pmi rational expressions & equations...

Alg II - Rationals ~1~ NJCTL.org

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 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 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 r2 and h. If 𝑉 = 24𝜋 when ℎ = 6 and 𝑟 = 2, find r when 𝑉 = 18𝜋 and ℎ = 2.

Variation

Homework

8. 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.


Reducing Rational Expressions

Class Work

Simplify.

15. 3x

6 16.

40b

12b 17.

12x2

4x 18.

18a3b2

14a5b6

19. 60j4k6m8

16j3k6m9 20. 12c2− 6

3 21.

8n2+ 4n

6n2 + 3n 22.

5h − 10

4h − 8

23. . v2 − 4v + 4

v2 − 4 24.

f2 + 7f + 12

f2 − 2f − 15 25.

4s3 − 20s2 + 24s

16 − 8s 26.

2d2 − 7d + 6

3d2 − 8d + 4

27. 4𝑥3 − 4𝑥2 − 15𝑥

2𝑥5 − 3𝑥4 − 5𝑥3 28. 54𝑥4 − 6𝑥2

54𝑥3 − 72𝑥2 + 18𝑥 29.

4𝑚3 − 4

2𝑚2 + 2𝑚 + 2 30.

3𝑝2 + 7𝑝 − 6

𝑝2 + 𝑝 − 6

31. 8𝑎2 + 4𝑎 − 60

4𝑎 − 10 32.

2𝑥3 + 2𝑦3

4𝑥2 − 4𝑦2 33. 12𝑎2𝑏2 − 12

4𝑎𝑏𝑐 − 4𝑐 34.

15𝑥3 + 7𝑥2 − 2𝑥

3𝑥4 + 2𝑥3

35. 3𝑝2 − 𝑝 − 2

4 − 4𝑝

36. 6𝑥3 + 15𝑥2+ 9𝑥

12𝑥 + 6𝑥2 − 6𝑥3


Reducing Rational Expressions

Homework

Simplify.

37. 9y

6 38.

48c

16c 39.

12x2

8x3 40.

28a3b2

14a4b8

41. 60j4k6m8

12j2k6m10 42.

12c2− 9

6 43.

10n2− 5n

15n2 − 35n 44.

12h − 8

9h − 6

45. v2 − 4v + 3

v2 − 9 46.

f2+ 12f + 27

f2 + f − 72 47.

5rs3 − 20rs2 + 15rs

15r − 5rs 48.

4d2 + 5d + 1

3d2 + 5d + 2

49. 𝑥4− 𝑥2

𝑥4 − 𝑥3 50. 2𝑎3 − 2𝑏3

12𝑎𝑐 − 12𝑏𝑐 51.

4𝑚2 − 4𝑚𝑛 + 4𝑛2

𝑚3 + 𝑛3 52. 8𝑚𝑝2− 2𝑚

4𝑝2 − 1

53. 12𝑥3 − 10𝑥2 + 2𝑥

2𝑥5 + 𝑥4 − 𝑥3 54.

6𝑘2𝑙2 − 5𝑘𝑙3 + 𝑙4

4𝑘2 − 𝑙2 55.

𝑥2𝑦 − 4𝑥𝑦

𝑥4𝑦3 − 2𝑥3𝑦3 56.

12𝑘3− 4𝑠𝑘3

4𝑘𝑠2 − 6𝑘𝑠 − 18𝑘

57. 6𝑝2− 15𝑝 + 6

18 − 30𝑝 − 12𝑝2 58. −𝑥 + 2

4𝑥2 − 7𝑥 − 2


Multiplying & Dividing Rational Expressions

Class Work

Perform the indicated operation. Write answer in simplified form.

59. 8𝑎 ∙11

12𝑎 60.

12𝑏

5𝑐∙

15𝑐2

8𝑏3 61.

𝑑 + 𝑒

𝑓 + 2∙

𝑓2 − 4

(𝑑 + 𝑒)2

62. 𝑔2− 7𝑔 + 12

𝑔2− 9∙

𝑔2+ 4𝑔 + 4

2𝑔2 + 6𝑔 + 4 63.

g + 5

h + 3∙

h2 − h − 12

g2 − 25∙

g2 − 3g − 10

h2− 5h + 4 64.

