCD : 4 x Team : x x+2x=4
3x3
=43
x=.133 (1hr . ,20min .)
9) If A can do a piece of work in 24 days and A and B together can do it in 6 days, how long would it take B to do the work alone?
A :24 124
(24 x )+ 1x
(24 x )=16(24 x )
B: x LCD :24 x Team :6 x+24=4 x
−x−x
243
=3 x3
8=x 8days
11) If Sam can do a certain job in 3 days, while it takes Fred 6 days to do the same job, how long will it take them, working together, to complete the job?
Sam :3 13
(6 x )+ 16
(6 x )=1x(6 x )
Fred :6 LCD :6 x Team : x 2 x+x=6
3x3
= 63
x=2days 13) Two people working together can complete a job in 6 hours. If one of them works twice as fast as
the other, how long would it take the faster person, working alone, to do the job?
A=x 1x
(6 x )+ 12 x
(6 x )=16(6 x )
B=2 x 6+3=x Team :6 9=x
A=9hr ,B=18hr
15) A water tank can be filled by an inlet pipe in 8 hours. It takes twice that long for the outlet pipe to empty the tank. How long will it take to fill the tank if both pipes are open?
¿ :8 18
(16 x )− 116
(16 x )=1x(16 x )
Out :−16 2 x−x=16 Team : x x=16hr
200
17) It takes 10 hours to fill a pool with the inlet pipe. It can be emptied in 15 hrs. with the outlet pipe. If the pool is half full to begin with, how long will it take to fill it from there if both pipes are open?
¿ :10 110
(30 x )− 115
(30 x )=1x(30 x)
Out :−15 3 x−2 x=30 Team : x x=30
12
(30 )=15hr
19) A sink has two faucets, one for hot water and one for cold water. The sink can be filled by a cold-
water faucet in 3.5 minutes. If both faucets are open, the sink is filled in 2.1 minutes. How long does it take to fill the sink with just the hot-water faucet open?
Hot : x 1x
(21x )+ 27
(21 x )=1021
(21 x)
Cold : 3.5=3510=72 LCD :21 x
Team :2.1=2110 21+6 x=10 x
−6 x−6 x
214
=4 x4
5.25hr=x
21) A tank can be emptied by any one of three caps. The first can empty the tank in 20 minutes while
the second takes 32 minutes. If all three working together could empty the tank in 8 859minutes,
how long would the third take to empty the tank?
First :20 120
(480x )+ 132
(480 x )+ 1x
(480x )= 59480
(480x )
Second :32 LCD : 480x Third : x 24 x+15 x+480=59 x
Team :8 859=48059 39 x+480=59x
201
−39 x−39 x
48020
=20 x20
24=x 24min .
23) Sam takes 6 hours longer than Susan to wax a floor. Working together they can wax the floor in 4 hours. How long will it take each of them working alone to wax the floor?
Sam : x+6 1x+6 (4 x ( x+6 ) )+ 1x (4 x ( x+6 ) )=14 (4 x ( x+6 ) )
Susan: x LCD : (4 x ( x+6 ) ) Team :4 4 x+4 ( x+6 )=x(x+6)
4 x+4 x+24=x2+6 x 8 x+24=x2+6 x −8 x−24−8 x−24
0=x2−2x−24 0=(x−6)(x+4)x−6=0 x+4=0
Susan=6 hr , Sam=12hr +6+6−4−4 x=6 x=−4
25) It takes Sally 10 12minutes longer than Patricia to clean up their dorm room. If they work together
they can clean it in 5 minutes. How long will it take each of them if they work alone?
Sally : x+10.5=x+ 212
=2 x+212
22x+21 (5 x (2x+21 ) )+ 1x (5x (2x+21 ) )=15 (5 x (2 x+21 ) )
Patricia : x LCD : (5 x (2x+21 ) ) Team :5 10 x+5 (2 x+21 )=x (2 x+21)
10 x+10 x+105=2x2+21 x20 x+105=2 x2+21 x
−20 x−105−20x−1050=2x2+x−1050=(2 x+15)( x−7)
2 x+15=0 x−7=0 −15−15+7+7
Patricia=7min . , Sally=17.5min . 2x2
=−152 x=7
x=−152
202
27) Secretary A takes 6 minutes longer than Secretary B to type 10 pages of manuscript. If they divide
the job and work together it will take them 8 34minutes to type 10 pages. How long will it take
each working alone to type the 10 pages?
A : x+6 1x+6 (35 x ( x+6 ))+ 1x (35 x ( x+6 ) )= 4
35 (35 x ( x+6 ) )
B: x LCD : (35 x (x+6 ) )
Team :8 34=354 35 x+35 ( x+6 )=4 x(x+6)
35 x+35x+210=4 x2+24 x70 x+210=4 x2+24 x
−70 x−210−70x−21002=4 x
2
2−46 x2
−2102
0=2x2−23 x−105 0=(2 x+7)(x−15)2 x+7=0 x−15=0 −7−7+15+15
2x2
=−72
x=15
x=−72
B:15hr , A :21hr
9.9
1)xyy
=72yx=72
y
( x+2 ) ( y−4 )
y−4= 128
y−4
x+2= 128y−4
y ( y−4 ) 72y
+2 y ( y−4 )= 128y−4
y ( y−4 )
LCD : y ( y−4 ) 72 ( y−4 )+2 y ( y−4 )=128 y 72 y−288+2 y2−8 y=128 y 2 y2+64 y−288=128 y −128 y−128 y
2 y2
2−64 y
2−2882
=0
y2−32 y−144=0 +144+144 y2−32 y=144
(32 ∙ 12 )2
=162=256
y2−32 y+256=144+256 √ ( y−16 )2=√400
y−16=±20 +16+16
y=36 ,−4
x=7236
=2 x= 72−4
=−18
203
(2 ,36 ) ,(48 ,−4)
3)xyy
=150y
x=150y
( x−6 ) ( y+1 )
y+1= 64
y+1
x−6= 64y+1
150 yy ( y ( y+1 ) )−6 ( y ( y+1 ) )= 64
y+1 ( y ( y+1 ) )
LCD : ( y ( y+1 ) ) 150 ( y+1 )−6 y ( y+1 )=64 y150 y+150−6 y2+6 y=64 y −6 y2+156 y+150=64 y +6 y2−156 y−150+6 y2−156 y−150
02=6 y
2
2−80 y
2−1502
0=3 y2−40 y−750=(3 y+5)( y−15)
3 y+5=0 y−15=0 −5−5+15+15
