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 r 2 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 r 2 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 Expressions
Class Work
Simplify.
15.
19.
3x
6
60j4 k6 m8
16j3 k6 m9
23. .
27.
31.
35.
16.
v2 − 4v + 4
v2
−4
4𝑥 3 − 4𝑥 2 − 15𝑥
2𝑥 5
− 3𝑥 4 − 5𝑥3
8𝑎2 + 4𝑎 − 60
4𝑎 − 10
3𝑝2 − 𝑝 − 2
4 − 4𝑝
Alg II - Rationals
20.
24.
28.
32.
36.
40b
17.
12b
12c2 − 6
21.
3
f2 + 7f + 12
f2
25.
− 2f − 15
54𝑥 4 − 6𝑥 2
54𝑥 3
29.
− 72𝑥 2 + 18𝑥
2𝑥 3 + 2𝑦 3
33.
4𝑥 2 − 4𝑦 2
12x2
4x
8n2 + 4n
6n2 + 3n
4s3 − 20s2 + 24s
16 − 8s
4𝑚3 − 4
2𝑚2 + 2𝑚 + 2
12𝑎2 𝑏 2 − 12
4𝑎𝑏𝑐 − 4𝑐
18.
22.
26.
30.
34.
18a3 b2
14a5 b6
5h − 10
4h − 8
2d2 − 7d + 6
3d2 − 8d + 4
3𝑝2 + 7𝑝 − 6
𝑝2 + 𝑝 − 6
15𝑥 3 + 7𝑥 2 − 2𝑥
3𝑥 4 + 2𝑥3
6𝑥 3 + 15𝑥 2 + 9𝑥
12𝑥 + 6𝑥 2 − 6𝑥 3
~2~
NJCTL.org
Reducing Rational Expressions
Homework
Simplify.
37.
41.
45.
49.
53.
57.
9y
38.
6
60j4 k6 m8
12j2 k6 m10
v2 − 4v + 3
v2
−9
𝑥4− 𝑥2
𝑥 4 − 𝑥3
12𝑥 3 − 10𝑥 2 + 2𝑥
2𝑥 5 + 𝑥 4 − 𝑥 3
6𝑝2 − 15𝑝 + 6
18 − 30𝑝 − 12𝑝2
Alg II - Rationals
42.
46.
50.
54.
58.
48c
39.
16c
12c2 − 9
43.
6
f2 + 12f + 27
f2
47.
+ f − 72
2𝑎3 − 2𝑏3
51.
12𝑎𝑐 − 12𝑏𝑐
6𝑘 2 𝑙2 − 5𝑘𝑙3 + 𝑙4
55.
4𝑘 2 − 𝑙2
12x2
40.
8x3
10n2 − 5n
15n2 − 35n
5rs3 − 20rs2 + 15rs
15r − 5rs
4𝑚2 − 4𝑚𝑛 + 4𝑛2
𝑚3
+ 𝑛3
𝑥 2 𝑦 − 4𝑥𝑦
𝑥 4 𝑦 3 − 2𝑥 3 𝑦3
44.
48.
52.
56.
28a3 b2
14a4 b8
12h − 8
9h − 6
4d2 + 5d + 1
3d2 + 5d + 2
8𝑚𝑝2 − 2𝑚
4𝑝2 − 1
12𝑘 3 − 4𝑠𝑘 3
4𝑘𝑠2 − 6𝑘𝑠 − 18𝑘
−𝑥 + 2
4𝑥 2
− 7𝑥 − 2
~3~
NJCTL.org
Multiplying & Dividing Rational Expressions
Class Work
Perform the indicated operation. Write answer in simplified form.
59.
62.
11
8𝑎 ∙ 12𝑎
𝑔2 − 7𝑔 + 12
𝑔2 − 9
65.
5𝑗 2 ÷
68.
2
n2 − 4
4
n−2
60.
𝑔2 + 4𝑔 + 4
∙ 2𝑔2 + 6𝑔 + 4
10𝑗
63.
66.
9
69.