14h

15÷ 6h

65. 5𝑗2 ÷10𝑗

9 66.

𝑘 + 𝑙

𝑘2+ 2𝑘𝑙 + 𝑙2 ÷2𝑘 + 3𝑙

𝑘2 − 𝑙2 67. 𝑚 + 5

𝑚2 + 7𝑚 + 10÷

𝑚2 + 5𝑚 + 6

𝑚2 − 4

68.

2

n2 − 44

n − 2

69. p2 + 4p + 3

p2 + 7p + 10÷

p2− 1

p2+ 2p + 1∙

p − 1

p2 + 5p + 6

70. 4q2r

12q5r3 ÷16q5r4

18q3r8 ÷8q6r3

24qr∙

q2r5

q5r2 ∙ 2

71. 𝑥3 − 27

𝑥2 − 9÷

4𝑥 − 2

2𝑥2 + 5𝑥 − 3 72.

3 − 𝑚

2𝑚 + 1∙

6𝑚2+ 𝑚 − 1

𝑚2− 9 73.

𝑚2 − 𝑛2

3𝑚2 − 5𝑚𝑛 + 2𝑛2 ∙3𝑚𝑛2 − 2𝑛3

𝑛

74. 4𝑎2− 4𝑎𝑏 + 4𝑏2

𝑎3+ 𝑏3 ∙𝑎2− 𝑏2

12𝑎 − 12𝑏 75.

6𝑚2+ 9𝑚 + 3

6𝑚2+ 24𝑚 + 18 ÷

2𝑚2+ 7𝑚 + 3

2𝑚2+ 3𝑚 − 9 76.

8 + 2𝑥 − 𝑥2

𝑥2 − 𝑥 − 12∙

𝑥 − 2

3𝑥2 + 5𝑥 − 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 Expressions

Homework


78. 9𝑎 ∙11

12𝑎 79.

18𝑏

5𝑐∙

20𝑐5

8𝑏4 80.

𝑑 + 𝑒

𝑓 − 2∙

𝑓2 − 4

(𝑑 + 𝑒)3

81. 𝑔2− 8𝑔 + 15

𝑔2− 25∙

𝑔2+ 10𝑔 + 25

2𝑔2− 4𝑔 − 6 82.

g + 4

h + 7∙

h2+ 3h − 28

g2− 16∙

g2+ 10g + 21

h2− 7h + 12 83.

10h

15÷ 8h

84. 6𝑗2 ÷12𝑗

9 85.

2𝑘 + 3𝑙

𝑘2− 2𝑘𝑙 + 𝑙2 ÷2𝑘 + 3𝑙

𝑘2 − 𝑙2 86. 𝑚 + 6

𝑚2+ 7𝑚 +12÷

𝑚2+ 5𝑚 − 6

𝑚2− 9

87.

n2− 1

n2− 4n + 4n − 1

n2− 4

88.

p2+ 4p + 4

p2+ 7p + 12÷

p2− 4

p2+ 5p + 4∙

p− 2

p2+ 4p + 3 89.

10q9r

15q2r7 ÷20q7r12

16q2r10 ÷8q7r5

25q3r∙

q3r4

q8r3 ∙ 𝑟

90. 27𝑥3− 𝑦3

18𝑥2+ 6𝑥𝑦 + 2𝑦2 ∙4𝑥2− 𝑦2

6𝑥2− 5𝑥𝑦 + 𝑦2 91. 𝑚𝑛3+ 𝑛4

3𝑚𝑛4− 𝑛5 ÷3𝑚2− 𝑚𝑛 − 4𝑛2

6𝑚2− 11𝑚𝑛 + 4𝑛2 92. 8𝑎𝑐 − 2𝑏𝑐

𝑎2− 𝑏2 ÷20𝑎𝑐2− 5𝑏𝑐2

25𝑎 − 25𝑏

93. 6𝑥2− 17𝑥 + 5

4𝑥2− 9𝑥 + 2÷

6𝑥2+ 7𝑥 − 3

3𝑥2− 𝑥 − 10 94.