3 y3
=−53
y=15
y=−53 x= 15015
=10
x=150−53
=150(−35 )=−90
(−90 ,−53 ) , (10 ,15 )
5)xyy
=45yx=45
y
( x+2 ) ( y+1 )
y+1= 70
y+1
x+2= 70y+1
45y ( y ( y+1 ) )+2 ( y ( y+1 ) )= 70
y+1 ( y ( y+1 ) )
LCD : ( y ( y+1 ) ) 45 ( y+1 )+2 y ( y+1 )=70 y 45 y+45+2 y2+2 y=70 y
2 y2+47 y+2 y=70 y −70 y−70 y
204
2 y2−23 y+45=0 (2 y−5 ) ( y−9 )=02 y−5=0 y−9=0 +5+5+9+9
2 y2
=52y=9
y=52x=45
9=5
x=45
52
=45 ∙ 25=18
(18 , 52 ) ,(5 ,9)
7)xyy
=90y
x=90y
( x−5 ) ( y+1 )
y+1= 120
y+1
x−5= 120y+1
90y ( y ( y+1 ) )−5 ( y ( y+1 ) )= 120
y+1 ( y ( y+1 ) )
LCD : ( y ( y+1 ) ) 90 ( y+1 )−5 y ( y+1 )=120 y 90 y+90−5 y2−5 y=120 y
−5 y2+85 y+90=120 y+5 y2−85 y−90+5 y2−85 y−90
0=5 y2
5+ 35 y5
−905
0= y2+7 y−180=( y+9)( y−2)
y+9=0 y−2=0 −9−9+2+2 y=−9 y=2
x= 90−9
=−10 x=902
=45
9)xyy
=12yx=12
y
( x+1 ) ( y−4 )
y−4= 16
y−4
x+1= 16y−4
12y ( y ( y−4 ) )+1 ( y ( y−4 ) )= 16
y−4 ( y ( y−4 ) )
LCD : ( y ( y−4 ) ) 12 ( y−4 )+( y ( y−4 ) )=16 y 12 y−48+ y2−4 y=16 y
y2+8 y−48=16 y −16 y−16 yy2−8 y−48=0( y−12 ) ( y+4 )=0y−12=0 y+4=0 +12+12−4−4 y=12 y=−4
x=1212=1x= 12
−4=−3
(1 ,12 ) ,(−3 ,−4 )
11)xyy
=45yx=45
y
( x−5 ) ( y+3 )
y+3= 160
y+3
x−5= 160y+3
45y ( y ( y+3 ) )−5 ( y ( y+3 ) )= 160y+3 ( y ( y+3 ))
45 ( y+3 )−5 y ( y+3 )=160 y 45 y+135−5 y2−15 y=160 y −5 y2+30 y+135=160 y
+5 y2−30 y−135+5 y2−30 y−135
0=5 y2
5+ 130 y
5−13550= y2+26 y−270=( y+27)( y−1)
205
y+27=0 y−1=0 −27−27+1+1 y=−27 y=1
x= 45−27
=−53
x= 451
(−53 ,−27) , (45 ,1 )
9.10
1) A merchant bought some pieces of silk for $900. Had he bought 3 pieces more for the same money, he would have paid $15 less for each piece. Find the number of pieces purchased.
npn
=900n
p=900n
(n+3 ) (p−15 )n+3
= 900n+3
n+3= 900n+3
900n (n (n+3 ) )−15 (n (n+3 ) )= 900
n+3n (n+3 )
LCD :n (n+3 )900 (n+3 )−15n (n+3 )=900n
900n+2700−15n2−45n=900n−15n2+855n+2700=900n+15n2−855n−2700+15n2−855n−2700
0=15n2
15+ 45n15
−270015
0=n2+3n−1800=(n+15)(n−12)
n+15=0n−12=0 −15−15+12+12 n=−15n=12
3) A merchant bought a number of barrels of apples for S120. He kept two barrels and sold the remainder at a profit of $2 per barrel making a total profit of $34. How many barrels did he originally buy?
npn
=120n
p=120n
(n−2 ) ( p+2 )
n−2= 154n−2
p+2= 154n−2
206
120n (n (n−2 ) )+2 (n (n−2 ))= 154n−2 (n (n−2 ) )
LCD : (n (n−2 ) )120 (n−2 )+2n (n−2 )=154 n120n−240+2n2−4 n=154 n
2n2+116n−240=154 n −154n −154n
2n2
2−38n2
−2402
=02
n2−17n−120=0 (n−24 ) (n+5 )=0n−24=0n+5=0 +24+24−5−5 n=24n=−5
5) A man bought a number of articles at equal cost for $500. He sold all but two for $540 at a profit of S5 for each item. How many articles did he buy?
npn
=500n
p=500n
(n−2 ) ( p+5 )n−2
= 540n−2
p+5= 540n−2
500n (n (n+2 ))+5 (n (n+2 ) )= 540
n−2 (n (n+2 ) )
LCD : (n (n+2 ) )500 (n−2 )+5 n (n−2 )=540n500n−1000+5n2−10n=540n
5n2+490n−1000=540n −540n−540n
5n2
5−50n5
−10005
=05
n2−10n−200=0 (n−20 ) (n+40 )=0n−20=0n+40=0 +20+20−40−40 n=20n=−40
207
7) A group of boys bought a boat for $450. Five boys failed to pay their share, hence each remaining boys were compelled to pay $4.50 more. How many boys were in the original group and how much had each agreed to pay?
npn
=450n
p=450n
(n−5 ) ( p+4.5 )n−5
= 450n−5
p+4.5= 450n−5
450n (2n (n−5 ) )+ 92 (2n (n−5 ) )= 450
n−5 (2n (n−5 ) )
LCD : (2n (n−5 ) )900 (n−5 )+9n (n−5 )=900n900n−4500+9n2−45n=900n
9n2+855n−4500=900n −900n−900n9n2
9−45n9
−45009
=09
n2−5n−500=0(n−25 ) (n+20 )=0
n−25=0n+20=0 +25+25−20−20 n=25n=−20
9) A factory tests the road performance of new model cars by driving them at two different rates of speed for at least 100 kilometers at each rate. The speed rates range from 50 to 70 km/hr in the lower range and from 70 to 90 km/hr in the higher range. A driver plans to test a car on an available speedway by driving it for 120 kilometers at a speed in the lower range and then driving 120 kilometers at a rate that is 20 km/hr faster. At what rates should he drive if he plans to
complete the test in 3 12hours?
rtr=120
rt=120
r(r+20 ) (3.5−t )
r+20= 120r+20
72 (2 r (r+20 ) )−120r (2 r (r+20 ) )= 120
r+20 (2r (r+20 ) )
LCD : (2 r (r+20 ) )7 r (r+20 )−240 (r+20 )=240 r7 r2+140 r−240 r−480=240 r 7 r2−100 r−480=240 r
208
−240 r−240r7 r2−340 r−4800=0
340±√ (340 )2−4 (7 ) (−4800 )2 (7 )
=340±√2500014
=340±50014
=60 ,−807
60mph∧80mph
11) The rate of the current in a stream is 3 km/hr. A man rowed upstream for 3 kilometers and then returned. The round trip required 1 hour and 20 minutes. How fast was he rowing?