12𝑏 15𝑐 2
5𝑐
∙
61.
8𝑏 3
g + 5 h2 − h − 12 g2 − 3g − 10
∙
h+3
g2
∙
− 25
𝑘+𝑙
2𝑘 + 3𝑙
𝑘 2 + 2𝑘𝑙 + 𝑙2
÷ 𝑘 2 − 𝑙2
p2 + 4p + 3
p2
h2 − 5h + 4
p2 − 1
÷ p2 + 2p + 1 ∙
+ 7p + 10
64.
67.
70.
𝑑+𝑒
𝑓2 − 4
∙
𝑓 + 2 (𝑑 + 𝑒)2
14h
15
÷ 6h
𝑚+5
𝑚2
÷
+ 7𝑚 + 10
4q2 r
12q5 r3
𝑚2 + 5𝑚 + 6
𝑚2 − 4
16q5 r4
÷ 18q3 r8 ÷
8q6 r3
24qr
q2 r5
∙ q5 r2 ∙ 2
p−1
p2 + 5p + 6
71.
74.
𝑥 3 − 27
𝑥2 − 9
4𝑥 − 2
÷ 2𝑥 2 + 5𝑥 − 3
4𝑎2 − 4𝑎𝑏 + 4𝑏2
𝑎3 + 𝑏3
𝑎2 − 𝑏2
∙ 12𝑎 − 12𝑏
72.
75.
3−𝑚
∙
2𝑚 + 1
6𝑚2 + 𝑚 − 1
𝑚2 − 9
6𝑚2 + 9𝑚 + 3
6𝑚2 + 24𝑚 + 18
2𝑚2 + 7𝑚 + 3
÷ 2𝑚2 + 3𝑚 − 9
73.
76.
𝑚2 − 𝑛 2
∙
3𝑚2 − 5𝑚𝑛 + 2𝑛2
8 + 2𝑥 − 𝑥 2
∙
3𝑚𝑛2 − 2𝑛3
𝑛
𝑥−2
𝑥 2 − 𝑥 − 12 3𝑥 2 + 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?
Alg II - Rationals
~4~
NJCTL.org
Multiplying & Dividing Rational Expressions
Homework
Perform the indicated operation. Write answer in simplified form.
78.
81.
11
9𝑎 ∙ 12𝑎
𝑔2 − 8𝑔 + 15 𝑔2 + 10𝑔 + 25
𝑔2 − 25
6𝑗 2 ÷
87.
n2 − 1
2
n − 4n + 4
n−1
n2 − 4
93.
∙
2𝑔2 − 4𝑔 − 6
12𝑗
84.
90.
79.
85.
9
88.
27𝑥 3 − 𝑦 3
4𝑥 2 − 𝑦 2
18𝑥 2 + 6𝑥𝑦 + 2𝑦 2
6𝑥 2 − 17𝑥 + 5
4𝑥 2 − 9𝑥 + 2
82.
∙ 6𝑥 2 − 5𝑥𝑦 + 𝑦 2
6𝑥 2 + 7𝑥 − 3
÷ 3𝑥 2 − 𝑥 − 10
91.
94.
18𝑏 20𝑐 5
5𝑐
∙
80.
8𝑏 4
g + 4 h2 + 3h − 28 g2 + 10g + 21
∙
h+7
g2 − 16
2𝑘 + 3𝑙
∙
h2 − 7h + 12
2𝑘 + 3𝑙
𝑘 2 − 2𝑘𝑙 + 𝑙2
p2 + 4p + 4
÷ 𝑘 2 − 𝑙2
p2 − 4
86.
p− 2
÷ p2 + 5p + 4 ∙ p2 + 4p + 3
p2 + 7p + 12
𝑚𝑛3 + 𝑛4
3𝑚𝑛4 − 𝑛5
3−𝑥
3𝑚2 − 𝑚𝑛 − 4𝑛2
÷ 6𝑚2 − 11𝑚𝑛 + 4𝑛2
2𝑥 2 − 7𝑥 − 4
∙
𝑥 2 + 𝑥 − 20 2𝑥 2 − 5𝑥 − 3
83.
