2
Table of Contents
Chapter 12-3 Undefined &Simplifying Rational Expressions – Day 1 …………………………………Page 3
Chapter 12-3 Undefined & Simplifying Rational Expressions (REVIEW) –…………………… Page 8
Chapter 12-4 Multiplying Rational Expressions – Day 2 ……………………………..……………… Page 9
Chapter 12-4 Dividing Rational Expressions – Day 3 ……………………………..…………… Page 13
Chapter 12-5 Adding & Subtracting Rational Expressions (Same Denominators) – Day 4 ………Page 19
Chapter 12-5 Adding & Subtracting Rational Expressions (Unlike Denominators) – Day 5 ……… Page 24
Chapter 12-7 Solving Rational Expressions (LCD) – Day 6 ……………………………..…………Page 29
3
a) 2x-10
2x b) 4x-8
4x c) 5x-15
5xa)
3x-15
x-53)
4x-6
2x-3
C h a p t e r 1 2 – 3 (Day 1) SWBAT: Simplify rational expressions and identify excluded values of rational expressions.
Warm – Up
Reduce to lowest terms.
A rational expression is an algebraic expression whose numerator and denominator are polynomials. The value of the
polynomial expression in the denominator cannot be zero since division by zero is undefined. This means that rational
expressions may have excluded values.
Example 1: Find the value for x that makes the fraction undefined (excluded values):
Practice: Find the value for x that makes the fraction undefined (excluded values):
Example 2: Reduce to lowest terms (factoring and cancelling).
c. 2x+1
x2- 15xa.
6
x+4b.
3
3x-6
a. 14x2
2x b. 3x5
6x3 c. 15x2
35x4a.
30x3y5
6x2y3
𝟒𝒙𝟐
𝒙𝟐 𝟓𝒙 𝟐𝟒
4
1) 10x-20
102)
3x-9
3x 3) 4x-12x
4xb)
6x-12
x-2
a) 3x+6
2x+4 b) 3x-2
6x-4c)
x2-9
3x+9
1) 2x-8
4x-162)
8x-12
4x-63)
x2-1
5x-5
a) x+5
x2+4x-5b)
x-1
x2-2x+1
Practice: Reduce to lowest terms.
Example #3: Reduce to lowest terms (factoring binomials in numerator and denominator).
Practice: Reduce to lowest terms.
Example # 4: Reduce to lowest terms (factoring trinomials and cancelling).
Practice: Reduce to lowest terms.
2) x+1
x2+5x+41)
x+3
x2-x-123)
x+4
x2+7x+12
5
2) 4x-x2
x2-16
Example 5: Reduce to lowest terms (factoring numerator and denominator)
Practice: Reduce to lowest terms
Challenge Problem:
Simplify.
Summary: Exit Ticket:
a) x2-4
x2-8x+12
b) x2-25
x2+10x+25c)
2x-4
x2+4x-12
1) x2-4
x2+4x+42)
x2-25
x2+2x-153)
5x+5
x2-4x-5
8
Simplifying Radical Expressions Review (Additional Practice)
Find for what value of x makes the following rational expressions undefined (excluded values).
1) 2) 3)
Reduce to lowest terms.
4) 5) 6)
7) 8) 9)
10) 11) 12)
13) 14)
9
Chapter 12 – 4 (Day 2) SWBAT: Multiply rational expressions
Warm – Up
Simplify.
1. 2.
Example #1: Find the product in lowest terms.
Practice: Find the product in lowest terms.
Example #2: Find the product in lowest terms.
27c4u3
9c2u4
5x + 20
3x + 12
5
10
2
4
Method 1
(Cancellation Method)
5
10
2
4
Method 2
5x2
3y
9y2
10x3
Method 1
5x2
3y
9y2
10x3
(Cancellation Method)
Method 2
2) 36
15
5
91)
8
12
30
363)
24x
35y
14y
8x2) 36
15
5
91)
8
12
30
363)
24x
35y
14y
8x
10
1) 24x
35y14y
8x
5) 30x2
18y6y
5x
Practice: Find the product in lowest terms.
Example #3: Find the product in lowest terms.
Practice: Find the product in lowest terms.
3) 6x2
5y210xy
6x2
4) 12x2
5y15y2
36x2 6) 24x3y2
7z321z2
12xy
x2-25
x2-1
x+1
2x+10
1) 4x2
x - 2
x2 - 4
122)
7m
2m + 14
3m + 21
9m 3) x2 - 1
x2
3x2 - 3x
15 3)
x2-1
x2
x2
3x2-3x
2) x2
832
3x
3a
6a + 6b
a + b
5a2
11
Challenge Problem: Find the product in lowest terms.
