math 2 honors - santowski 1 lesson 24 – operations with rational expressions 12/24/2015 math 2...
TRANSCRIPT
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
04/21/23Math 2 Honors - Santowski
(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
04/21/23Math 2 Honors - Santowski
(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
04/21/23Math 2 Honors - Santowski
(A) Multiplying and Dividing With Rational Expressions –Example 2
5
Simplify and state restrictions for
y
a
y
a
2
3
2 4
9
04/21/23Math 2 Honors - Santowski
(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
( ) ( )
( ); ;
; ;
04/21/23Math 2 Honors - Santowski
(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
04/21/23Math 2 Honors - Santowski
(A) Multiplying and Dividing With Rational Expressions – Solution for Example 3
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
(A) Multiplying and Dividing With Rational Expressions – Solution for Example 3
9 04/21/23Math 2 Honors - Santowski
(A) Multiplying and Dividing With Rational Expressions – Example 4
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
04/21/23Math 2 Honors - Santowski
(A) Multiplying and Dividing With Rational Expressions – Solution for Example 4
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 - Santowski
(A) Multiplying and Dividing With Rational Expressions – Example 5
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
04/21/23Math 2 Honors - Santowski13
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
04/21/23Math 2 Honors - Santowski
15
(B) Adding & Subtracting With Rational Expressions – Example 1
Simplify and state restrictions 3 4
a b
04/21/23Math 2 Honors - Santowski
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
( )
( )
( )
( )
; ,
04/21/23Math 2 Honors - Santowski
17
(B) Adding & Subtracting With Rational Expressions – Example 2
Simplify and state restrictions
Graphically verify your work
4
1
3
1
w w
04/21/23Math 2 Honors - Santowski
18
(B) Adding & Subtracting With Rational Expressions – Solution for Example 2
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
( )
( )( )
( )
( )( )
( ) ( )
( )( )
( )( );
04/21/23Math 2 Honors - Santowski
(B) Adding & Subtracting With Rational Expressions – Solution for Example 2
19 04/21/23Math 2 Honors - Santowski
20
(B) Adding & Subtracting With Rational Expressions – Example 3
Simplify and state restrictions for
1 1 22 2a b a b
b
a b
04/21/23Math 2 Honors - Santowski
21
(B) Adding & Subtracting With Rational Expressions – Solution for Example 3
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
( )
( )( )
( )
( )( ) ( )( )
( ) ( )
( )( )
( )( )
( )
( )( );
04/21/23Math 2 Honors - Santowski
22
(B) Adding & Subtracting With Rational Expressions – Example 4
Simplify and state restrictions
1
6 9
1
92 3x x x x
04/21/23Math 2 Honors - Santowski
23
(B) Adding & Subtracting With Rational Expressions – Solution for Example 4
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
( )( ) ( )
( )( ) ( )( )
( )
( )( )( )
( )
( )( )( )
( )( )
( )( ); ,
04/21/23Math 2 Honors - Santowski
24
(B) Adding & Subtracting With Rational Expressions – Example 5
Simplify and state restrictions
x
x x
x
x x
1
2 3
2
4 52 2
04/21/23Math 2 Honors - Santowski
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
( )( ) ( )( )
( )( )
( )( )( )
( )( )
( )( )( )
( ) ( )
( )( )( )
( )( )( )
( )( ); , ,
04/21/23Math 2 Honors - Santowski
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?
04/21/23Math 2 Honors - Santowski
(C) Adding & Subtracting With Rational Expressions – Solution to Example 6
04/21/23Math 2 Honors - Santowski27
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
04/21/23Math 2 Honors - Santowski
Lesson 25 – Solving Rational Equations Math 2Honors - Santowski 1/8/2016 1 Math 2 Honors - Santowski
12/7/2015 Math SL1 - Santowski 1 Lesson 16 – Modeling with Exponential Functions Math SL - Santowski