89.
92.
95.
𝑑+𝑒
𝑓2 − 4
∙
𝑓 − 2 (𝑑 + 𝑒)3
10h
15
÷ 8h
𝑚+6
𝑚2 + 7𝑚 +12
10q9 r
÷
𝑚2 + 5𝑚 − 6
20q7 r12
𝑚2 − 9
8q7 r5
q3 r4
÷ 16q2 r10 ÷ 25q3 r ∙ q8 r3 ∙ 𝑟
15q2 r7
8𝑎𝑐 − 2𝑏𝑐
𝑎2 − 𝑏2
÷
20𝑎𝑐 2 − 5𝑏𝑐 2
8 − 2𝑚 − 3𝑚2
5𝑚2 − 3𝑚3
25𝑎 − 25𝑏
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?
Alg II - Rationals
~5~
NJCTL.org
Adding and Subtracting Rational Expressions
Class Work
Perform the indicated operation. Write answer in simplified form.
97.
5
2x
100.
103.
106.
109.
112.
3
+ 2x
98.
4w + 7
2w + 1
2w + 6
− 2w + 1
3
101.
2
+ x−4
x+4
2
104.
3
x−3
4
+ x2 − 7x + 12 + x − 4
5
x2 − 5x + 6
4𝑥 − 2
107.
4
3
+ x2 + 3x − 10 − 𝑥 2 + 2𝑥 − 15
−
2𝑥 2 − 5𝑥 + 3
Alg II - Rationals
6y
𝑥2
3𝑥 − 5
+
2𝑥 2 + 𝑥 − 6
𝑥2
+𝑥−2
110.
113.
4
−
2y
99.
4
5v + 1
v+8
2v + 3
+ 2v + 3
5
2x
x2 − 9
− x−3
𝑥2 − 2
3𝑥 2
2𝑚 + 4
𝑚−6
4𝑦 − 9
3𝑦 + 2
~6~
𝑥2 − 3
− 3𝑥 − 1
−𝑥
+
z−3
102.
105.
108.
3𝑚 − 10
6−𝑚
111.
6
+
2z
6
3(u + 2)
2u − 1
−
5
2𝑥 2 − 𝑥
4−𝑥
2u − 1
6
x2 + 4x + 4
3𝑥 −
5(u + 1)
+ x2 − 4
𝑥−4
𝑥 2 + 4𝑦
2𝑥 2
− 𝑥−4−
𝑥2
𝑥
− 5𝑦
NJCTL.org
Adding and Subtracting Rational Expressions
Homework
Perform the indicated operation. Write answer in simplified form.
114.
117.
120.
123.
126.
129.
4
8
3x
+ 3x
5w + 4
115.
2w + 2
3w + 2
− 3w + 2
3
2
3t − 1
2
x+3
4𝑥
𝑥−3
+ 2t + 2
121.
3
4
+ x2 − 3x − 18 + x − 6
+
6𝑥 − 5
2𝑥 2 − 6𝑥
3𝑚 − 2
16𝑚2
118.
−
−1
Alg II - Rationals
6𝑚 + 5
8𝑚2
− 6𝑚 + 1
124.
127.
130.
7y
5
−
2y
3v + 7
4v + 6
116.
5
v+9
+ 4v + 6
5
119.
2x
x2 − 5x + 6
5
− x−3
4
122.
3
x2 − 4x + 3
+ x2 + x − 12 − 𝑥 2 + 3𝑥 − 4
6𝑝 − 2
𝑝− 1
4−𝑝
−
128.
𝑝− 4
6
+
4(2u + 1)
2u − 1
5z
6
−
5
2u − 1
6
x2 + 6x + 9
4−
2(u + 8)
+ x2 − 9
2𝑥 − 3
3𝑥 − 2
8𝑏 + 4
3 − 2𝑏
−
6𝑏 + 2
2𝑏 − 3
𝑦2 − 𝑦
2𝑦 − 7
−
6𝑦 2 + 𝑦 −1
125.
