PMI Rational Expressions & Equations Unit Variation Class Work 1. y varies inversely with x. If y=12 when x=4 , find y when x=−6. 2. y varies inversely with x. If y=8 when x=3 , find x when y=−2. 3. y varies inversely with x 2 . If y=3 when x=5, find y when x=3. 4. Circumference varies directly with area of a circle and inversely with the radius. Find the constant of variation if C=18 when A =27 and r=3. What is radius when C=25 and A =50? 5. y varies jointly with x and z. If y=24 when x=4 and z=2, find y when x=−6 and z=2. 6. y varies jointly with x and z. If y=32 when x=4 and z=2, find x when y=48 and z=2. 7. V varies jointly with r 2 and h. If V =24 πwhen h=6 and r=2, find r when V =18 π and h=2. Variation Homework 8. y varies inversely with x. If y=9 when x=4, find y when x=−6. 9. y varies inversely with x. If y=8 when x=5, find x when y=−2. 10. y varies inversely with x 2 . If y=3, when x=4, find y when x=3. Alg II - Rationals ~1~ NJCTL.org
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PMI Rational Expressions & Equations Unit
VariationClass Work1. y varies inversely with x. If y=12 when x=4 , find y when x=−6.
2. y varies inversely with x. If y=8 when x=3 , find x wheny=−2.
3. y varies inversely with x2. If y=3 whenx=5, find y whenx=3.
4. Circumference varies directly with area of a circle and inversely with the radius. Find the constant of variation if C=18 when A=27 and r=3. What is radius whenC=25 and A=50?
5. y varies jointly with x and z. If y=24 when x=4 and z=2, find y when x=−6 and z=2.
6. y varies jointly with x and z. If y=32 when x=4 and z=2, find x when y=48 and z=2.
7. V varies jointly with r2 and h. If V=24πwhen h=6 and r=2, find r when V=18 π and h=2.
VariationHomework8. y varies inversely with x. If y=9 whenx=4, find y whenx=−6.
9. y varies inversely with x. If y=8 when x=5, find x when y=−2.
10. y varies inversely with x2. If y=3, when x=4, find y when x=3.
11. Area of a triangle varies jointly with its height and base. Find the constant of variation if A=16 when h=4 and b=8. What is base when A=9 and h=3?
12. y varies jointly with x and z. If y=24 when x=6 and z=2, find y when x=−6 and z=3.
13. y varies jointly with x and z. If y=40 when x=−4 and z=2, find x wheny=60 and z=2.
14. V varies jointly with r2 and h. V=36π when h=4 and r=3, find r whenV=80π and h=5.
Alg II - Rationals ~1~ NJCTL.org
Reducing Rational ExpressionsClass WorkSimplify.
15.3x6
16. 40b12b 17. 12x
2
4 x18.
18a3b2
14a5b6
19.60 j4 k6m8
16 j3 k6m9 20. 12c2−63
21.8n2+4 n6n2+3n
22.5h−104h−8
23. . v2−4v+4v2−4
24.f 2+7 f +12f 2−2 f−15
25. 4 s3−20 s2+24 s16−8 s
26.2d2−7d+63d2−8d+4
27.4 x3−4 x2−15 x2x5−3 x4−5 x3
28.54 x4−6 x2
54 x3−72 x2+18 x29.
4m3−42m2+2m+2
30.3 p2+7 p−6p2+ p−6
31. 8a2+4 a−604 a−10
32.2x3+2 y3
4 x2−4 y233. 12a
2b2−124 abc−4 c
34.15x3+7 x2−2 x3 x4+2 x3
35. 3 p2−p−24−4 p
36.6 x3+15 x2+9 x12 x+6 x2−6 x3
Alg II - Rationals ~2~ NJCTL.org
Reducing Rational ExpressionsHomeworkSimplify.
37.9 y6
38.48c16c 39.
12x2
8x340.
28a3b2
14a4b8
41.60 j4 k6m8
12 j2k6m10 42. 12c2−96
43.10n2−5n15n2−35n
44.12h−89h−6
45.v2−4v+3v2−9
46.f 2+12 f +27f 2+f−72
47. 5 rs3−20rs2+15 rs15 r−5rs
48.4 d2+5d+13d2+5d+2
49.x4−x2
x4−x350. 2a3−2b3
12ac−12bc51.
