Math 2 Honors - Santowski 1 Lesson 24 – Operations with Rational Expressions 12/24/2015 Math 2...

Math 2 Honors - Santowski

1

Lesson 24 – Operations with Rational Expressions

04/21/23Math 2 Honors - Santowski

(A)Multiplying and Dividing With Fractions

2

recall how to divide and multiply rational numbers - fractions

ex. (6/5) x (25/42) - multiply the numerator and then multiply the denominator (150/210) = 5/7

OR look to cancel COMMON FACTORS between the numerator and denominators prior to multiplying

(6/5) x (5x5)/(6x7) = (1/1) x (5/7) = 5/7

ex. (5/12) (15/64) - multiply by the reciprocal (5/12) x (64/15) (5/4x3) ÷ (4x16/3x5) (16/3x3) 16/9


(A) Multiplying and Dividing With Fractions – Example 1

3

the strategy that will be used - we must try to factor the expressions in the numerator and denominator

Example: Simplify

But are there restrictions in this case? Why?

2

2

5

21

3

7

a

ac

c

ab


(A) Multiplying and Dividing With Fractions – Solution for Example 1

4

Example: Simplify

0,0;5

495

7

1

7

0,0;5

37

3

75

21

3

72

2

cabc

cb

caaa

cca

c

baa

ac

c

ab

2

2

5

21

3

7

a

ac

c

ab


(A) Multiplying and Dividing With Rational Expressions –Example 2

5

Simplify and state restrictions for

y

a

y

a

2

3

2 4

9


(A) Multiplying and Dividing With Rational Expressions – Solution for Example 2

6

Simplify y

a

y

a

2

3

2 4

9

y

a

y

ay

a

a

ya y

a y

2

3

2 4

91 2

3

3 3

2 20 2

3

20 2

( ) ( )

( ); ;

; ;


(A) Multiplying and Dividing With Rational Expressions – Example 3

7

Simplify and state restrictions for:

Graphically, verify your answer

a a

a a

a a

a a

2

2

2

2

2 3

6 5

7 10

6



8

Simplify a a

a a

a a

a a

2

2

2

2

2 3

6 5

7 10

6

a a

a a

a a

a aa a

a a

a a

a aa

a

a

2

2

2

2

2 3

6 5

7 10

63 1

5 1

5 2

3 25 1 3 2

1

15 1 3 2

1 5 1 3 2

( )( )

( )( )

( )( )

( )( ); , , ,

; , , ,

; , , ,04/21/23Math 2 Honors - Santowski


9 04/21/23Math 2 Honors - Santowski


10

Simplify and state restrictions

2 3 2

3 8 4

2 3 1

3 2

2

2

2

2

d d

d d

d d

d d



11

Simplify 2 3 2

3 8 4

2 3 1

3 2

2

2

2

2

d d

d d

d d

d d

2 3 2

3 8 4

2 3 1

3 22 1 2

3 2 2

3 2

1 2 1

11 2

2

3

1

20

2

2

2

2

d d

d d

d d

d dd d

d d

d d

d d

d

dd

( )( )

( )( )

( )

( )( )

; , , , ,



04/21/23Math 2 Honors - Santowski12

We introduce an idea called a complex fraction:

Consider the 2 expressions

Is there any difference?

How do you then simplify a complex fraction?

321 and

32

122

2

2

2

2

xx

x

x

x

xxx

xx

(A) Multiplying and Dividing With Rational Expressions – Solution to Example 5


Simplify the complex fraction

1,0,1,3;1

3

1

3

1,0,1,3;13

11

321

32

1

2

2

22

2

2

2

2

xx

xx

x

xx

xx

xx

xx

x

xx

x

x

x

xxx

xx

14

(B) Adding & Subtracting With FractionsTo add and subtract fractions:

find a common denominatorconvert fractions to equivalent fractions &

operate 2/5 + 3/7 = 2(7) / 5(7) + 3(5) / 7(5) = 14/35 + 15/35 = 29/35

3/13 - 5/4 = 3(4) / 13(4) – 5(13) / 4(13) = 12/52 – 65/52 = -53/52


15

(B) Adding & Subtracting With Rational Expressions – Example 1

Simplify and state restrictions 3 4

a b


16

(B) Adding & Subtracting With Rational Expressions – Solution for Example 1

Simplify and state restrictions 3 4

a b

3 4

3 4

3 40 0

a bb

a b

a

b a

b a

aba b

( )