5z − 4
2𝑦 2 − 𝑦 − 1
~7~
NJCTL.org
Solving Rational Equations
Class Work
Solve for x. Check for extraneous solutions.
131.
133.
135.
137.
139.
141.
143.
2
3
= x−2
x+3
2x − 1
2
2
+
x+3
10
5
x+3
132.
=
6x
134.
5
1
+ 2 = x+3
3
136.
4
5
− x − 1 = x2 + x − 2
x+2
2
x2 − 4
5
2𝑥
1
3
138.
5
+ x − 2 = x + 2 − x2 − 4x + 4
7
+3=
3
𝑥 3 + 27
+
Alg II - Rationals
2
3𝑥
−
140.
4
142.
6
2
=
𝑥+3
4
6
2x − 1
5
−
2x
2
x−3
= x+5
x+3
x
+
3
=4
4x
x2 − 9
x
=
−1
x+3
2
1
− x − 3 = x2 + 2x − 15
x+5
30
5𝑥
6
+3=𝑥
4
3𝑥 − 2
+
2
𝑥−1
=
12
3𝑥 2 − 5𝑥 + 2
5
𝑥2
− 3𝑥 + 9
~8~
NJCTL.org
Solving Rational Equations
Homework
Solve for x. Check for extraneous solutions.
144.
146.
148.
150.
152.
154.
156.
2
5
x−1
= x+4
3x + 1
+
3
2
7
4−x
2
5
x−3
x+5
−
147.
6
2
=
x−2
149.
3
3
+
3
4𝑥
151.
x2 + 3x − 10
1
2𝑥 − 1
+
5x
1
2
4
=
+ 3 = x−3
3
5
+ x + 3 = x + 1 − x2 + 6x + 9
x2 + 4x + 3
𝑥
145.
4
2𝑥 + 1
=
Alg II - Rationals
−2
3
=
+
153.
3
155.
4𝑥 2 − 1
5
6
3x + 4
5
−
3x
3
x+3
x
x
+
−4
𝑥−1
16
+ x2 − 4 = x + 2
2
2x + 1
4
3
=2
2x
x−2
2𝑥
= 3x + 6
x+2
− x − 1 = 2x2 − x − 1
5
𝑥
+
=
7
𝑥
2
𝑥3 − 1
=
8
𝑥2 + 𝑥 + 1
5
𝑥
~9~
NJCTL.org
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
158.
𝑔(𝑥) =
−3
𝑥+2
159.
ℎ(𝑥) =
𝑥+1
𝑥 2 −1
x-intercepts: ____________________
x-intercepts: ____________________
x-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Holes: _________________________
Holes: _________________________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
160.
𝑥−1
𝑓(𝑥) = (𝑥−1)(𝑥+2)
161.
𝑔(𝑥) =
𝑥 2 +5𝑥+6
𝑥 2 +3𝑥+2
162.
ℎ(𝑥) =
𝑥 2 −𝑥−6
𝑥 2 −5𝑥+6
x-intercepts: ____________________
x-intercepts: ____________________
x-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Holes: _________________________
Holes: _________________________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
Alg II - Rationals
~10~
NJCTL.org
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
164.
𝑔(𝑥) =
−3
𝑥−4
165.
𝑥+2
𝑓(𝑥) = (𝑥−1)(𝑥+2)
x-intercepts: ____________________
x-intercepts: ____________________
x-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Holes: _________________________
Holes: _________________________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
166.
ℎ(𝑥) =
𝑥−2
𝑥 2 −4
167.
𝑔(𝑥) =
𝑥 2 +9𝑥+18
𝑥 2 +7𝑥+6
168.
ℎ(𝑥) =
𝑥 2 +5𝑥−14
𝑥 2 +6𝑥−7
x-intercepts: ____________________
x-intercepts: ____________________
x-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Holes: _________________________
Holes: _________________________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
Horizontal asymptotes: ____________
Alg II - Rationals
~11~
NJCTL.org
Unit Review - Multiple Choice
1. Simplify
a.
b.
c.
d.