1 13=43
(r+3 ) tr+3
= 3r+3
t= 3r+3
(r−3 )( 43−t )r−3
= 3r−3
43−t= 3
r−3433 (r+3 ) (r−3 )− 3
r+33 (r+3 ) (r−3 )= 3
r−33 (r+3 ) (r−3 )
LCD :3 (r+3 ) (r−3 )4 (r2−9 )−9 (r−3 )=9(r+3)4 r2−36−9 r+27=9 r+27
4 r2−9 r−9=9 r+27 −9 r−27−9 r−274 r 2
2−18 r2
−362
=0
2 r2−9 r−18=0(2 r+3 )(r−6)=02 r+3=0 r−6=0 −3−3+6+6
2r2
=−32
r=6
r=−32
13) Two drivers are testing the same model car at speeds that differ by 20 km/hr. The one driving at the slower rate drives 70 kilometers down a speedway and returns by the same route. The one driving at the faster rate drives 76 kilometers down the speedway and returns by the same route.
Both drivers leave at the same time, and the faster car returns 12 hour earlier than the slower car.
At what rates were the cars driven?
209
rtr=140
rt=140
r
(r+20 )(t−12 )r+20
= 152r+20
t−12= 152r+20
140r (2 r (r+20 ) )−12 (2r (r+20 ) )= 152
r+20 (2 r (r+20 ) )
LCD : (2 r (r+20 ) )280 (r+20 )−r (r+20 )=304 r280 r+560−r2−20 r=304 r −r2+260 r+5600=304 r
+r2−260 r−5600+r2−260 r−56000=r2+44 r−56000=(r−56)(r+100)r−56=0r+100=0
56km /hr∧76km /hr +56+56−100−100 r=56 r=−100
15) An automobile goes to a place 72 miles away and then returns, the round trip occupying 9 hours. His speed in returning is 12 miles per hour faster than his speed in going. Find the rate of speed in both going and returning.
rtr=72
rt=72
r(r+12 ) (9−t )
r+12= 72r+12
9−t= 72r+12
9 (r (r+12 ) )−72r ( r (r+12 ) )= 72r+12 (r (r+12 ) )
LCD :r (r+12)9 r (r+12 )−72 (r+12 )=72r
9 r2+108 r−72 r−864=72r9 r2+36 r−864=72r −72r−72 r9 r2
9−36 r9
−8649
=09
r2−4 r−96=0(r−12 ) (r+8 )=0
210
r−12=0 r+8=0 +12+12−8−8
12mph∧24mph r=12r=−817) The rate of a stream is 3 miles an hour. If a crew rows downstream for a distance of 8 miles and
then back again, the round trip occupying 5 hours, what is the rate of the crew in still water?
(r+3 ) tr+3
= 8r+3
t= 8r+3
(r−3 ) (5−t )r−3
= 8r−3
5−t= 8r−3
5 (r+3 ) (r−3 )− 8r+3
(r+3 ) (r−3 )= 8r−3
(r+3 ) (r−3 )
5 (r2−9 )−8 (r−3 )=8(r+3)5 r2−45−8 r+24=8 r+24 5 r2−8 r−21=8 r+24
−8 r−24−8 r−24 5 r2−16 r−46=0 (5 r+9 ) (r−5 )=0 5 r+9=0 r−5=0 −9−9+5+5
5 r5
=−95
r=5mph
r=−95
19) By going 15 miles per hour faster, a train would have required 1 hour less to travel 180 miles. How fast did it travel?
rtr=180
rt=180
r(r+15 ) ( t−1 )
r+15= 180r+15
t−1= 180r+15
180r (r (r+15 ) )−1 ( r (r+15 ) )= 180
r+15 (r (r+15 ) )
LCD :r (r+15)180 (r+15 )−r (r+15 )=180 r180 r+2700−r2−15 r=180 r
−r2+165 r+2700=180 r+r2−165−2700+r 2−165−2700
211
0=r2+15 r−27000=(r+60)(r−45)r+60=0 r−45=0 −60−60+45+45 r=−60 r=45mph
21) If a train had traveled 5 miles an hour faster, it would have needed 1 12 hours less time to travel
150 miles. Find the rate of the train.
rtr=150
rt=150
r
(r+5 )(t−32 )r+5
= 150r+5
t−32= 150r+5
150r (2 r (r+5 ) )−32 (2r (r+5 ) )= 150r+5 (2 r (r+5 ) )
LCD : (2 r (r+5 ) )300 (r+5 )−3 r (r+5 )=300 r300 r+1500−3r 2−15 r=300 r−3 r2+285 r+1500=300 r
+3 r2−285−1500+3 r2−285r−1500
0=3 r2
3+15 r3
−15003
0=(r+25 )(r−20)r+25=0 r−20=0 −25−25+20+20 r=−25 r=20mph
9.11
1) y=x2−2 x−8 y−inter :(0 ,−8)
212 (−2 ,0) (4 ,0)
(0 ,−8)
x−inter :0=x2−2x−8 0=(x−4)(x+2)x−4=0 x+2=0 +4+4−2−2 x=4 x=−2 (4 ,0 ) ,(−2 ,0)
vertex : x=22 (1 )
=22=1
y= (1 )2−2 (1 )−8y=1−2−8y=−9(1 ,−9)
3) y=2x2−12 x+10 y−inter :(0 ,10) x−inter :0=2x2−12 x+10 (0 ,10 )
0=2(x2−6 x+5) 0=2(x−5)(x−1) (1 ,0 )(3 ,0)x−5=0 x−1=0 +5+5+1+1 x=5 x=1(5 ,0 )(1 ,0)
vertex : x=122 (2 )
=124
=3 (3 ,−8)
y=2 (3 )2−12 (3 )+10y=2 (9 )−36+10y=18−36+10y=−8(3 ,−8 )