Summary:
Exit Ticket:
4) x2 - 9
3
12
2x - 65)
4a - 6
4a + 8
6a + 12
5a - 156)
x2 - 5x + 6
3x
2
4x - 124) x2 - 9
3
12
2x - 65)
4a - 6
4a + 8
6a + 12
5a - 156)
x2 - 5x + 6
3x
2
4x - 125)
3a-6
4a+86a+12
5a-15
x2 + 12x + 36
x2 - 36
36 - x2
36 + x2
12
Homework (Day 2) Multiply and reduce to lowest terms.
3)2)1) 8x2
3
9
16x
3x
14x2
7x2
15x
9
11
2
3
5) 6)4) 3x7
5x4
10x5
6x5
12x2
8x
12x
6x2
x3
8
2
x2
9)8)7) 8x
2x + 6
x + 3
x2
ab - a
b2
b3 - b2
a
12 - 4
b
b3
12
11)10) 12) x2 - 7x - 8
2x + 2
5
x - 8
x2 - x - 2
3
21
x2 - 4
1
x2 - 1
2x + 2
6
13) 14)x2 - 3x + 2
2x2 - 2
2x
x - 2
x2 - 81
(x + 9)2
10x + 90
5x - 45
13
Chapter 12 – 4 (Day 3) SWBAT: Divide rational expressions
Warm – Up
Find the product in lowest terms.
Example 1: Divide and express the quotient in lowest terms (Keep, change, flip).
Practice: Divide and express the quotient in lowest terms.
Example 2: Divide and express the quotient in lowest terms.
Practice: Divide and express the quotient in lowest terms.
1) 2)
x2 - 25
x2 - 4
5x2 - 20
x + 5
8
7
5
3
1) 7
10
21
5 2)
12
35
4
7 3)
x
9
x
3
3x
5y
21x
2y
7ab2
10cd
14b3
5c2d2
16c3
21d2
24c4
14d3
14
3) 4) 5)
Example 3: Divide and express the quotient in lowest terms.
Practice: Divide and express the quotient in lowest terms.
1) 2) 3)
4) 5)
6a2b2
8c 3ab
xy2
x2y
x
y3
24a3b2
7c3
12ab
21c2
4x+4
9
8x+8
3
2x-2
5
x2-1
10
x2-25
4
x-5
8
m - 5
8
m - 5
16
2x - 4
2x
x2 - 4
x
x2 - 36
x2 - 49
x + 6
x + 7
15
Example 4: Divide and express the quotient in lowest terms.
Practice: Divide and express the quotient in lowest terms.
1) 2)
3) 4)
8x + 24
x2 - 25
4x
x2 + 8x + 15
3x - 3
x2 + 4x - 5
x2 - 25
x - 5
x2 - 5x + 6
5
x - 3
15
x2 + 6x + 8
x2 + 4x + 4
x + 4
x + 2
x2 + 7x + 12
x2 + 3x - 10
x2 - 9
x2 - 25
x2 - 5x + 4
2x
2x - 2
8x2
16
Challenge Problem:
Divide and express the quotient in lowest terms.
Summary:
Exit Ticket:
x - 1
x + 1
2x + 2
x + 2
4x - 4
x + 2
17
Homework (Day 3)
Divide and express the quotient in lowest terms.
1) 7
3
21
6 2)
25
4
5
4
3) 7b2
10
14b3
15 4)
6x2
5
10
x3
5) 15x2 5x4
6 6)
x3
8
x2
16
7) 3
a2 6a4 8)
5x3
x2
3x4
10x
18
Divide and express the quotient in lowest terms.
1. 2.
3. 4.
5. 6.
7. 8.
x - 1
x + 4
x + 3
x + 4
10x + 20
x2 - 4
6x + 12
x - 2
3x + 9
x ( x + 3)
2x - 8
x2 - 36
4 - x
x + 6
x2 - x - 6
x + 5
x2 - 4
x + 5
x2 + x - 6
x2 - x - 12
2x - 4
x - 4
3x+12
5x
x+4
10x
19
Chapter 12 – 5 (Day 4)
SWBAT: To add and subtract rational expressions with the same denominators.
Steps: adding and subtracting with same denominator
1. Add or subtract across the numerators keeping the denominator the same.
2. Reduce the fraction to lowest terms.
Example 1: Add the fractions and reduce to lowest terms.
c) 2m + 4
m2 - 9 +
2
m2 - 9
a) 7
12 -
1
12b)
7x
12 -
x
12c)
7
12x -
1
12x
2. Subtract and simplify your answer.
c) 5
6 +
1
6a)
4
9 +
2
9b)
4x
9 +
2x
9
1. Add and simplify your answer.
Warm _Up
a) 3b
b2 +
5b
b2
20
Practice: Add the fractions and reduce to lowest terms.