2. Simplify
a.
b.
c.
2𝑥 2 − 10𝑥 + 12
4𝑥 2 − 12𝑥
x−2
2
x−2
2x
(x − 3)(x − 2)
2x2 − 6x
(x − 6)(x + 1)
2x(x − 3)
x2 + 15x + 56
x2 −49
(x − 7)(x − 3)
∙
x2 − 10x + 21
x2 + 11x + 24
(x + 7)(x + 3)
(x + 7)(x − 3)
(x − 7)(x + 3)
x−3
x+3
d. 1
3. Simplify
a.
b.
c.
d.
4. Simplify
a.
b.
c.
d.
5. Simplify
a.
b.
c.
d.
6𝑚6 𝑛3
4𝑚2 𝑛9
4m3
÷
9𝑚4 𝑛2
8𝑚3 𝑛7
3n
4m5
3n11
3n
4m3
3n11
4m5
2
3x2
−1
−
5
6x
6x2
4−5x
6x2
−1
6x
−1
3x2
2
+
4
x2 − 16
x2 + 8x + 16
6x−24
(x − 4)(x + 4)(x + 4)
6
(x + 4)(x + 4)
6x−8
(x − 4)(x + 4)(x + 4)
6x
(x − 4)(x + 4)(x + 4)
Alg II - Rationals
~12~
NJCTL.org
6. The function ℎ(𝑥) =
4𝑥 2 − 3𝑥 − 1
4𝑥 2 − 1
has which of the following discontinuities?
a. Vertical asymptotes at 𝑥 = ±
1
2
b. Removable discontinuity at 𝑥 = ±
1
2
1
Vertical asymptote at 𝑥 = ; removable discontinuity at 𝑥 = −
c.
2
1
1
2
1
2
2
d. Vertical asymptote at 𝑥 = − ; removable discontinuity at 𝑥 =
7. h varies inversely with t. If ℎ = 8 when 𝑡 = 6, find t when ℎ = 16.
a. 2
b. 3
8
c.
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
a.
b.
c.
d.
10. Solve:
a.
b.
c.
d.
11. Solve:
a.
b.
c.
d.
12. Solve:
a.
b.
c.
d.
𝑥+3
−
4
𝑥+5
𝑥−5
𝑥 2 −3𝑥+3
+
3𝑥−7
𝑥 2 −25
𝑥 2 −25
𝑥 2 −2𝑥−15
𝑥 2 −25
4𝑥−8
𝑥 2 −25
𝑥 2 −3𝑥−42
𝑥 2 −25
3
x−2
=
4
x+2
4
8
14
no solution
4x
x2 − 1
+
4
x−1
=
2
x+1
-1
1
6
no solution
2
x2 − 9
+
3
x2 − x − 6
=
4
x2 + 5x + 6
-25
-8
8
no solution
Alg II - Rationals
~13~
NJCTL.org
Extended Response
1. At math camp the lap pool is a rectangle that is (𝑥 2 − 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.
𝑥 2 +𝑥−6
𝑓(𝑥) = −4𝑥2 −16𝑥−12
x-intercepts: ________________________
y-intercepts: ________________________
holes: _____________________________
vertical asymptotes: __________________
horizontal asymptotes: ________________
Alg II - Rationals
~14~
NJCTL.org
PMI Rational Expressions, Equations and Functions- SOLUTIONS:
Variation:
Classwork:
1.
2.
3.
4.
5.
6.
7.
Reducing Rational Expressions:
Classwork:
𝑘
𝑘
𝑘
𝑘
𝑘
𝑘
𝑘
Variation:
Homework:
8. 𝑘
9. 𝑘
10. 𝑘
11. 𝑘
12. 𝑘
13. 𝑘
14. 𝑘
= 48; 𝑦 = −8
= 24; 𝑥 = −12
= 75; 𝑦 = 8.3
0.6
= 54
= 3; 𝑦 = −36
= 4; 𝑥 = 6
= 𝜋; 𝑟 = 3
15.