5) y=−2x2+12x−18 y−inter :(0 ,10) x−inter :0=−2x2+12x−18
0=−2(x2−6 x+9)
213
(−2 ,0) (4 ,0)
(0 ,−8)
0=−2 ( x−3 )2 (3,0)x−3=0 +3+3 x=3 (3 ,0) (0 ,−18)
vertex : x=−122 (−2 )
=−12−4
=3
y=−2 (3 )2+12 (3 )−18y=−2 (9 )+36−18y=−18+36−18y=0(3 ,0)
7) y=−3x2+24 x−45 y−inter :(0 ,−45) x−inter :0=−3x2+24 x−45
0=−3(x2−8 x+15) 0=−3(x−5)(x−3) (4,3)x−5=0 x−3=0 (5,0) +5+5+3+3 (3,0) x=5 x=3(5 ,0 )(3,0)
vertex : x=−242 (−3 )
=−24−6
=4
y=−3 (4 )2+24 (4 )−45y=−3 (16 )+96−45 (−45,0)y=−48+96−45y=3(4 ,3)
9) y=−x2+4 x+5 (2 ,9)
214
y−inter :(0 ,9) (0,5) x−inter :0=−x2+4 x+5
0=−1(x2−4 x−5) 0=−1(x−5)(x+1) (−1,0) (5,0)x−5=0 x+1=0 +5+5−1−1 x=5 x=−1 (5 ,0 )(−1 ,0)
vertex : x=−42 (−1 )
=−4−2
=2
y=−(2 )2+4 (2 )+5y=−4+8+5y=9(2 ,9)
11) y=−x2+6 x−5 y−inter :(0 ,−5) x−inter :0=−x2+6 x−5
0=−1(x2−6 x+5) (3,4) 0=−1(x−1)(x−5)x−1=0x−5=0 (1,0) (5,0) +1+1+5+5 x=1 x=5 (1 ,0 )(5,0)
vertex : x=−62 (−1 )
=−6−2
=3 (0 ,−5)
y=−(3 )2+6 (3 )−5y=−9+18−5y=4
215
(3 ,4)
13) y=−2x2+16 x−24 y−inter :(0 ,−24) (4,8) x−inter :0=−2x2+16 x−24
0=−2(x2−8 x+12) 0=−2(x−2)(x−6) (2,0) (6,0)x−2=0 x−6=0 +2+2+6+6 x=2 x=6 (2 ,0 )(6 ,0)
vertex : x=−162 (−2 )
=−16−4
=4
y=−2 (4 )2+16 (4 )−24 (0 ,−24)y=−2 (16 )+64−24y=−32+64−24y=8
(4 ,8)
15) y=3 x2+12x+9 y−inter :(0 ,9) (0 ,9) x−inter :0=3x2+12x+9
0=3(x2+4 x+3) 0=3(x+1)( x+3)x+1=0 x+3=0 (−3,0) (−1,0) −1−1−3−3 x=−1 x=−3 (−2 ,−3) (−1 ,0 )(−3 ,0)
216
vertex : x=−122 (3 )
=−126
=−2
y=3 (−2 )2+12 (−2 )+9y=3 (4 )−24+9y=12−24+9y=−3(−2 ,−3)
17) y=5 x2−40x+75 (0,75) y−inter :(0 ,75) x−inter :0=5x2−40 x+75
0=5¿ 0=5(x−3)(x−5) (0,3) (0,5)x−3=0 x−5=0 +3+3+5+5 x=3 x=5 (4 ,−5)
vertex :402 (5 )
= 4010
=4
y=5 (4 )2−40 (4 )+75y=5 (16 )−160+75y=80−160+75y=−5(4 ,−5)
19) y=−5x2−60 x−175 (−6 ,5) y−inter :(0 ,−175) x−inter :0=−5x2−60x−175
0=−5(x2+12x+35) (−7,0 )(−5,0) 0=−5(x+5)(x+7)
217
x+5=0 x+7=0 −5−5−7−7 x=−5 x=−7 (−5 ,0 )(−7 ,0)
vertex : x=602 (−5 )
= 60−10
=−6 (0 ,−175)
y=−5 (−6 )2−60 (−6 )−175y=−5 (36 )+360−175y=−180+360−175y=5(−6 ,5)
218
Chapter 10: Functions
219
10.1
1) A) B) C) D) E) y=3 x−7
Yes Yes No No
F) y2−x2=1 G)√ y+x=2 H) x2+ y2=1 +x2+x2 −x−x −x2−x2
√ y2=√(x2+1) (√ y )2=(2−x )2 √ y2=√(1−x2) y=±√ x2+1 y= (2−x )2 y=±√1−x2
No Yes No
3) f ( x )=√5−4 x 5−4 x ≥0 −5−5
−4 x−4
≥−5−4
x≤ 54
5) f ( x )=x2−3 x−4 All Real Numbers R
7) f ( x )=√x−16 x−16≥0 +16+16 x≥16
9) h ( x )=√3 x−12x2−25
x2−25≠03x−12≥0 ( x−5 ) ( x+5 )≠0+12+12
x−5=0 x+5=0 3x3
≥ 123
+5+5−5−5 x≥ 4 x=5 x=−5 x≥ 4 , x≠5
11) g ( x )=4 x−4 g (0) g (0 )=4 (0 )−4 g (0 )=0−4 g (0 )=−4
13) f ( x )=|3x+1|+1 f (0) f (0 )=|3 (0 )+1|+1 f (0 )=|0+1|+1 f (0 )=|1|+1 f (0 )=1+1 f (0 )=2
15) f (n )=−2|−n−2|+1 f (−6) f (−6 )=−2|− (−6 )−2|+1 f (−6 )=−2|6−2|+1 f (−6 )=−2|4|+1 f (−6 )=−2 (4 )+1 f (−6 )=−8+1 f (−6 )=−7
220
17) f (rt )=3t−2 f (−2) f (−2 )=3−2−2
f (−2 )= 132
−2
f (−2 )=19−2
f (−2 )=19−189
f (−2 )=−179
19) f (t )=|t+3|f (10) f (10 )=¿10+3∨¿ f (10 )=¿13∨¿ f (10 )=13
21) w (n )=4n+3w (2) w (2 )=4 (2 )+3 w (2 )=8+3 w (2 )=11
23) w ( n )=2n+2w(−2) w (−2 )=2−2+2
w (−2 )=20
w (−2 )=1
25) p (n )=−3|n|p (7) p (7 )=−3∨7∨¿ p (7 )=−3(7) p (7 )=−21
27) p (t )=−t 3+t p (4 ) p (4 )=−43+4 p (4 )=−64+4 p (4 )=−60
29) k (n )=|n−1|k (3) k (3 )=¿3−1∨¿ k (3 )=¿2∨¿ k (3 )=2
31) h ( x )=x3+2h(−4 x) h (−4 x )=(−4 x )3+2 h (−4 x )=−64 x3+2
33) h ( x )=3 x+2h(−1+x) h (−1+x )=3 (−1+x )+2 h (−1+x )=−3+3 x+2 h (−1+x )=3 x−1
35) h ( t )=2|−3 t−1|+2h(n2) h (n2 )=2|−3n2−1|+2
37) g ( x )=x+1g(3x ) g (3 x )=3 x+1
39) g ( x )=5x g(−3−x)
g (−3−x )=5−3− x
10.2