4m2−4mn+4 n2
m3+n352.
8m p2−2m4 p2−1
53.12x3−10 x2+2x2x5+x4−x3
54.6k2l2−5k l3+l4
4 k2−l255.
x2 y−4 xyx4 y3−2x3 y3
56.12k3−4 s k3
4 k s2−6ks−18k
57.6 p2−15 p+618−30 p−12 p2
58.−x+2
4 x2−7 x−2
Alg II - Rationals ~3~ NJCTL.org
Multiplying & Dividing Rational ExpressionsClass WorkPerform the indicated operation. Write answer in simplified form.
59. 8a ∙ 1112a 60.
12b5c
∙ 15c2
8b361.
d+ef +2
∙ f2−4
(d+e )2
62.g2−7 g+12
g2−9∙ g
2+4 g+42 g2+6 g+4
63.g+5h+3
∙ h2−h−12g2−25
∙ g2−3 g−10h2−5h+4
64.14h15
÷6h
65. 5 j2÷ 10 j9 66.k+l
k2+2kl+l2÷ 2k+3 lk2−l2
67.m+5
m2+7m+10÷ m
2+5m+6m2−4
68.
2n2−44
n−2
69.p2+4 p+3p2+7 p+10
÷ p2−1p2+2 p+1
∙ p−1p2+5 p+6
70.4q2r12q5r 3
÷ 16q5r 4
18q3r 8÷ 8q
6 r3
24 qr∙ q
2r5
q5r 2∙2
71.x3−27x2−9
÷ 4 x−22x2+5 x−3
72.3−m2m+1
∙ 6m2+m−1m2−9
73.m2−n2
3m2−5mn+2n2∙ 3mn2−2n3
n
74.4 a2−4 ab+4b2
a3+b3∙ a2−b2
12a−12b75.
6m2+9m+36m2+24m+18
÷ 2m2+7m+3
2m2+3m−976.
8+2 x−x2
x2−x−12∙ x−23 x2+5 x−2
77. A rectangular shaped “dartboard” has dimensions (3x +3) by (2x+1) inches. On the board is a square with sides (x+1) inches. What is the probability that a randomly thrown dart that hits board lands in the square?
Alg II - Rationals ~4~ NJCTL.org
Multiplying & Dividing Rational ExpressionsHomeworkPerform the indicated operation. Write answer in simplified form.
78. 9a ∙ 1112a 79.
18b5c
∙ 20 c5
8b480.
d+ef−2
∙ f2−4
(d+e )3
81.g2−8 g+15g2−25
∙ g2+10g+252g2−4 g−6
82.g+4h+7
∙ h2+3h−28g2−16
∙ g2+10 g+21h2−7h+12
83.10h15
÷8h
84. 6 j2÷ 12 j9 85.
2k+3 lk2−2kl+l2
÷ 2k+3lk 2−l2
86.m+6
m2+7m+12÷ m
2+5m−6m2−9
87.
n2−1n2−4 n+4
n−1n2−4
88.p2+4 p+4p2+7 p+12
÷ p2−4p2+5 p+4
∙ p−2p2+4 p+3
89.10q9r15q2 r7
÷ 20q7 r12
16q2r 10÷ 8q
7 r5
25q3 r∙ q
3 r4
q8 r3∙ r
90.27x3− y3
18x2+6 xy+2 y2∙ 4 x2− y2
6 x2−5 xy+ y291.
mn3+n4
3mn4−n5÷ 3m2−mn−4n2
6m2−11mn+4 n292.
8ac−2bca2−b2
÷ 20ac2−5bc2
25a−25b
93.6 x2−17x+54 x2−9 x+2
÷ 6 x2+7 x−3
3 x2−x−1094.
3−xx2+x−20
∙ 2 x2−7x−4
2 x2−5 x−395.
8−2m−3m2
5m2−3m3 ÷ 4−3m3m5+m4−10m3
96. A rectangular shaped “dartboard” has dimensions (4x+6) by (2x+3) inches. On the board is a square with sides (2x+3) inches. What is the probability that a randomly thrown dart that hits board lands in the square?
