algebra - white plains middle school · chapter 12-3 undefined &simplifying rational...

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Algebra

Chapter 12 Rational Expressions

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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 ……… 4

Chapter 12-7 Solving Rational Expressions (LCD) – Day 6 ……………………………..…………Page 29

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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).


Example # 4: Reduce to lowest terms (factoring trinomials and cancelling).


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

6

Homework (Day1)

Reduce to lowest terms (factoring and cancelling).

7

Reduce to lowest terms (factoring and cancelling).

16) 17)

18) 19)

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.


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




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.


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)



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



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)


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


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

22

Summary:

Exit Ticket:

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


c)


2) 4)


d)


5) 6)


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


7) 8)


f)


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

27

Summary:

Exit Ticket:

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



4)



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



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

32

Homework (Day 6)

Solve. Check your answer.

1) 2)

3) 4)

5) 6)

7) 8)

6

15

4

1

3

8

xxx

3

1

3

11

xx 6

1

12

11

x

xx 2

5

12

13

6

5

3

21

xx

2

174

xx2

311

xx

xx

x 1

3

1

3

algebra - white plains middle school · chapter 12-3 undefined &simplifying rational...

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