CHAPTER
7.6,7.7,9.4-9.6
Rational Expressions
Algebra Trigonometry Chapter 9
Page 161
Algebra Trigonometry Chapter 9
Page 162
7.6 Function Operations
1.
E Street Denim has marked down its merchandise by 25%. It later decreases by $5 the price
of items that have not sold.
a. Write a function ƒ(x) to represent the price after the 25% markdown.
f(x) = 0.75(x)
b. Write a function g(x) to represent the price after the $5 markdown.
g(x) = x - 5
c. Use a composition function to find the price of a $50 item after both price adjustments.
g(f(x))= g(.75(50)) = g(37.5) = $32.50
d. Does the order in which the adjustments are applied make a difference? Explain.
Yes. If we look at the general cases….
Let ƒ(x) = 4x  1 and g(x) = 2x2 + 3. Perform each function operation and then find the
domain.
2. ƒ(x) + g(x)
3. ƒ(x)  g(x)
4x-1 + 2x2 + 3
4x-1 – (2x2 + 3)
2x2 + 4x + 2
5.
g ( x) 2 x 2  3

f ( x) 4 x  1
4. ƒ(x)  g(x)
(4x-1)(2x2 +3)
4x – 1 – 2x2 – 3
8x3 + 12x – 2x2 - 3
-2x2 + 4x – 4
8x3 -2x2 + 12x - 3
6. g(x)  ƒ(x)
2x2 + 3 –( 4x-1 )
7.
f ( x) 4 x  1

g ( x) 2 x 2  3
2x2 + 3 – 4x + 1
2x2 – 4x + 4
Algebra Trigonometry Chapter 9
Page 163
Let ƒ(x) = 3x + 2, g(x) =
x
5
, h(x) = –2x2 + 9, and j(x) = 5 – x. Find each value or expression.
8. (f o j)(3)
9. (j o h)(4)
10. (h o g)(x)
f(j(3))
j( h(-4))
f(5-3)
j(-2(-4)2 + 9)
h ( 5x )
f(2)
j(-2(16)+9)
–2( )2 + 9,
j(-32+9)
2  x 
  9
1 5
f(2) =-3(2)+2 = -4
h ( g(x))
2
j(-23)
5 – (-23) = 28
2  x 2 
 9
1  25 
2 x 2
9
25
Let g(x) = x2 – 5 and h(x) = 3x + 2. Perform each function operation.
11. (h o g)(x)
h(g(x))
h(x2 – 5)
3(
)+2
3(x2 – 5) + 2
3x2 – 15 + 2
3x2 – 13
12. g(h(x))
13. 2g(x) + h(x)
g( 3x+2)
(
)2 -5
-2( x2 – 5) + 3x + 2
-2x2 + 10 + 3x + 2
(3x+2)2- 5
(3x+2)(3x+2) - 5
9x2 + 6x + 6x + 4 - 5
-2x2 + 13x + 2
9x2 + 12x - 1
14. Anthropology prices merchandise by adding 80% to its cost. It later decreases by 25% the
price of items that don’t sell quickly.
a. Write a function ƒ(x) to represent the price after the 80% markup.
f(x) = x + 0.8x = 1.8x
b. Write a function g(x) to represent the price after the 25% markdown.
g(x) = 0.75x or 0.75(f(x))
c. Use a composition function to find the price of an item after both price adjustments that
originally costs the boutique $150.
F(150) = 1.8(150) = 270
g(270) = .75(270) = $202.50
d. Does the order in which the adjustments are applied make a difference? Explain.
Order will not matter, since all operations are multiplication
Algebra Trigonometry Chapter 9
Page 164
7.7 Notes
Given each function and its inverse.
a) Graph each relation and its inverse.
b) Find the domain and range of each.
c) Note generalizations about a function and its inverse?