3 − 𝑥

𝑥2+ 𝑥 − 20∙

2𝑥2− 7𝑥 − 4

2𝑥2− 5𝑥 − 3 95.

8 − 2𝑚 − 3𝑚2

5𝑚2− 3𝑚3 ÷4 − 3𝑚

3𝑚5 + 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 Expressions

Class Work


97. 5

2x+

3

2x 98.

6y

4−

2y

4 99.

z − 3

6+

2z

6

100. 4w + 7

2w + 1−

2w + 6

2w + 1 101.

5v + 1

2v + 3+

v + 8

2v + 3 102.

3(u + 2)

2u − 1−

5(u + 1)

2u − 1

103. 3

x + 4+

2

x − 4 104.

5

x2− 9−

2x

x − 3 105.

5

x2 + 4x + 4+

6

x2− 4

106. 2

x − 3+

3

x2− 7x + 12+

4

x − 4 107.

𝑥2 − 2

3𝑥2 − 𝑥−

𝑥2 − 3

3𝑥 − 1

108. 3𝑥 − 𝑥 − 4

𝑥2 + 4𝑦

109. 5

x2− 5x + 6+

4

x2+ 3x − 10−

3

𝑥2+ 2𝑥 − 15 110.

2𝑚 + 4

𝑚 − 6+

3𝑚 − 10

6 − 𝑚 111.

2𝑥2 − 𝑥

4 − 𝑥−

2𝑥2

𝑥 − 4−

𝑥2

𝑥

112. 4𝑥 − 2

2𝑥2 − 5𝑥 + 3−

3𝑥 − 5

2𝑥2 + 𝑥 − 6+

𝑥2

𝑥2 + 𝑥 − 2 113.

4𝑦 − 9

3𝑦 + 2− 5𝑦


Adding and Subtracting Rational Expressions

Homework


114. 4

3x+

8

3x 115.

7y

5−

2y

5 116.

5z − 4

6+

5z

6

117. 5w + 4

3w + 2−

2w + 2

3w + 2 118.

3v + 7

4v + 6+

v + 9

4v + 6 119.

4(2u + 1)

2u − 1−

2(u + 8)

2u − 1

120. 3

3t − 1+

2

2t + 2 121.

5

x2− 5x + 6−

2x

x − 3 122.

5

x2+ 6x + 9+

6

x2− 9

123. 2

x + 3+

3

x2− 3x − 18+

4

x − 6 124.

5

x2− 4x + 3+

4

x2+ x − 12−

3

𝑥2+ 3𝑥 − 4 125. 4 −

2𝑥 − 3

3𝑥 − 2

126. 4𝑥

𝑥 − 3+

6𝑥 − 5

2𝑥2 − 6𝑥 127.

6𝑝 − 2

4 − 𝑝−

𝑝 − 1

𝑝 − 4 128.

8𝑏 + 4

3 − 2𝑏−

6𝑏 + 2

2𝑏 − 3

129. 3𝑚 − 2

16𝑚2 − 1−

6𝑚 + 5

8𝑚2 − 6𝑚 + 1 130.

2𝑦 − 7

6𝑦2 + 𝑦 −1−

𝑦2 − 𝑦

2𝑦2 − 𝑦 − 1


Solving Rational Equations

Class Work

Solve for x. Check for extraneous solutions.

131. 2

x + 3=

3

x − 2 132.

4

2x − 1=

6

x + 5

133. 2x − 1

2+

x + 3

10=

6x

5 134.

5

2x−

x + 3

x=

3

4

135. 2

x + 3+

5

2=

1

x + 3 136.

2

x − 3+

4x

x2− 9=

−1

x + 3

137. 3

x + 2−

4

x − 1=

5

x2+ x − 2 138.

x

x + 5−

2

x − 3=

1

x2+ 2x − 15

139. 2

x2− 4+

1

x − 2=

3

x + 2−

5

x2− 4x + 4 140.

30

5𝑥+ 3 =

6

𝑥

141. 5

2𝑥+

7

3=

2

3𝑥−

4

6 142.