( )

( )

( )

; ,


17



Graphically verify your work

4

1

3

1

w w


18


Simplify and state restrictions 4

1

3

1

w w

4

1

3

14 1

1 1

3 1

1 1

4 1 3 1

1 1

7 1

1 11

w ww

w w

w

w w

w w

w w

w

w ww

( )

( )( )

( )

( )( )

( ) ( )

( )( )

( )( );



19 04/21/23Math 2 Honors - Santowski

20


Simplify and state restrictions for

1 1 22 2a b a b

b

a b


21


Simplify and state restrictions for 1 1 22 2a b a b

b

a b

1 1 2

1 1 2

1 1 2

2 2

2 2

2 2a b a b

b

a ba b

a b a b

a b

a b a b

b

a b a b

a b a b b

a b a b

a b

a b a b

a b

a b a b a ba b

( )

( )( )

( )

( )( ) ( )( )

( ) ( )

( )( )

( )( )

( )

( )( );


22



1

6 9

1

92 3x x x x


23


Simplify and state restrictions 1

6 9

1

92 3x x x x

1

6 9

1

91

3 3

1

9

1

3 3

1

3 3

3

3 3 3

3

3 3 3

3 3

3 3

2 3

3 30 3

2 3

2

2

2

2

2

x x x x

x x x x

x x x x x

x x

x x x x

x

x x x x

x x x

x x x

x x

x x xx

( )( ) ( )

( )( ) ( )( )

( )

( )( )( )

( )

( )( )( )

( )( )

( )( ); ,


24



x

x x

x

x x

1

2 3

2

4 52 2


25

(B) Adding & Subtracting With Rational Expressions – Solutions for Example 5

Simplify and state restrictions x

x x

x

x x

1

2 3

2

4 52 2

x

x x

x

x xx

x x

x

x x

x x

x x x

x x

x x x

x x x x

x x x

x

x x x

x xx

1

2 3

2

4 51

3 1

2

5 1

1 5

3 1 5

2 3

5 1 3

6 5 5 6

3 1 5

1

3 1 5

1

3 55 3 1

2 2

2 2

( )( ) ( )( )

( )( )

( )( )( )

( )( )

( )( )( )

( ) ( )

( )( )( )

( )( )( )

( )( ); , ,


26

(C) Adding & Subtracting With Rational Expressions – Example 6

Mr S. and Mr. E set off at the same time on a 30K run. Mr. E. runs 1.4 km/h faster than Mr. S. Mr. E. finishes 2 hrs ahead of Mr. S. in spite of a 20 minute rest in the race. How fast was each one running and how long did it take each person to run the race?


(C) Adding & Subtracting With Rational Expressions – Solution to Example 6


Mr S. and Mr. E set off at the same time on a 30K walk. Mr. E. runs 1.4 km/h faster than Mr. S. Mr. E. finishes 2 hrs ahead of Mr. S. in spite of a 20 minute rest in the race. How fast was each one running and how long did it take each person to run the race?

Equation isSolution is vs = 3.6

km/h and it took me 8 hours & 20 min

3

12

4.1

3030

ss vv

(D) Internet Links

28

Adding and Subtracting Rational Expressions from West Texas A&M

Adding & Subtracting Rationals Lesson - I from Purple Math

College Algebra Tutorial on Multiplying and Dividing Rational Expressions from West Texas A&M

Multiplying & Dividing Rationals Lesson - I from Purple Math 04/21/23Math 2 Honors - Santowski

(E) Homework

29

Textbook, S8.4, p502, Q13,17-25odds, 31,33,44,46

Textbook, S8.5, p509, Q15,19,21,29,36,37,43,45,48-50


Math 2 Honors - Santowski 1 Lesson 24 – Operations with Rational Expressions 12/24/2015 Math 2...

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