1) g (a )=a3+5a2
f (a )=2a+4 g (3 )+ f (3) g (3 )=33+5 (3 )2
g (3 )=27+5(9)
g (3 )=27+45 g (3 )=72 f (3 )=2 (3 )+4 f (3 )=6+4 f (3 )=10
221
g (3 )+ f (3 )=72+10=82
3) g (a )=3a+3 f (a )=2a−2 (g+ f ) (a )=g (9 )+ f (9) g (9 )=3 (9 )+3 g (9 )=27+3 g (9 )=30 f (9 )=2 (9 )−2 f (9 )=18−2 f (9 )=16 g (9 )+ f (9 )=30+16=46
5) g ( x )=x+3 f ( x )=−x+4 (g− f ) (3 )=g (3 )−f (3) g (3 )=3+3 g (3 )=6 f (3 )=−3+4 f (3 )=−1 g (3 )+ f (3 )=6−1=5
7) g ( x )=x2+2 f ( x )=2x+5 (g− f ) (0 )=g (0 )− f (0) g (0 )=02+2 g (0 )=0+2 g (0 )=2 f (0 )=2 (0 )+5 f (0 )=0+5 f (0 )=5 g (0 )−f (0 )=2−5=−3
9) g (t )=t−3
h (t )=−3 t 3+6 t g (1 )+h(1) g (1 )=1−3 g (1 )=−2 h (1 )=−3 (1 )3+6 (1) h (1 )=−3 (1 )+6 h (1 )=−3+6 h (1 )=3 g (1 )+h (1 )=−2+3=1
11) h ( t )=t+5 g ( t )=3 t−5 (h ∙g ) (5 )=h(5) ∙ g (5) h (5 )=5+5 h (5 )=10 g (5 )=3 (5 )−5 g (5 )=15−5 g (5 )=10 h (5 ) ∙ g (5 )=10∙10=100
13) h (n )=2n−1 g (n )=3n−5 h(0)÷ g(0) h (0 )=2 (0 )−1 h (0 )=0−1 h (0 )=−1 g (0 )=3 (0 )−5 g (0 )=0−5 g (0 )=−5
h (0 )÷g (0 )=−1÷−5=−15
15) f (a )=−2a−4 g (a )=a2+3
( fg ) (7 )= f (7 )g (7 )
f (7 )=−2 (7 )−4
222
f (7 )=−14−4 f (7 )=−18 g (7 )=72+3 g (7 )=49+3 g (7 )=52
f (7 )g (7 )
=−1852
=−926
17) g ( x )=−x3−2 h ( x )=4 x (g−h ) (x )=g (x )−h(x) g ( x )−h ( x )=(−x3−2 )−(4 x ) g ( x )−h ( x )=−x3−2−4 x
19) f ( x )=−3x+2 g ( x )=x2+5 x ( f−g ) ( x )= f ( x )−g (x) f ( x )−g ( x )= (−3x+2 )−(x2+5x ) f ( x )−g ( x )=−3x+2−x2−5 x f ( x )−g ( x )=−x2−8x+2
21) g ( x )=4 x+5
h ( x )=x2+5x g ( x ) ∙ h ( x ) g ( x ) ∙ h ( x )=(4 x+5)(x2+5) g ( x ) ∙ h ( x )=4 x3+20 x2+5x2+25 x g ( x ) ∙ h ( x )=4 x3+25 x2+25
23) f ( x )=x2−5 x g ( x )=x+5 ( f +g ) (x )=f ( x )+g (x) f ( x )+g ( x )=(x2−5 x )+(x+5)
f ( x )+g ( x )=x2−4 x+5
25) g (n )=n2+5 f (n )=3n+5
g (n )÷ f (n )= g (n )f (n )
g (n )f (n )
= n2+53n+5
27) g (a )=−2a+5 f ( a )=3a+5
( gf ) (a )=g (a )f (a )
g (a )f (a )
=−2a+53 a+5
29) h (n )=n3+4 n g (n )=4n+5 h (n )+g(n) h (n )+g (n )=(n3+4n )+(4 n+5) h (n )+g (n )=n3+8n+5
31) g (n )=n2−4n h (n )=n−5 g (n2 ) ∙ h (n2 ) g (n2 ) ∙ h (n2 )=[ (n2 )2−4 (n2 )] [(n2 )−5 ] g (n2 ) ∙ h (n2 )=(n4−4n2)(n2−5) g (n2 ) ∙ h (n2 )=n6−5n4−4 n4+20n2
g (n2 ) ∙ h (n2 )=n6−9n4+20n2
33) f ( x )=2x g ( x )=−3 x−1 ( f +g ) (−4−x )=f (−4−x )+g(−4−x) f (−4−x )+g (−4−x )=[2 (−4−x ) ]+[−3 (−4−x )−1]
223
f (−4−x )+g (−4−x )=(−8−2 x )+(12+3 x−1) f (−4−x )+g (−4−x )=x+3
35) f (t )=t 2+4 t g ( t )=4 t+2 f ( t2 )+g (t 2 ) f ( t2 )+g (t 2 )=[ (t 4 )+4 ( t2 ) ]+[4 (t 2 )+2] f ( t2 )+g (t 2 )=t 4+8 t 2+2
37) g (a )=a3+2a h (a )=3a+4
( gh ) (−x )=g (−x )h (−x )
g (−x )h (−x )
=(−x )3+2 ( x )3 ( x )+4
=−x3−2x−3 x+4
39) f (n )=−3n2+1 g (n )=2n+1
( f−g )( n3 )=f ( n3 )−g ( n3 ) f ( n3 )−g ( n3 )=[−3(n3 )
2
+1]−[2( n3 )+1] f ( n3 )−g ( n3 )=[−3(n29 )+1]−[ 2n3 +1] f ( n3 )−g ( n3 )=(−n2
3+1)−2n3 −1
f ( n3 )−g ( n3 )=−n2−2n3
43) f ( x )=−4 x+1
g ( x )=4 x+3 ( f ∘ g ) (x )=f (g (9 ) ) g (9 )=4 (9 )+3 g (9 )=36+3 g (9 )=39 f (39 )=−4 (39 )+1 f (39 )=−156+1 f (39 )=−155
45) h (a )=3a+3 g (a )=a+1 (h∘g ) (5 )=h (g (5 ) ) g (5 )=5+1 g (5 )=6 h (6 )=3 (6 )+3 h (6 )=18+3 h (6 )=21
45) g ( x )=x+4 h ( x )=x2−1 (g∘h ) (10 )=g (h (10 ) ) h (10 )=102−1 h (10 )=100−1 h (10 )=99 g (99 )=99+4 g (99 )=103
47) f (n )=−4n+2 g (n )=n+4 ( f ∘ g ) (9 )=f (g (9 ) ) g (9 )=9+4 g (9 )=13 f (13 )=−4 (13 )+2 f (13 )=−52+2 f (13 )=−50
224
49) g ( x )=2 x−4 h ( x )=2 x3+4 x2
(g∘h ) (3 )=g (h (3 ) ) h (3 )=2 (33 )+4 (32) h (3 )=2 (27 )+4 (9) h (3 )=54+36 h (3 )=90 g (90 )=2 (90 )−4 g (90 )=180−4 g (90 )=176