4

3𝑥 − 2+

2

𝑥 − 1=

12

3𝑥2 − 5𝑥 + 2

143. 3

𝑥3+ 27+

2

𝑥 + 3=

5

𝑥2 − 3𝑥 + 9


Solving Rational Equations

Homework

Solve for x. Check for extraneous solutions.

144. 2

x − 1=

5

x + 4 145.

5

3x + 4=

6

3x + 6

146. 3x + 1

3+

4 − x

2=

5x

6 147.

5

3x−

x + 3

x=

3

2

148. 2

x − 3+

5

3=

1

x − 3 149.

3

x − 2+

2x

x2− 4=

16

x + 2

150. 7

x + 5−

2

x − 2=

3

x2+ 3x − 10 151.

x

2x + 1−

2

x − 1=

x+2

2x2− x − 1

152. 2

x2+ 4x + 3+

1

x + 3=

3

x + 1−

5

x2+ 6x + 9 153.

2𝑥

4+

5

𝑥=

7

𝑥

154. 3

2𝑥 − 1+

4

2𝑥 + 1=

3

4𝑥2− 1 155.

−4

𝑥 − 1+

2

𝑥3 − 1=

8

𝑥2 + 𝑥 + 1

156. 4

𝑥+

3

4𝑥=

−2

3+

5

𝑥


Graphing Rational Expressions

Class Work

Graph 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) =2

𝑥−1

x-intercepts: ____________________ y-intercepts: ____________________

Holes: _________________________

Vertical asymptotes: ______________

Horizontal asymptotes: ____________

158. 𝑔(𝑥) =−3

𝑥+2

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



159. ℎ(𝑥) =𝑥+1

𝑥2−1

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



160. 𝑓(𝑥) =𝑥−1

(𝑥−1)(𝑥+2)

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



161. 𝑔(𝑥) =𝑥2+5𝑥+6

𝑥2+3𝑥+2

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



162. ℎ(𝑥) =𝑥2−𝑥−6

𝑥2−5𝑥+6

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________




Graphing Rational Expressions

Homework

Graph 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) =2

𝑥+3

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



164. 𝑔(𝑥) =−3

𝑥−4

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



165. 𝑓(𝑥) =𝑥+2

(𝑥−1)(𝑥+2)

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



166. ℎ(𝑥) =𝑥−2

𝑥2−4

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



167. 𝑔(𝑥) =𝑥2+9𝑥+18

𝑥2+7𝑥+6

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________



168. ℎ(𝑥) =𝑥2+5𝑥−14

𝑥2+6𝑥−7

x-intercepts: ____________________

y-intercepts: ____________________

Holes: _________________________




Unit Review - Multiple Choice

1. Simplify 2𝑥2− 10𝑥 + 12

4𝑥2− 12𝑥

a. x − 2

2

b. x − 2

2x

c. (x − 3)(x − 2)

2x2− 6x

d. (x − 6)(x + 1)

2x(x − 3)

2. Simplify x2+ 15x + 56

x2−49∙

x2− 10x + 21

x2 + 11x + 24

a. (x − 7)(x − 3)

(x + 7)(x + 3)

b. (x + 7)(x − 3)

(x − 7)(x + 3)

c. x − 3

x + 3

d. 1

3. Simplify 6𝑚6𝑛3

4𝑚2𝑛9 ÷9𝑚4𝑛2

8𝑚3𝑛7

a. 4m3

3n

b. 4m5

3n11

c. 3n

4m3

d. 3n11

4m5

4. Simplify 2

3x2 −5

6x

a. −1

6x2

b. 4−5x

6x2

c. −1

6x

d. −1

3x2

5. Simplify 2

x2− 16+

4

x2+ 8x + 16

a. 6x−24

(x − 4)(x + 4)(x + 4)

b. 6

(x + 4)(x + 4)

c. 6x−8

(x − 4)(x + 4)(x + 4)

d. 6x

(x − 4)(x + 4)(x + 4)