16.
17.
18.
19.
20.
21.
= 36; 𝑦 = −6
= 40; 𝑥 = −20
= 48; 𝑦 = 5.3
= 0.5; 𝑏 = 6
= 2; 𝑦 = −36
= −5; 𝑥 = −6
= 𝜋; 𝑟 = 4
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
Alg II - Rationals
𝑥
Reducing Rational Expressions:
Homework:
37.
2
10
38.
3
39.
3𝑥
9
7𝑎2 𝑏 4
15𝑗
40.
4𝑚
41.
2(2𝑐2 – 1)
4
42.
3
5
43.
4
𝑣−2
44.
𝑣+2
45.
𝑓+ 4
𝑓− 5
𝑠2 − 3𝑠
–
46.
2
2𝑑 − 3
47.
3𝑑 − 2
2𝑥 + 3
48.
𝑥 2 (𝑥
+ 1)
𝑥(3𝑥 + 1)
49.
3(𝑥 −1)
50.
2(𝑚 − 1)
3𝑝 − 2
𝑝− 2
2(𝑎 + 3)
𝑥 2 − 𝑥𝑦 + 𝑦 2
51.
52.
53.
2(𝑥 − 𝑦)
3(𝑎𝑏 + 1)
54.
𝑐
5𝑥 − 1
55.
𝑥2
−
3𝑝 + 2
4
2𝑥 + 3
− 2(𝑥 − 2)
~15~
56.
3𝑦
2
3
3
2𝑥
2
𝑎𝑏 6
5𝑗 2
𝑚2
4𝑐 2 −3
2
2𝑛−1
3𝑛−7
4
3
𝑣−1
𝑣+3
𝑓+3
𝑓− 8
𝑠(𝑟𝑠−1)(𝑟𝑠−3)
3−𝑠
4𝑑 + 1
3𝑑 + 2
𝑥+1
𝑥
𝑎2 + 𝑎𝑏 + 𝑏2
4
6𝑐
𝑚+𝑛
2𝑚
2(3𝑥 − 1)
𝑥 2 (𝑥 + 1)
𝑙2 (3𝑘 − 𝑙)
2𝑘 + 𝑙
𝑥−4
𝑥 2 𝑦 2 (𝑥 − 2)
2𝑘 2
− 2𝑠 + 3
𝑝−2
57.
−
58.
− 4𝑥 + 1
2(𝑝 + 3)
1
NJCTL.org
Multiplying & Dividing Rational
Expressions
Class Work:
59.
60.
61.
62.
63.
64.
65.
66.
22𝑥 2
3
3𝑏𝑐
2𝑏5𝑐
𝑓−2
𝑑+𝑒
(𝑔+2)(𝑔−4)
2(𝑔+1)(𝑔+3)
𝑔+2
ℎ−1
7
45
9𝑗 2
2
𝑘−1
2𝑘+3𝑙
𝑚−2
67.
(𝑚+3)(𝑚+2)
1
68.
69.
70.
71.
2(𝑛+2)
(𝑝+1)2
(𝑝+5)(𝑝+2)2
9𝑟 3
4𝑞 13
𝑥 2 + 3𝑥 + 9
2
3𝑚 − 1
− 𝑚+3
73. 𝑛(𝑚 + 𝑛)
72.
74.
75.
1
3
2𝑚 − 3
2(𝑚 + 3)
𝑥−2
76. −
(𝑥 + 3)(3𝑥 −1)
3(2𝑥+1)
77.
𝑥+1
Alg II - Rationals
Multiplying & Dividing
Rational Expressions
Homework
33
1
78.
𝑜𝑟 8 4
4
79.
9𝑐 4
82.
83.
84.
85.
86.
87.
90.
2(𝑔+1)
(ℎ+3)(ℎ+7)
93.
98.
𝑦
103.
(𝑔−4)(ℎ−3)
104.
1
12
9𝑗
2
105.
𝑘+1
𝑘−1
106.
𝑚−3
107.