51) g ( x )=x2−5 x h ( x )=4 x+4 (g∘h ) ( x )=g(h (x )) g (4 x+4 )=(4 x+4 )2−5 (4 x+4 ) g (4 x+4 )=16x2+32 x+16−20 x−20 g (4 x+4 )=16x2+12 x−4
53) f (a )=−2a+2 g (a )=4 a ( f ∘ g ) (a )=f (g (a ) ) f (4a )=−2 (4 a )+2 f ( 4a )=−8a+2
55) g ( x )=4 x+4 f ( x )=x3−1 (g∘ f ) (x )=g ( f ( x ) ) g (x3−1 )=4 (x3−1 )+4 g (x3−1 )=4 x3−4+4 g (x3−1 )=4 x3
57) g ( x )=−x+5
f ( x )=2x−3 (g∘ f ) (x )=g ( f ( x ) ) g (2x−3 )=−(2 x−3 )+5 g (2x−3 )=−2 x+3+5 g (2x−3 )=−2 x+8
59) f (t )=4 t+3 g ( t )=−4 t−2 ( f ∘ g ) (t )= f (g (t )) f (−4 t−2 )=4 (−4 t−2 )+3 f (−4 t−2 )=−16 t−8+3 f (−4 t−2 )=−16 t−5
10.3
1) g ( x )=−x5−3 f ( x )= 5√−x−3 (g∘ f ) (x )=g ( f ( x ) ) g ( 4√−x−3 )=−( 5√−x−3 )5−3
¿−(−x−3 )−3 ¿ x+3−3 ¿ x
yes
3) f ( x )=−x−1x−2
g ( x )=−2x+1−x−1
( f ∘ g ) (x )=f (g ( x ) )
225
f (−2x+1−x−1 )=−(−2x+1−x−1 )−1
−2x+1−x−1
−2
¿((−x−1 ) 2x−1
−x−1−1 (−x−1 ))
(−x−1 ) −2 x+1−x−1
−2 (−x−1 )
¿2 x−1+x+1
−2x+1+2 x+2
¿3x3
=x
yes
5) g ( x )=−10 x+5
f ( x )= x−510
(g∘ f ) (x )=g ( f ( x ) )
g( x−510 )=−10( x−510 )+5 ¿−x+5+5
¿−x+10No
7) f ( x )= −2x+3
g ( x )=3x+2x+2
( f ∘ g ) (x )=f (g ( x ) )
f ( 3x+2x+2 )= −2(x+2)
( x+2 ) 3 x+2x+2
+3 (x+2)
¿−2x−4
3x+2+3 x+6
¿ −2x−46 x+12
=−2 ( x+2 )6 ( x+2 )
=−26
=−13
No
9) g ( x )=5√ x−12
f ( x )=2x5+1 (g∘ f ) (x )=g ( f ( x ) )
g (2 x5+1 )=5√ 2 x5+1−12
¿ 5√ 2 x52 ¿ 5√ x5 ¿ xyes
11) f ( x )= (x−2 )5+3 y= (x−2 )5+3 x=( y−2 )5+3 −3−3 5√ x−3= 5√ y−2 5√ x−3= y−2 +2+2 5√ x−3+2= y f−1 (x )= 5√x−3+2
13) g ( x )= 4x+2
y= 4x+2
x= 4y+2
( y+2 ) x= 4y+2
( y+2)
x ( y+2 )
x=4
x
y+2= 4x −2−2 y=4 x−2
g−1 ( x )=4x−2
15) f ( x )=−2 x−2x+2
226
y=−2 x−2x+2
x=−2 y−2y+2
( y+2 ) x=−2 y−2y+2
( y+2)
xy+2 x=−2 y−2 +2 y−2 x+2 y−2x xy+2 y=−2−2x
y ( x+2 )x+2
=−2−2 xx+2
y=−2−2xx+2
f−1 (x )=−2−2xx+2
17) f ( x )=10−x5
y=10−x5
5 ( x )=10− y5
(5)
5 x=10− y −10−10
5 x−1
− 10−1
=− y−1
−5 x+10= y f−1 (x )=−5x+10
19) g ( x )=−( x−1 )3
y=−( x−1 )3
x−1
=− ( y−1 )3
−1 3√−x=3√ ( y−1 )3
3√−x= y−1 +1+1
3√−x+1= y g−1 ( x )=3√−x+1
21) f ( x )= (x−3 )3 y= (x−3 )3
3√ x=3√ ( y−3 )3
3√ x= y−3 +3+3 3√ x+3= y f−1 (x )= 3√x+3
23) g ( x )= xx−1
y= xx−1
( y−1 ) x= yy−1
( y−1)
xy−x= y −xy−xy −x= y−xy
−x1−x
= y (1−x )1−x
−x1−x
= y
g−1 ( x )= −x1−x
25) f ( x )= x−1x+1
y= x−1x+1
( y+1 ) x= y−1y+1
( y+1 )
xy+x= y−1 − y−x− y−x xy− y=−1−x
y ( x−1 )x−1
=−1−xx−1
y=−1−xx−1
227
f−1 (x )=−1−xx−1
27) g ( x )=8−5 x4
y=8−5 x4
4 ( x )=8−5 y4
(4)
4 x=8−5 y −8−8
4 x−8−5
=−5 y−5
4 x−8−5
= y
g−1 ( x )=4 x−8−5
29) g ( x )=−5 x+1 y=−5x+1 x=−5 y+1 −1−1
x−1−5
=−5 y−5
x−1−5
= y
g−1 ( x )= x−1−5
31) g ( x )=−1+x3
y=−1+x3
x=−1+ y3
+1+1 3√ x+1=3√ y3
y= 3√x+1 g−1 ( x )=3√ x+1
33) h ( x )=4−3√4 x2
y= 4−3√4 x2
(2 ) x=4−3√4 y2
(2)
2 x=4− 3√4 y −4−4
2x−4−1
=−3√4 y−1
(−2 x+4 )3=( 3√4 y )3
(−2x+4 )3
4=4 y4
(−2x+4 )3
4= y
h−1 ( x )= (−2 x+4 )3
4
35) f ( x )= x+1x+2
y= x+1x+2
( y+2 ) x= y+1y+2
( y+2)
xy+2 x= y+1 − y−2x− y−2x xy− y=1−2 x
y ( x−1 )x−1
=1−2 xx−1
y=1−2 xx−1
f−1 (x )=1−2 xx−1
37) f ( x )=7−3 xx−2
y=7−3 xx−2
( y−2 ) x=7−3 yy−2
( y−2)
xy−2 x=7−3 y +3 y+2 x+2 x+3 y xy+3 y=7+2x
y ( x+3 )x+3
= y+2xx+3
228
y= y+2 xx+3
f−1 (x )= y+2 xx+3
39) g ( x )=−x
y=−x
x
−1=(− y )/−1
−x= y g−1 ( x )=−x
10.4
1) 31−2n=31−3n
1−2n=1−3n +3n+3n 1+n=1 −1−1 n=0
3) 42a=1 42a=10
2aa
=02
a=0
5) ( 125 )−k
=125−2k−2
(5−2 )−k=(53 )−2k−2
52k=5−6k−6
2k=−6k−6 +6 k+6k
8k8
=−68
k=−34
7) 62m+1= 136
62m+1=6−2
2m+1=−2 −1−1
2m2
=−32
m=−32
9) 6−3x=36 6−3x=62
−3 x−3
= 2−3
x=−23
11) 64b=25
(26 )b=25
26b=25
6b6
=56
b=56
13) ( 14 )x