6. The function ℎ(𝑥) =4𝑥2− 3𝑥 − 1

4𝑥2 − 1 has which of the following discontinuities?

a. Vertical asymptotes at 𝑥 = ±1

2

b. Removable discontinuity at 𝑥 = ±1

2

c. Vertical asymptote at 𝑥 =1

2; removable discontinuity at 𝑥 = −

1

2

d. Vertical asymptote at 𝑥 = −1

2; removable discontinuity at 𝑥 =

1

2

7. h varies inversely with t. If ℎ = 8 when 𝑡 = 6, find t when ℎ = 16.

a. 2

b. 3

c. 8

3

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. 1

b. 3

c. 6

d. 9

9. Simplify 𝑥+3

𝑥+5−

4

𝑥−5+

3𝑥−7

𝑥2−25

a. 𝑥2−3𝑥+3

𝑥2−25

b. 𝑥2−2𝑥−15

𝑥2−25

c. 4𝑥−8

𝑥2−25

d. 𝑥2−3𝑥−42

𝑥2−25

10. Solve: 3

x − 2=

4

x + 2

a. 4

b. 8

c. 14

d. no solution

11. Solve: 4x

x2− 1+

4

x − 1=

2

x + 1

a. -1

b. 1

c. 6

d. no solution

12. Solve: 2

x2− 9+

3

x2 − x − 6=

4

x2+ 5x + 6

a. -25

b. -8

c. 8

d. 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

−4𝑥2−16𝑥−12

x-intercepts: ________________________

y-intercepts: ________________________

holes: _____________________________

vertical asymptotes: __________________

horizontal asymptotes: ________________

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PMI Rational Expressions, Equations and Functions- SOLUTIONS:

Variation: Classwork:

1. 𝑘 = 48; 𝑦 = −8 2. 𝑘 = 24; 𝑥 = −12 3. 𝑘 = 75; 𝑦 = 8.3

4. 𝑘 =0.6

54

5. 𝑘 = 3; 𝑦 = −36 6. 𝑘 = 4; 𝑥 = 6 7. 𝑘 = 𝜋; 𝑟 = 3

Variation: Homework:

8. 𝑘 = 36; 𝑦 = −6 9. 𝑘 = 40; 𝑥 = −20 10. 𝑘 = 48; 𝑦 = 5.3 11. 𝑘 = 0.5; 𝑏 = 6 12. 𝑘 = 2; 𝑦 = −36 13. 𝑘 = −5; 𝑥 = −6 14. 𝑘 = 𝜋; 𝑟 = 4

Reducing Rational Expressions: Classwork:

15. 𝑥

2

16. 10

3

17. 3𝑥

18. 9

7𝑎2𝑏4

19. 15𝑗

4𝑚

20. 2(2𝑐2 – 1)

21. 4

3

22. 5

4

23. 𝑣 − 2

𝑣 + 2

24. 𝑓 + 4

𝑓 − 5

25. –𝑠2− 3𝑠

2

26. 2𝑑 − 3

3𝑑 − 2

27. 2𝑥 + 3

𝑥2(𝑥 + 1)

28. 𝑥(3𝑥 + 1)

3(𝑥 −1)

29. 2(𝑚 − 1)

30. 3𝑝 − 2

𝑝 − 2

31. 2(𝑎 + 3)

32. 𝑥2− 𝑥𝑦 + 𝑦2

2(𝑥 − 𝑦)

33. 3(𝑎𝑏 + 1)

𝑐

34. 5𝑥 − 1

𝑥2

35. −3𝑝 + 2

4

36. −2𝑥 + 3

2(𝑥 − 2)

Reducing Rational Expressions: Homework:

37. 3𝑦

2

38. 3

39. 3

2𝑥

40. 2

𝑎𝑏6

41. 5𝑗2

𝑚2

42. 4𝑐2−3

2

43. 2𝑛−1

3𝑛−7

44. 4

3

45. 𝑣 − 1

𝑣 + 3

46. 𝑓 + 3

𝑓 − 8

47. 𝑠(𝑟𝑠−1)(𝑟𝑠−3)

