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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


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


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?


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?


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


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


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


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


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: _________________________



159. h ( x )= x+1x2−1

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



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: _________________________








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: _________________________



164. g ( x )= −3x−4

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



165. f ( x )= x+2( x−1 )(x+2)

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



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: ____________________


Holes: _________________________



Holes: _________________________



Holes: _________________________



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


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


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?


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.

f ( x )= x2+x−6−4 x2−16 x−12

x-intercepts: ________________________y-intercepts: ________________________holes: _____________________________vertical asymptotes: __________________horizontal asymptotes: ________________

PMI Rational Expressions, Equations and Functions- SOLUTIONS:Variation: Classwork:

1. k=48 ; y=−82. k=24 ; x=−12

Reducing Rational Expressions:Classwork:

Reducing Rational Expressions:Homework:


https://s3.amazonaws.com/grapher/exports/b8eidhdaz2.png

3. k=75 ; y=8.3

4. k=0.654

5. k=3 ; y=−366. k=4 ;x=67. k=π;r=3

Variation: Homework:

8. k=36 ; y=−69. k=40 ; x=−2010. k=48 ; y=5.311. k=0.5 ;b=612. k=2 ; y=−3613. k=−5 ; x=−614. k=π;r=4

15.x2

16.103

17. 3 x

18.9

7a2b4

19.15 j4m

20. 2(2c 2– 1)

21.43

22.54

23.v−2v+2

24. f +4f−5

25. – s2−3 s2

26.2d−33d−2

27.2x+3x2(x+1)

28.x(3x+1)3(x−1)

29. 2(m−1)

30.3 p−2p−2

31. 2(a+3)

32.x2−xy+ y2

2(x− y )

33.3(ab+1)

c

34. 5x−1x2

35. −3 p+24

36. −2 x+32(x−2)

37.3 y2

38. 3

39. 32x

40. 2ab6

41. 5 j2

m2

42. 4 c2−32

43. 2n−13n−7

44. 43

45. v−1v+3

46. f +3f−8

47.s (rs−1)(rs−3)

3−s

48. 4 d+13d+2

49. x+1x

50. a2+ab+b2

6 c

51. 4

m+n52. 2m

53.2(3 x−1)x2(x+1)

54. l2 (3k−l )2k+ l

55.x−4

x2 y2(x−2)

56. −2k2

2 s+3

57.−p−22( p+3)

58.−14 x+1



Multiplying & Dividing Rational ExpressionsClass Work:

1. 22x2

3

2.3bc2b5c

3.f−2d+e

4.(g+2)(g−4)2(g+1)(g+3)

5. g+2h−1

6.745

7. 9 j2

2

8.k−12 k+3l

9.m−2

(m+3 ) (m+2 )

10.1

2 (n+2 )

11.( p+1 )2

(p+5 ) (p+2 )2

12.9 r3

4 q13

13. x2+3 x+92

14.−3m−1m+3

15. n(m+n)

16.13

17. 2m−32(m+3)

18.−x−2

( x+3 ) (3 x−1 )

19.3 (2 x+1 )x+1

Multiplying & Dividing Rational ExpressionsHomework

20.334

∨8 14

21.9c4

b3

22. f +2

(d+e )2

23. g+52 (g+1 )

24.(h+3 ) (h+7 )

(g−4 ) (h−3 )

25. 112

26. 9 j2

27.k+1k−1

28. m−3

(m+4)(m−1)

29. (n+1 ) (n+2 )

n−2

30.p+2

(p+3 )2

31.5

3 r10q7

32. 2x+ y2

33. 2m−n

n(3m−n)

34. 10

c (a+b)35.(2 x−5)(3 x−5)(4 x−1)(2x+3)

36. −1x+5

37. −m (m+2 )2

38.12

Adding & Subtracting Rational ExpressionsClasswork

39. 4x

40. y

41. z−12

42. 143. 344. -1

45. 5 x−4

( x+4 ) ( x−4 )

46. −2x2−6 x+5

( x−3 ) ( x+3 )

47. 11 x+2

( x+2 )2 ( x−2 )

48.6 x−17

( x−4 ) (x−3 )

49. 6 x−19

( x−2 ) (x−3 ) (x+5 )

50.3x2+12 xy−x+4

x2+4 y

51.−2 x2+7x(3x−1)

52.−m+14m−6

53.−5x (x−1)

x−4

54. 2 x3−2x2+14 x−9(2 x−3)(x−1)(x+2)