=16
(4−1 )x=42
4−x=42
−x−1
= 2−1
229
x=−2
15) 43a=43
3a3
=33
a=1
17) 363 x=2162 x+1 (62)3x=(63 )2x+1
66 x=66 x+1
6 x=6 x+1 −6 x−6 x 0=1 false NoSolution
19) 92n+3=243 (32 )2n+3
=35
34n+6=35
4 n+6=5 −6−6
4n4
=−14
n=−14
21) 33 x−2=33x+1 3 x−2=3 x+1 −3 x−3 x −2=1 false NoSolution
23) 3−2x=33
−2x−2
= 3−2
x=−32
25) 5m+2+5−m
m+2=−m −m−m
2
−2=−2m
−2 −1=m
27) ( 136 )b−1
=216
(6−2 )b−1=63
6−2b+2=63
−2b+2=3 −2−2
−2b−2
= 1−2
b=−12
29) r 62−2 x=62 2−2x=2 −2−2
−2x−2
= 0−2
x=0
31) 4 ∙2(−3n−1)=14
22∙2−3n−1=2−2
2−3n+1=2−2
−3n+1=−2 −1−1
−3n−3
=−3−3
n=1
33) 43k−3 ∙42−2k=16−k
4k−1=(42)−k
4k−1=4−2 k
k−1=−2 k −k−k
−1−3
=−3k−3
230
13=k
35) 9−2x ( 1243 )3x
=243− x
(32 )−2x (3−5 )3x=(35 )−x
3−4x ∙3−15 x=3−5x
3−19 x=3−5 x −19 x=−5 x +19 x+19x
014
=14 x14
0=x
37) 64n−2 ∙16n+ 2=( 14 )3n−1
(43 )n−2 ∙ (42)n+2=(4−1 )3n−1
43n−6 ∙42n+4=4−3n+1
45n−2=4−3n+1
5n−2=−3n+1 +3n+3n 8n−2=1
+2+2
8n8
=38
n=38
39) 5−3n−3 ∙52n=1 5−n−3=50
−n−3=0 +3+3
−n−1
=3/−1
n=−3
10.5
1) log 981=2 92=81
3) log7149
=−2
7−2= 149
5) log13169=2 132=169
7) 80=1
log 81=0
9) 152=225 log15225=2
11) 6416=2
log64 2=16
13) log1255=x 125x=5
(53 )x=51 53 x=51
3x3
=13
x=13
15) log34317=x
343x= 17
(73)x=7−1
231
73 x=7−1
3x3
=−13
x=−13
17) log 416=x 4 x=16 4 x=42
x=2
19) log636=x 6x=36 6x=62 x=2
21) log264=x 2x=64 2x=26
x=6
23) log5 x=1 51=x 5=x
25) log2 x=−2 2−2= x
122
=x
14=x
27) log11 k=2 112=k 121=k
29) log 9 (n+9 )=4 94=n+9 6561=n+9 −9−9 6552=n
31) log5 (−3m )=3 53=−3m
125−3
=−3m−3
−1253
=m
33) log11 ( x+5 )=−1 11−1=x+5
111
=x+5
−5−5
−5411
=x
35) log 4 (6b+4 )=0 40=6b+4 1=6 b+4 −4−4
−36
= 6b6
−12
=b
37) log5 (−10 x+4 )=4 54=−10 x+4 625=−10 x+4 −4−4
621−10
=−10x−10
−62110
=x
39) log2 (10−5 a )=3 23=10−5a 8=10−5a −10−10
−2−5
=−5a−5
25=a
10.6
1) Find each of the following:a. S500 invested at 4% compounded annually for 10 years.
A=500(1+ .041 )1 ∙10
232
A=500 (1.04 )10
A=500(1.48) A=$740.12
b. S600 invested at 6% compounded annually for 6 years.
A=600(1+ .061 )1 ∙6
A=600 (1.06 )6
A=600(1.42) A=$ 851.11
c. S750 invested at 3% compounded annually for 8 years.
A=750(1+ .031 )1 ∙8
A=750 (1.03 )8
A=750(1.27) A=$ 950.08
d. S1500 invested at 4% compounded semiannually for 7 years.
A=1500(1+ .042 )2 ∙7
A=1500 (1.02 )14
A=1500(1.32) A=$1979.22
e. S900 invested at 6% compounded semiannually for 5 years.
A=900(1+ .062 )2 ∙5
A=900 (1.03 )10
A=900(1.34) A=$1209.52
f. $950 invested at 4% compounded semiannually for 12 years.
A=950(1+ .042 )(2 ∙12)
A=950 (1.02 )24
A=950(1.61) A=$1528.02
233
g. $2000 invested at 5% compounded quarterly for 6 years.
A=200(1+ .052 )4 ∙ 6
A=200 (1.0125 )24
A=200(1.35) A=$2694.70
h. $2250 invested at 4% compounded quarterly for 9 years.
A=2250(1+ .044 )4 ∙9
A=2250 (1.01 )36
A=2250(1.43) A=$3219.23
i. $3500 invested at 6% compounded quarterly for 12 years.
A=3500(1+ .064 )4 ∙ 12
A=3500 (1.015 )48
A=3500(2.04) A=$7,152.17
j. All of the above compounded continuously. A=500 e.04 ∙10 A=500 e.4
A=500(1.49) A=¿$745.91
A=600e.06 ∙ 6
A=600e.36
A=600(1.43) A=$ 860.00
A=750 e.03 ∙ 8
A=750 e.24
A=750(1.27) A=$ 953.44
A=1500 e.04 ∙7
A=1500e.28
234
A=1500(1.32) A=$1984.69
A=900 e .06 ∙5
A=900 e .30
A=900(1.35) A=$1214.87
A=950 e .04 ∙12
A=950 e .48
A=950(1.62) A=$1535.27
A=2000e.05 ∙ 6
A=2000e.3
A=2000(1.35) A=$2699.72
A=2250 e.04 ∙9
A=2250e.36
A=2250(1.43) A=$3224.99
A=3500 e.06 ∙ 12
A=3500e.72
A=3500(2.05) A=$7190.52
3) What principal will amount to S3500 if invested at 4% interest compounded quarterly for 5 years?