(𝑚 + 4)(𝑚 −1)
(𝑛+1)(𝑛+2)
108.
𝑛−2
109.
110.
3𝑟 10 𝑞7
2𝑥 + 𝑦
111.
2
2𝑚 − 𝑛
91.
92.
4
𝑥
100.
101.
102.
𝑝+2
88.
(𝑝+3)2
5
89.
97.
99.
𝑏3
𝑓+2
80.
(𝑑+𝑒)2
𝑔+5
81.
Adding & Subtracting Rational Expressions
Classwork
112.
𝑛(3𝑚 − 𝑛)
10
𝑐(𝑎 + 𝑏)
(2𝑥 − 5)(3𝑥 − 5)
113.
𝑧−1
2
1
3
-1
5𝑥−4
(𝑥+4)(𝑥−4)
−2𝑥 2 −6𝑥+5
(𝑥−3)(𝑥+3)
11𝑥+2
(𝑥+2)2 (𝑥−2)
6𝑥−17
(𝑥−4)(𝑥−3)
6𝑥−19
(𝑥−2)(𝑥−3)(𝑥+5)
3𝑥 2 +12𝑥𝑦 − 𝑥 + 4
𝑥 2 + 4𝑦
+7
−2𝑥 2
𝑥(3𝑥−1)
−𝑚 + 14
𝑚−6
−5𝑥(𝑥 − 1)
𝑥−4
2𝑥 3 − 2𝑥 2 + 14𝑥 − 9
(2𝑥 − 3)(𝑥 − 1)(𝑥 + 2)
15𝑦 2 + 6𝑦 + 9
−
3𝑦 + 2
(4𝑥 − 1)(2𝑥 + 3)
1
−𝑥+5
95. −𝑚(𝑚 + 2)2
94.
96.
1
2
~16~
NJCTL.org
Adding and Subtracting Rational
Expressions
Homework
114.
4𝑥
115.
𝑦
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129.
130.
5𝑧−2
3
1
2(𝑣+4)
2𝑣+3
6(𝑢−2)
2𝑢−1
2(3𝑡+1)
(𝑡+1)(3𝑡+1)
−2𝑥 2 +4𝑥+5
(𝑥−2)(𝑥−3)
11𝑥+3
(𝑥+3)2 (𝑥−3)
3(2𝑥+1)
(𝑥+3)(𝑥−6)
6𝑥+25
(𝑥−1)(𝑥+4)(𝑥−3)
10𝑥 − 5
Solving Rational
Equations
Class Work
131.
−13
132.
133.
134.
13
𝑜𝑟
4
136.
−7
−16
5
5±√69
2
19±√2811
4
140.
No Solution
141.
− 18
2
143.
3𝑥 − 2
14𝑥 − 5
11
11 ± √73
4
3
2
7
147.
3(5±√105)
20
148.
149.
150.
12
5
38
11
27
5
No
Solution
151.
152.
153.
154.
155.
156.
2𝑥(𝑥 − 3)
−7𝑝 + 3
13
145.
146.
3
139.
142.
144.
17
−
138.
1
34
−2
4 3
,
5 2
135.
137.
Solving Rational
Equations
Homework
−7±𝑖√55
4
±2
2
7
−3 ± √15
3
2
8
𝑝−4
2(7𝑏 − 3)
−
2𝑏 − 3
18𝑚2 + 33𝑚 + 3
− (4𝑚 − 1)(4𝑚 + 1)(2𝑚 − 1)
−3𝑦 3 + 6𝑦2 −10𝑦 + 7
(2𝑦 + 1)(3𝑦 − 1)(𝑦 − 1)
Alg II - Rationals
~17~
NJCTL.org
Graphing Rational Equations Classwork:
157.
𝑓(𝑥) =
2
𝑥−1
158.
𝑔(𝑥) =
−3
𝑥+2
x-intercepts: None
y-intercepts: 𝑦 = 2
x-intercepts: None
y-intercepts: 𝑦 = −1.5
Holes: None
Vertical asymptotes: 𝑥 = 1
Horizontal asymptotes: 𝑦 = 0
Holes: None
Vertical asymptotes: 𝑥 = −2
Horizontal asymptotes: 𝑦 = 0
159.