55. −15 y2+6 y+93 y+2

Adding and Subtracting Rational ExpressionsHomework

59. 4 x60. y

61. 5 z−23

62. 1

63. 2 ( v+4 )2v+3

64. 6 (u−2 )2u−1

65. 2 (3t+1 )

( t+1 ) (3 t+1 )

66. −2x2+4 x+5

( x−2 ) ( x−3 )

67. 11x+3

( x+3 )2 ( x−3 )

68.3 (2 x+1 )

( x+3 ) ( x−6 )

69. 6 x+25

( x−1 ) ( x+4 ) ( x−3 )

70.10x−53x−2

71.14 x−52x (x−3)

72.−7 p+3p−4

73.−2(7b−3)2b−3

74.−18m2+33m+3

(4m−1 ) (4m+1 ) (2m−1 )

75. −3 y3+6 y2−10 y+7(2 y+1)(3 y−1)( y−1)

Solving Rational EquationsClass Work

76. −13

77. 134

∨3 14

78. −2

79. 45, 32

80.−175

81. −37

82. −16

83. 5±√692

84. 19±√28114

85. No Solution

86.−1118

87. 2

88. 11±√734

Solving Rational EquationsHomework

89.133

90. 291. 7

92. 3(5±√105)

20

93.125

94. 3811

95.275

96. No Solution

97. −7±i √554

98. ±2

99.27

100. −3±√152

101.38


Graphing Rational Equations Classwork:

157. f ( x )= 2x−1

x-intercepts: None

y-intercepts: y=2

Holes: None

Vertical asymptotes: x=1

Horizontal asymptotes: y=0

158. g ( x )= −3x+2

x-intercepts: None

y-intercepts: y=−1.5

Holes: None

Vertical asymptotes: x=−2


159. h ( x )= x+1x2−1

160. f ( x )= x−1( x−1 )(x+2)


https://s3.amazonaws.com/grapher/exports/0hofutbeku.png

https://s3.amazonaws.com/grapher/exports/teki4zxwa5.png

x-intercepts: None

y-intercepts: y=−1

Holes: x=−1



x-intercepts: None

y-intercepts: y=12

Holes: x=1



161. g ( x )= x2+5 x+6x2+3 x+2

x-intercepts: x=−3

y-intercepts: y=3

Holes: x=−2



162. h ( x )= x2−x−6x2−5x+6



Holes: x=3



Graphing Rational Equations Homework:

163. f ( x )= 2x+3

164. g ( x )= −3x−4


https://s3.amazonaws.com/grapher/exports/8zhwzftqbk.png

https://s3.amazonaws.com/grapher/exports/xcsmvl7fit.png

https://s3.amazonaws.com/grapher/exports/s7ic1d2yhj.png

https://s3.amazonaws.com/grapher/exports/ww08hjltxn.png

x-intercepts: None

y-intercepts: y=23

Holes: None



x-intercepts: None

y-intercepts: y=34

Holes: None



165. f ( x )= x+2( x−1 )(x+2)

x-intercepts: None


Holes: x=−2



166. h ( x )= x−2x2−4

x-intercepts: None

y-intercepts:y=12

Holes: x=2



167. g ( x )= x2+9 x+18x2+7 x+6

168. h ( x )= x2+5 x−14x2+6 x−7


https://s3.amazonaws.com/grapher/exports/9u0g36a5jp.png

https://s3.amazonaws.com/grapher/exports/42hjcw8zt5.png

https://s3.amazonaws.com/grapher/exports/atlkhzmxgy.png

https://s3.amazonaws.com/grapher/exports/kvikab0l86.png


y-intercepts: y=3

Holes: x=−6



x-intercepts: x=2

y-intercepts: y=2

Holes: x=−7



Unit Review - Multiple Choice

1. B.2. C. 3. A.4. A.

5. C. 6. A.7. B. 8. D.

9. D. 10. C.11. D. 12. A.

Extended Response

1. a.

( x−4 )(x+3 )x+4

b. x3+4 x2−16 x−64

c. x+3


https://s3.amazonaws.com/grapher/exports/2jdtseq5wk.png

https://s3.amazonaws.com/grapher/exports/j2bs1zchjp.png

2.

x-intercepts: (2 ,0 )

y-intercepts: (0 , 12 )Hole: x=−3Vertical Asymptote: x=2

Horizontal Asymptote: y=−14


https://s3.amazonaws.com/grapher/exports/4tlcopphut.png

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