Alg II - Rationals ~5~ NJCTL.org
Adding and Subtracting Rational ExpressionsClass WorkPerform the indicated operation. Write answer in simplified form.
97.52x
+ 32 x
98.6 y4
−2 y4
99.z−36
+ 2 z6
100.4w+72w+1
−2w+62w+1 101.
5v+12v+3
+ v+82v+3 102.
3(u+2)2u−1
−5 (u+1)2u−1
103.3
x+4+ 2x−4 104.
5x2−9
− 2xx−3 105.
5x2+4 x+4
+ 6x2−4
106.2
x−3+ 3x2−7 x+12
+ 4x−4 107.
x2−23x2−x
− x2−33 x−1
108. 3 x− x−4x2+4 y
109.5
x2−5 x+6+ 4x2+3 x−10
− 3x2+2x−15
110.2m+4m−6
+3m−106−m 111. 2x2−x
4−x− 2x2
x−4− x2
x
112.4 x−2
2x2−5 x+3− 3 x−52x2+x−6
+ x2
x2+x−2113.
4 y−93 y+2
−5 y
Alg II - Rationals ~6~ NJCTL.org
Adding and Subtracting Rational ExpressionsHomeworkPerform the indicated operation. Write answer in simplified form.
114.43x
+ 83 x
115.7 y5
−2 y5
116.5 z−46
+ 5 z6
117.5w+43w+2
−2w+23w+2 118.
3v+74 v+6
+ v+94v+6 119.
4 (2u+1)2u−1
−2(u+8)2u−1
120.3
3t−1+ 22t+2 121.
5x2−5x+6
− 2 xx−3 122.
5x2+6 x+9
+ 6x2−9
123.2
x+3+ 3x2−3x−18
+ 4x−6
124.5
x2−4 x+3+ 4x2+x−12
− 3x2+3 x−4
125. 4−2 x−33 x−2
126.4 xx−3
+ 6 x−52 x2−6 x
127.6 p−24−p
− p−1p−4
128.8b+43−2b
− 6b+22b−3
129.3m−216m2−1
− 6m+58m2−6m+1
130.2 y−7
6 y2+ y−1− y2− y2 y2− y−1
Alg II - Rationals ~7~ NJCTL.org
Solving Rational EquationsClass WorkSolve for x. Check for extraneous solutions.
131.2
x+3= 3x−2
132.4
2x−1= 6x+5
133.2x−12
+ x+310
=6 x5 134.
52x
− x+3x
= 34
135.2
x+3+ 52= 1x+3 136.
2x−3
+ 4 xx2−9
= −1x+3
137.3x+2
− 4x−1
= 5x2+x−2
138.x
x+5− 2x−3
= 1x2+2x−15
139.2
x2−4+ 1x−2
= 3x+2
− 5x2−4 x+4
140.305x
+3=6x
141.52x
+ 73= 23 x
−46
142.4
3x−2+ 2x−1
= 123 x2−5 x+2
143.3
x3+27+ 2x+3
= 5x2−3 x+9
Alg II - Rationals ~8~ NJCTL.org
Solving Rational EquationsHomeworkSolve for x. Check for extraneous solutions.
144.2
x−1= 5x+4
145.5
3x+4= 63 x+6
146.3x+13
+ 4−x2
=5 x6 147.
53x
− x+3x
=32
148.2
x−3+ 53= 1x−3 149.
3x−2
+ 2xx2−4
= 16x+2
150.7x+5
− 2x−2
= 3x2+3 x−10
151.x
2x+1− 2x−1
= x+22 x2−x−1
152.2
x2+4 x+3+ 1x+3
= 3x+1
− 5x2+6 x+9
153.2x4
+ 5x=7x
154.3
2x−1+ 42x+1
= 34 x2−1
155.−4x−1
+ 2x3−1
= 8x2+x+1
156.4x+ 34 x
=−23
+ 5x
Alg II - Rationals ~9~ NJCTL.org
Graphing Rational ExpressionsClass WorkGraph the following functions. Clearly indicate any x-intercepts, y-intercepts, asymptotes and holes (discontinuities) on the graph, noting them in the spaces provided below.