ℎ(𝑥) =
𝑥+1
𝑥 2 −1
160.
𝑥−1
𝑓(𝑥) = (𝑥−1)(𝑥+2)
x-intercepts: None
y-intercepts: 𝑦 = −1
Holes: 𝑥 = −1
x-intercepts: None
y-intercepts: 𝑦 =
1
2
Vertical asymptotes: 𝑥 = 1
Holes: 𝑥 = 1
Horizontal asymptotes: 𝑦 = 0
Vertical asymptotes: 𝑥 = −2
Horizontal asymptotes: 𝑦 = 0
Alg II - Rationals
~18~
NJCTL.org
161.
𝑔(𝑥) =
𝑥 2 +5𝑥+6
𝑥 2 +3𝑥+2
162.
ℎ(𝑥) =
𝑥 2 −𝑥−6
𝑥 2 −5𝑥+6
x-intercepts: 𝑥 = −3
x-intercepts: 𝑥 = −2
y-intercepts: 𝑦 = 3
y-intercepts: 𝑦 = −1
Holes: 𝑥 = −2
Holes: 𝑥 = 3
Vertical asymptotes: 𝑥 = −1
Vertical asymptotes: 𝑥 = 2
Horizontal asymptotes: 𝑦 = 1
Horizontal asymptotes: 𝑦 = 1
Graphing Rational Equations Homework:
163.
𝑓(𝑥) =
2
164.
𝑥+3
−3
𝑥−4
x-intercepts: None
x-intercepts: None
y-intercepts: 𝑦 =
𝑔(𝑥) =
y-intercepts: 𝑦 =
2
3
4
3
Holes: None
Holes: None
Vertical asymptotes: 𝑥 = 4
Vertical asymptotes: 𝑥 = 3
Horizontal asymptotes: 𝑦 = 0
Horizontal asymptotes: 𝑦 = 0
Alg II - Rationals
~19~
NJCTL.org
165.
𝑥+2
𝑓(𝑥) = (𝑥−1)(𝑥+2)
166.
ℎ(𝑥) =
𝑥−2
𝑥 2 −4
x-intercepts: None
x-intercepts: None
y-intercepts:𝑦 =
y-intercepts: 𝑦 = −1
1
2
Holes: 𝑥 = −2
Holes: 𝑥 = 2
Vertical asymptotes: 𝑥 = 1
Vertical asymptotes: 𝑥 = −2
Horizontal asymptotes: 𝑦 = 0
Horizontal asymptotes: 𝑦 = 0
167.
𝑔(𝑥) =
𝑥 2 +9𝑥+18
168.
𝑥 2 +7𝑥+6
ℎ(𝑥) =
𝑥 2 +5𝑥−14
𝑥 2 +6𝑥−7
x-intercepts: 𝑥 = −3
x-intercepts: 𝑥 = 2
y-intercepts: 𝑦 = 3
y-intercepts: 𝑦 = 2
Holes: 𝑥 = −6
Holes: 𝑥 = −7
Vertical asymptotes: 𝑥 = −1
Vertical asymptotes: 𝑥 = 1
Horizontal asymptotes: 𝑦 = 1
Horizontal asymptotes: 𝑦 = 1
Alg II - Rationals
~20~
NJCTL.org
Unit Review - Multiple Choice
1.
2.
3.
4.
5.
6.
7.
8.
B.
C.
A.
A.
9. D.
10. C.
11. D.
12. A.
C.
A.
B.
D.
Extended Response
(x - 4)(x + 3)
x+4
3
2
b. x + 4x -16x - 64
c. x + 3
1. a.
2.
x-intercepts: (2, 0)
1
y-intercepts: (0, 2)
Hole: 𝑥 = −3
Vertical Asymptote: 𝑥 = 2
1
Horizontal Asymptote: 𝑦 = − 4
Alg II - Rationals
~21~
NJCTL.org