Chapter 9
Algebra 2 Honors
Rational Expressions
Rational expression:
Examples:
Rational Equations:
Examples:
Finding Domains
Find the domain:
x3
2x  1
f (x) 
2
x2
x2  1
f (x) 
x2
f (x) 
3x
x 4
f (x) 
2
1
Chapter 9
Algebra 2 Honors
Simplify Rational Expressions
Simplify vs. dividing
7
2
16
2
Simplifying expressions:
Step 1. Look to see if something divides evenly into ALL terms.
8y  2
2
3y  5
3
3y 3  12y 2
6y
16x 4 (y  8)
12y 2 (y  8)
(y  1)(y  3)
(y  3)(2y  1)
2(8  y)
x(y  8)
2
Chapter 9
Algebra 2 Honors
Step 2. Factor, completely, the numerator and denominator. Then simplfy.
3y  6
y 2  2y
x 2  x  20
x 2  2x  15
x 2  2x
x 2  2x
x 4  y4
x 2  y2
Homework
Page 476
14 - 20
3
Chapter 9
Algebra 2 Honors
Multiplying Rational Expressions
2 9
 
3 8
2x 9y


3y2 8x 4
x  1 (y  5) 2


y  5 2(x  1)
x 2  2x (x  2)(x  1)


3x(x  1)
x2  4
2x 2  6x 6x  12


3x  6 8x 3  12x 2
x 2  x  20 x 2  x  6


x 2  4x  4 12  x  x 2
Homework
Page 476 23, 29, 31, 33
4
Chapter 9
Algebra 2 Honors
Dividing Rational Expressions
2 7
 
3 6
2
3
5
6
12x 2 y2  24xy2 4x 3y  8x 2 y


5z 2
3z 4
6x 2  3xy 16x 2 y2  8xy3


10ab4
15a 2 b2
6x 2  17x  12 4x 2  8x  3
 2

3x 2  x  2
5x  x  4
Homework
Page 476 25, 27, 35, 37, 39, 41
5
Chapter 9
Algebra 2 Honors
Adding and Subtracting Rational Expressions
2 4
 
3 5
4 5
 
3 6
2x 1


3 4x
2
5


2
4x y 3xy
2
3


x 1 x 1
4x
x


x 1 x 1
3
7


4x  8 x  2
3 7x


x x2
6
Chapter 9
x 1
4


2
x 4 x2
x
4x
 2

2x  4 x  2x
Algebra 2 Honors
3x
3x  6
 2

2x  3 2x  x  6
6x  3
3x
5



2
2x  x  6 2x  3 x  2
Homework
Page 482
23 – 41 odd
7
Chapter 9
Algebra 2 Honors
Complex Fractions
Simplify:
5
2
3
8
2
3
2 ways:
1.
1
x
1
1 2
x
1
2.
y2
4
y
y
8
Chapter 9
4
2
x2
10
5
x2
Algebra 2 Honors
3 10
1  2
x x
4 18
1  2
x x
Homework
Page 483
40, 41
Graphing Rational Functions
Rational function:
Using your calculator graph f (x) 
What is unusual about the graph?
Why is this happening?
2x
.
x 1
9
Chapter 9
Graph y 
Algebra 2 Honors
2
(x  2)2
Continuity:
Asymptote:
Graph g(x) 
Hole:
x2  4
x2
10
Chapter 9
Graph y 
Algebra 2 Honors
3x  3
2x 2  x  3
Homework
Page 489 17, 19, 21
Variation
Direct variation:
If the ratio of the two
If y varies directly with as x, and y = 12 when x = -3, find x when y = -64.
equation
Joint variation:
ratio
11
Chapter 9
Algebra 2 Honors
If y varies jointly as x and w, find y when x = 8 and w = 3, if y = 16 when w = 2 and x = 5.
Inverse variation:
If P varies inversely with x, find P when x is 8 if P = 5 when x = 4.
The circumference of a circle is an example of
The area of a circle is an example of
The area of a triangle is an example of
Homework
Page 496
27 – 37 odd, 38 – 43, 47, 49
12
Chapter 9
Algebra 2 Honors
13
Solving Rational Equations
Solve for x:
x  1 2x

4
3
x x 15
 
2 4 4
(2 ways)
NOTE: You must always check your answer(s) when the variable(s) is in the
____________________________ of the original equation. Think DOMAIN!
Solve for x:
7
6
5
x
5
7
2 
x 3
x 3
Chapter 9
4 13
3x  
x 2
8
4

x  2 x 1
Algebra 2 Honors
4
x6
 1
x2
x 1
6
8x 2
4x
 2

x 3 x 9 x 3
Homework
Page 510 11, 13, 23 – 29 odd
14
Chapter 9
Algebra 2 Honors
Solving Rational Inequalities
Solve fore x:
4
1
w2
1
5 1


4x 8x 2
Homework
Page 510 15, 19 – 22
A landscaper with a large mower can mow the grass in the cemetery in 10 hours. Another
landscaper, with a smaller mower, can mow the grass in 15 hours. If they work together how
many hours will it take to mow the grass?
15
Chapter 9
Algebra 2 Honors
After 2 hours of mowing the large mower broke. The small mower had to finish mowing all that
was left. How long did the entire job take?
A welder can do a job in 25 hours. The welder and his assistant work on the job for 10 hours
when the welder decides to quit. The assistant finishes the job in 17 hours. How long would it
have taken the assistant to do the entire job by herself?
With the faucets wide open a bathtub can be filled in 10 minutes. With the drain wide open that
same bathtub can be drained in 15 minutes. If the faucets and the drain are all wide open, how
long, if ever, will it take to fill the tub?
Homework
Page 510 34, 35, review above problems
16