3−𝑠

48. 4𝑑 + 1

3𝑑 + 2

49. 𝑥 + 1

𝑥

50. 𝑎2 + 𝑎𝑏 + 𝑏2

6𝑐

51. 4

𝑚 + 𝑛

52. 2𝑚

53. 2(3𝑥 − 1)

𝑥2(𝑥 + 1)

54. 𝑙2(3𝑘 − 𝑙)

2𝑘 + 𝑙

55. 𝑥 − 4

𝑥2𝑦2(𝑥 − 2)

56. −2𝑘2

2𝑠 + 3

57. − 𝑝 − 2

2(𝑝 + 3)

58. −1

4𝑥 + 1


Multiplying & Dividing Rational Expressions Class Work:

59. 22𝑥2

3

60. 3𝑏𝑐

2𝑏5𝑐

61. 𝑓−2

𝑑+𝑒

62. (𝑔+2)(𝑔−4)

2(𝑔+1)(𝑔+3)

63. 𝑔 + 2

ℎ − 1

64. 7

45

65. 9𝑗2

2

66. 𝑘−1

2𝑘+3𝑙

67. 𝑚−2

(𝑚+3)(𝑚+2)

68. 1

2(𝑛+2)

69. (𝑝+1)2

(𝑝+5)(𝑝+2)2

70. 9𝑟3

4𝑞13

71. 𝑥2 + 3𝑥 + 9

2

72. −3𝑚 − 1

𝑚 + 3

73. 𝑛(𝑚 + 𝑛)

74. 1

3

75. 2𝑚 − 3

2(𝑚 + 3)

76. −𝑥 − 2

(𝑥 + 3)(3𝑥 −1)

77. 3(2𝑥+1)

𝑥+1

Multiplying & Dividing Rational Expressions Homework

78. 33

4𝑜𝑟 8

1

4

79. 9𝑐4

𝑏3

80. 𝑓+2

(𝑑+𝑒)2

81. 𝑔+5

2(𝑔+1)

82. (ℎ+3)(ℎ+7)

(𝑔−4)(ℎ−3)

83. 1

12

84. 9𝑗

2

85. 𝑘+1

𝑘−1

86. 𝑚 − 3

(𝑚 + 4)(𝑚 −1)

87. (𝑛+1)(𝑛+2)

𝑛−2

88. 𝑝+2

(𝑝+3)2

89. 5

3𝑟10𝑞7

90. 2𝑥 + 𝑦

2

91. 2𝑚 − 𝑛

𝑛(3𝑚 − 𝑛)

92. 10

𝑐(𝑎 + 𝑏)

93. (2𝑥 − 5)(3𝑥 − 5)

(4𝑥 − 1)(2𝑥 + 3)

94. −1

𝑥 + 5

95. −𝑚(𝑚 + 2)2

96. 1

2

Adding & Subtracting Rational Expressions Classwork

97. 4

𝑥

98. 𝑦

99. 𝑧−1

2

100. 1 101. 3 102. -1

103. 5𝑥−4

(𝑥+4)(𝑥−4)

104. −2𝑥2−6𝑥+5

(𝑥−3)(𝑥+3)

105. 11𝑥+2

(𝑥+2)2(𝑥−2)

106. 6𝑥−17

(𝑥−4)(𝑥−3)

107. 6𝑥−19

(𝑥−2)(𝑥−3)(𝑥+5)

108. 3𝑥2+12𝑥𝑦 − 𝑥 + 4

𝑥2 + 4𝑦

109. −2𝑥2 + 7

𝑥(3𝑥−1)

110. −𝑚 + 14

𝑚 − 6

111. −5𝑥(𝑥 − 1)

𝑥 − 4

112. 2𝑥3− 2𝑥2+ 14𝑥 − 9

(2𝑥 − 3)(𝑥 − 1)(𝑥 + 2)

113. −15𝑦2+ 6𝑦 + 9

3𝑦 + 2


Adding and Subtracting Rational Expressions Homework

114. 4𝑥

115. 𝑦

116. 5𝑧−2

3

117. 1

118. 2(𝑣+4)

2𝑣+3

119. 6(𝑢−2)