1) 2)
3) 4)
5) 6)
Example 2: Subtract the fractions and reduce to lowest terms.
3
6
3
2
xx
x
5a + 2
a2 - 4 -
2a - 4
a2 - 4
b)
3m - 6
m2 + m - 6 -
-m + 2
m2 + m - 6
c)
8
4
8
2 xx
9
3
9 22 xx
x
25
7
25
222 xx
x
d
c
d
c
12
9
12
19
4
6
4
322 xx
x
10x
5y -
2x
5y
a)
21
Practice: Subtract the fractions and reduce to lowest terms.
7) 8)
9) 10)
11) 12)
2323 279
6
279
45
xx
x
xx
x
Challenge Practice:
13) 14)
6
8
6
42
2
2
2
rr
r
rr
rr
y
b
y
b
3
4
3
11
5
3
5
2 dc
3
12
3
45 xx
62
64
62
48
x
x
x
x
1
65
1
5622 x
x
x
x
yx
yxy
yx
xyx
2
2
2
2 22
23
Homework - Chapter 12 – 5 (Day 4)
Add or Subtract. Simply your answer.
1) 2)
3) 4)
5) 6)
7) 8)
9) 10)
22 3
5
3 x
x
x
x
16
4
16 22 xx
x
33
44
y
y
y
y
3
3
3
122
xx
x
33 2
3
2
7
xx
6
6
6
2
x
x
x
x
22
22
22
3 22
x
xx
x
xx
254
8
254
322 x
c
x
x
65
1
65
13422 xxxx
x
23
65
23
2722 xx
x
xx
x
24
Chapter 12 – 5 (Day 5)
SWBAT: To add and subtract fractions with unlike denominators.
Warm Up
Add and simplify your answer.
Prior Knowledge: Add.
Example 1: Add or subtract and simplify your answers.
a) b)
Practice: Add or subtract and simplify your answers.
1) 2)
7
2
4
3
9
72
9
8322 x
x
x
x
48
3 xx
xx
x
6
5
3
1
5
2
10
7 xx
xx 8
3
4
3
25
Example 2: Add or subtract and simplify your answers.
c)
Practice: Add or subtract and simplify your answers.
2) 4)
Example 3: Add or subtract and simplify your answers.
d)
Practice: Add or subtract and simplify your answers.
5) 6)
Example 4: Add or subtract and simplify your answers.
e)
3
3
2
2 xx
5
3
2
4 xx
x
x
x
x
4
32
5
23
xx 5
4
5
7
yy
y
5
10
5
2
xx
x
63
1
36
2
3
5
2
1
xx
26
Practice: Add or subtract and simplify your answers.
7) 8)
Example 5: Add or subtract and simplify your answers.
f)
Practice: Add or subtract and simplify your answers.
9)
Challenge Problem: Add or subtract and simplify your answers.
dcdc
11
3
2
2
4
xx
4
8
2 2xx
x
9
18
3 2yy
y
hf
g
fh 232
5
28
Homework- (Day 5)
Add or subtract. Simplify your answer.
1) 2) 3)
4) 5) 6)
7) 8) 9)
10) 11) 12)
xx 3
43
6
4
2
2 yy
ba
46
6
5
2
53 xx
5
4
10
32x
3
52
9
7 x
5
5
5
2
ww
xx 3
5
3
1
4
6
2
2
xx
1
3
1
22 xx
x
2
7
4
52 xx
baba
11
29
Chapter 12 – 7 Solving Rational Equations (Day 6)
SWBAT: Solve Rational Expressions by Using the LCD.
Warm Up
1. Solve. Check your answer. 2. Divide.
Solving a Rational Equation
Step 1: Find the LCD of all the denominators.
Step 2: Multiply each side of the equation by the LCD.
Step 3: Simplify the terms and solve the resulting equation.
Example 1: Solve. Check your answer.
Practice: Solve. Check your answer.
1) 2)
(6x4 + 12x3 - 9x2) 3x2
1
2
7
4
7
2
x
2
7
5
3
10
xx
y + 3
2y =
2
3
2
1
3
1
6
x2
2
1
8
5
x
30
Example 2: Solve. Check your answer.
Practice: Solve. Check your answer.
4)
Example 3: Solve. Check your answer.
Practice: Solve. Check your answer.
5) 6)
xx
1
9
1
3
5
5
2
5
3 x
x
2
1
6
11
x
x
x 1
6
5
6
xx
x
3
8
2
3
6
31
Example 4: Solve. Check your answer.
Practice: Solve. Check your answer.
7)
Challenge Problem: Solve. Check your answer.
Summary: Exit Ticket:
9
x +
9
x - 2 = 12
6
88
3
2
2
15
xxx
10
52
4
7
5
14
xxx