157. f ( x )= 2x−1
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
158. g ( x )= −3x+2
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
159. h ( x )= x+1x2−1
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
160. f ( x )= x−1( x−1 )(x+2)
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
161. g ( x )= x2+5 x+6x2+3 x+2
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
162. h ( x )= x2−x−6x2−5x+6
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Alg II - Rationals ~10~ NJCTL.org
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Graphing Rational ExpressionsHomeworkGraph the following functions. Clearly indicate any x-intercepts, y-intercepts, asymptotes and holes (discontinuities) on the graph, noting them in the spaces provided below.
163. f ( x )= 2x+3
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
164. g ( x )= −3x−4
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
165. f ( x )= x+2( x−1 )(x+2)
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
166. h ( x )= x−2x2−4
x-intercepts: ____________________
y-intercepts: ____________________
167. g ( x )= x2+9 x+18x2+7 x+6
x-intercepts: ____________________
y-intercepts: ____________________
168. h ( x )= x2+5 x−14x2+6 x−7
x-intercepts: ____________________
y-intercepts: ____________________
Alg II - Rationals ~11~ NJCTL.org
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Unit Review - Multiple Choice
1. Simplify 2x2−10 x+124 x2−12 x
a.x−22
b.x−22x
c.( x−3 ) ( x−2 )2 x2−6 x
d.( x−6 ) ( x+1 )2 x ( x−3 )
2. Simplify x2+15 x+56x2−49
∙ x2−10 x+21x2+11 x+24
a.( x−7 ) ( x−3 )( x+7 ) ( x+3 )
b.( x+7 ) ( x−3 )( x−7 ) ( x+3 )
c.x−3x+3
d. 1
3. Simplify 6m6n3
4m2n9÷ 9m
4n2
8m3n7
a. 4m3
3n
b.4m5
3n11
c.3n4m3
d.3n11
4m5
Alg II - Rationals ~12~ NJCTL.org
4. Simplify 23x2
− 56 x
a.−16 x2
b.4−5 x6 x2
c.−16 x
d.−13x2
5. Simplify 2
x2−16+ 4x2+8 x+16
a.6 x−24
( x−4 ) (x+4 ) ( x+4 )
b.6
( x+4 ) ( x+4 )
c.6 x−8
( x−4 ) (x+4 ) ( x+4 )
d.6 x
( x−4 ) (x+4 ) ( x+4 )
6. The function h ( x )=4 x2−3 x−14 x2−1
has which of the following discontinuities?
a. Vertical asymptotes at x=± 12
b. Removable discontinuity at x=± 12
c. Vertical asymptote at x=12
; removable discontinuity at x=−12
d. Vertical asymptote at x=−12
; removable discontinuity at x=12
7. h varies inversely with t. If h=8 when t=6, find t when h=16.a. 2b. 3
c.83
d. 48
Alg II - Rationals ~13~ NJCTL.org
8. The volume of a cone varies jointly to its height and the square of the radius of its base. If V = 18π when h=6 and r=3. What is the radius when V = 12π and h = 4?
a. 1b. 3c. 6d. 9
9. Simplify x+3x+5
− 4x−5
+ 3 x−7x2−25
a.x2−3 x+3x2−25
b.x2−2 x−15x2−25
c.4 x−8x2−25
d.x2−3x−42x2−25
10. Solve: 3
x−2= 4x+2
a. 4b. 8c. 14d. no solution
11. Solve: 4 xx2−1
+ 4x−1
= 2x+1
a. -1b. 1c. 6d. no solution
12. Solve: 2
x2−9+ 3x2−x−6
= 4x2+5 x+6
a. -25b. -8c. 8d. no solution
Extended Response
1. At math camp the lap pool is a rectangle that is (x2−16 ) ft by (x+3) ft, the wading pool is a square with sides
(x+4 ) ft.a. How many times larger is the lap pool than the wading pool?
Alg II - Rationals ~14~ NJCTL.org
b. If the wading pool is (x−4) ft deep, what is the pool’s volume?
c. If the lap pool has a depth of (x+4 ) ft, how many times larger is the volume of the lap pool to the wading pool?
2. Determine each of the following for the graph of the rational function and graph the function.