2𝑢−1

120. 2(3𝑡+1)

(𝑡+1)(3𝑡+1)

121. −2𝑥2+4𝑥+5

(𝑥−2)(𝑥−3)

122. 11𝑥+3

(𝑥+3)2(𝑥−3)

123. 3(2𝑥+1)

(𝑥+3)(𝑥−6)

124. 6𝑥+25

(𝑥−1)(𝑥+4)(𝑥−3)

125. 10𝑥 − 5

3𝑥 − 2

126. 14𝑥 − 5

2𝑥(𝑥 − 3)

127. −7𝑝 + 3

𝑝 − 4

128. −2(7𝑏 − 3)

2𝑏 − 3

129. −18𝑚2+ 33𝑚 + 3

(4𝑚 − 1)(4𝑚 + 1)(2𝑚 − 1)

130. −3𝑦3 + 6𝑦2 −10𝑦 + 7

(2𝑦 + 1)(3𝑦 − 1)(𝑦 − 1)

Solving Rational Equations Class Work

131. −13

132. 13

4𝑜𝑟 3

1

4

133. −2

134. 4

5,

3

2

135. −17

5

136. −3

7

137. −16

138. 5±√69

2

139. 19±√2811

4

140. No Solution

141. −11

18

142. 2

143. 11 ± √73

4

Solving Rational Equations Homework

144. 13

3

145. 2 146. 7

147. 3(5±√105)

20

148. 12

5

149. 38

11

150. 27

5

151. No Solution

152. −7±𝑖√55

4

153. ±2

154. 2

7

155. −3 ± √15

2

156. 3

8


Graphing Rational Equations Classwork:

157. 𝑓(𝑥) =2

𝑥−1

x-intercepts: None y-intercepts: 𝑦 = 2

Holes: None

Vertical asymptotes: 𝑥 = 1

Horizontal asymptotes: 𝑦 = 0

158. 𝑔(𝑥) =−3

𝑥+2

x-intercepts: None

y-intercepts: 𝑦 = −1.5

Holes: None

Vertical asymptotes: 𝑥 = −2


159. ℎ(𝑥) =𝑥+1

𝑥2−1

x-intercepts: None

y-intercepts: 𝑦 = −1

Holes: 𝑥 = −1

Vertical asymptotes: 𝑥 = 1 Horizontal asymptotes: 𝑦 = 0

160. 𝑓(𝑥) =𝑥−1

(𝑥−1)(𝑥+2)

x-intercepts: None

y-intercepts: 𝑦 =1

2

Holes: 𝑥 = 1



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161. 𝑔(𝑥) =𝑥2+5𝑥+6

𝑥2+3𝑥+2

x-intercepts: 𝑥 = −3

y-intercepts: 𝑦 = 3

Holes: 𝑥 = −2



162. ℎ(𝑥) =𝑥2−𝑥−6

𝑥2−5𝑥+6



Holes: 𝑥 = 3



Graphing Rational Equations Homework:

163. 𝑓(𝑥) =2

𝑥+3

x-intercepts: None


3

Holes: None



164. 𝑔(𝑥) =−3

𝑥−4

x-intercepts: None


4

Holes: None



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165. 𝑓(𝑥) =𝑥+2

(𝑥−1)(𝑥+2)

x-intercepts: None


Holes: 𝑥 = −2



166. ℎ(𝑥) =𝑥−2

𝑥2−4

x-intercepts: None

y-intercepts:𝑦 =1

2

Holes: 𝑥 = 2



167. 𝑔(𝑥) =𝑥2+9𝑥+18

𝑥2+7𝑥+6



Holes: 𝑥 = −6



168. ℎ(𝑥) =𝑥2+5𝑥−14

𝑥2+6𝑥−7

x-intercepts: 𝑥 = 2


Holes: 𝑥 = −7



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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 + 4x2 -16x-64

c. x+3

2.

x-intercepts: (2, 0)

y-intercepts: (0,1

2)

Hole: 𝑥 = −3

Vertical Asymptote: 𝑥 = 2

Horizontal Asymptote: 𝑦 = −1

4

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