3500=P (1+ .044 )4 ∙ 5
3500=P (1.01 )20
35001.22
=P (1.22 )1.22
$2868.41=P
5) What principal will amount to S2500 if invested at 5% interest compounded semiannually for 7.5 years?
2500=P (1+ .052 )2 ∙7.5
235
2500=P (1.025 )1.5
25001.45
=P (1.45 )1.45
$1726.16=P
7) A thousand dollars is left in a bank savings account drawing 7% interest, compounded quarterly for 10 years. What is the balance at the end of that time?
A=1000(1+ .074 )4 ∙ 10
A=100 (1.0175 )40
A=1000(2.00) A=$2001.60
9) $1750 is invested in an account earning 13.5% interest compounded monthly for a 2 year period. What is the balance at the end of 9 years?
A=1750(1+ .13512 )12∙ 2
A=1750 (1.01125 )24
A=1750(1.31) A=$2288.98
11) A $10, 000 Treasury Bill earned 16% compounded monthly. If the bill matured in 2 years, what was it worth at maturity?
A=10,000(1+ .1612 )12 ∙2
A=10,000 (1.01 )24
A=10,000(1.37)A=$13,742.19
13) A savings institution advertises 7% annual interest, compounded daily, How much more interest would you earn over the bank savings account or credit union in problems 7 and 8?
A=1000(1+ .07365 )365∙ 10
A=1000 (1.00019 )3650
A=1000(2.01) A=$2013.62 ¿7 :2001.60−2013.62=$12.02
236
¿8 :2009.66−2013.62=$−3,96
15) You lend $100 at 10% continuous interest. If you are repaid 2 months later, what is owed?
A=100 e.1( 16 )
A=100 e.0167
A=100 (1.02 ) A=$101.68
10. 7
1) cos71=0.3256
3) sin 75=0.9659
5) sinθ= 725
7) sin θ= 716
9) sin θ= 88√2 (√2√2 )
sin θ=¿( 8√28 ∙2 )¿sin θ=√2
2
11) (13 ) tan 51= x13
(13 )
13 tan 51=x 13 (1.235 )=x 16.05=x
13) ( x ) sin24=13x
( x )
x sin 24sin 24
= 13sin 24
x= 13sin 24
x= 130.407
x=31.96
15) (9 ) tan71= x9(9)
9 tan71=x9 (2.904 )=x26.14=x
17) (6 )cos68= x6(6)
6cos68=x 6 ( .3746 )=x 2.25=x
19) (6 ) tan71.4= x6
237
θ24 25
7
7 3√2
θ 16
θ 8 8√2
8
x 13 51
x 13 24
x 71 9
x
68 6
6 71.4
x
6 tan71.4=x6 (2.9714 )=x17.83=x
21) (5 ) cos38= x5
(5 )
5cos38=x 5 ( .7880 )=x3.94=x
23) ( x ) cos67= 4x
( x )
xcos67xcos67
= 4cos 67
x= 4cos67
x= 4.3907
x=10.24
25) (4 ) tan 67.2= x4
(4)
4 tan 67.2=x 4 (2.3789 )=x 9.52=x
27) ( x ) cos32=4x
( x )
xcos32cos32
= 4cos32
x= 4cos32
x= 40.8480
x=4.72
29) (2.4 ) tan 22= x2.4
(2.4)
2.4 tan 22=x2.4 (0.4040 )=x
0.97=x
31) (3 ) cos61= x3
(3)
3cos61=x3 ( .4848 )=x 1.45=x
33) (11) sin 30= x11
(11)
11sin 30=x 11 (0.5 )=x 5.5= x
35) (11) tan 95= x11
(11 )
11 tan95=x11 (3.7321 )=x41.05=x
37) ( x ) sin37.1=11x
(x )
x sin 37.1sin 37.1
= 11sin37.1
x= 11sin 37.1
x= 11.6032
x=18.24
39) ( x ) cos40=13.1(x)
xcos 40cos 40
= 13.1cos40
x= 13.1cos 40
x= 13.1.7660
x=17.1
238
38 5 x
x 67 4
x 67.2 4
x 32 4
2.4 22 x
3 x 61
11 30 x
x 11 75
11 37.1 x
13.1 40 x
θ14
15
10.8
1) sin z=0.4848 z=29 °
3) sin y=0.6561 y=41°
5) cos x=3235
cos x=.9143 x=24 °
7) cos x=3031 cos x=.9677 x=15 °
9) tan x=36
tan x=.5 x=27 °
11) tanθ=118
tanθ=.7273 θ=36 °
13) cosθ=47
cosθ=0.5714 θ=55.2°
15) cosθ=47
cosθ=0.5714 cosθ=55.2°
17) tanθ=1610
tanθ=1.6 tanθ=58 °
19) cosθ= 9.313.2
cosθ=0.7045 cosθ=45.2 °
21) tanθ=54
tanθ=1.25 tanθ=51.3 °
23) sin θ=1012
sin θ=0.8333 sin θ=56.4 °
25) cosθ= 915.7
cosθ=0.5732 cosθ=55 °
27) tanθ=1514
tanθ=1.0714 tanθ=47 °
29) sin θ= 714
sin θ=0.5 sin θ=30°
31) ( x ) tan 62=28.4x
(x )
x tan62tan 62
= 28.4tan62
x= 28.4tan 62
x= 28.41.8507
239
θ10
16
35 ? 32
30 ? 31
3 ? 6
?8
11
θ8
11
7
θ 4
9.3 θ
13.2
5 θ 4
10
θ 12
9 θ 15.7
7 θ 14
C 28.4 B Z
X Y
62
A
2.9 7
Y X Z
x=15.115.12+28.42= y2
228.01+806.56= y2
√1034.57=√ y2 32.2= yz=90−62z=28 °
33) tan x=2.97
tan x=.4143 x=22.5 °y=90−22.5y=67.5°2.92+72=z2
8.41+49=z2
√57.41=√ z2 7.6=z
35) sin x=37
sin x=.4286 x=25.4 °y=90−25.4y=64.6°32+z2=72
9+z2=49 −9−9
√ z2=√40 z=6.3
37) (16 ) sin 52= x16
(16 )
16sin 52=x 16 ( .7880 )=x 12.6=x12.62+ y2=162
158.76+ y2=256 −158.76−158.76
√ y2=√97.24y=9.9
z=90−52z=38 °
39) (8 ) tan 45= x8
(8 )
8 tan 45=x 8 (1 )=x 8=x82+82= y2
64+64= y2
√128=√ y2 11.3= yz=90−45z=45°
240
Y3 7 X Z
X Z
Y 16 52
Z Y X
45° 8
241