1.
x3
f ( x) 
g(x)=3x -3
3
1. f(x) = x  3
3
1
x 1
3
3.
f(x) = 4x2
2.
m( x)  8 x3
1
n( x )  3 x
2
g(x) = 3x-3
g(x) = 
x
2
Algebra Trigonometry Chapter 9
Page 165
Given a graph of a relation, r, we should be able to determine the graph of its inverse.
4)
5)
Algebra Trigonometry Chapter 9
Page 166
We can use Composition of Inverse Functions to determine if functions are inverses of each
other.
If f and f -1 are inverse functions, then (f -1 f)(x) = x
and (f f -1)(x) =x
Composition of Inverse Functions
If f and f -1 are inverse functions, then (f -1 f)(x) = x
and (f f -1)(x) =
x
Example Problems
6) Are f(x) and g(x) inverse functions?
x  4
3
g ( x)  4  3 x
f ( x) 
Check if f( g(x)) = x
AND
g( f(x) ) = x
7. Describe HOW to show 2 functions are inverses of each other:
Algebra Trigonometry Chapter 9
Page 167
Graph each relation. Are the relations functions?
Find the inverse of each function: Is the inverse a function?
To find the inverse:
a) Switch the x for y and the y for x.
b) Solve for y.
3.
y = 2x + 5
4.
a) x = 2y + 5
b) x – 5 = 2y
y = x2 + 2
a) x = y2 + 2
b) x – 2 = y2
x5
y
2
 x2  y
Having both equations as ‘y=’ is a bit confusing. Many people use function notation,
So: f(x) and g(x) or f(x) and f-1(x).
Find the inverse of each function and then graph each on the same coordinate plan:
Is the inverse a function?
8) y = 6x + 2
9. f(x) = 5x2
x = 6y + 2
Same as: y = 5x2
x – 2 = 6y
x = 5y2
1
( x  2)  y
6
10.
x
 y2
5
f(x) = (x + 4)2  2
x = (y+4)2 - 2
x+ 2 = (y + 4)2
Algebra Trigonometry Chapter 9
 x2  y4
4  x  2  y  f 1 ( x)
Page 168
7.
f(x) =
x 1
7a) Domain of f(x): [1, )
7b) Range of f(x): [0, )
7c) Domain of f-1(x): [0, )
7d) Range of f-1(x) [1, )
8) What do we notice about the domain and range of the original and its inverse?
They are flipped. So, use the domain of the original as the range of
the inverse. Use the range of the original as the domain of the
inverse.
9) Go back to each of the graphs and graph the line: y = x.
Observation:
The inverse is the reflection of the original across the line y = x
Algebra Trigonometry Chapter 9
Page 169
Find the inverse of each relation. Graph the given relation and its inverse.
10.
11.
x
-3
-2
-1
0
x
-3
-1
0
-2
f-1(x)
-2
-1
0
1
f-1(x)
0
1
2
3
Algebra Trigonometry Chapter 9
Page 170
7.6 -7.7 Review
Let f(x) = 3x2 – 2x and g(x) = x – 6. Find each value.
1.
(f o g)(2)
2.  g f  (4)
f( g(2) )
f( 2 – 6 )
f(-4)
3( )2 – 2(
)
2
3(-4) – 2( -4 )
3(16)+8
g( f(-4) )
g( 3(-4)2 – 2(-4) )
g( 48 + 8 )
g(56)
56 - 6
56
50
3.
(g ◦ f)(x)
g( f(x) )
g( 3(x)2 – 2(x) )
(3(x)2 – 2(x))- 6
3x2 – 2x – 6
3(
4. (f ◦ g)(x)
f( g(x) )
f( x – 6 )
)2 - 2 (
)
3(x-6)2 - 2 (x-6 )
3( x – 6) (x – 6) – 2(x-6)
3( x2 – 6x – 6x + 36) – 2x + 12
3(x2 – 12x + 36) – 2x + 12
3x2 – 36x + 108 – 2x + 12
3x2 – 38x + 120
Graph each relation and its inverse.
5. y =
x3 1
 x 1
3
3
f-1(x) =3(x-1)= 3x - 3
Algebra Trigonometry Chapter 9
6. y =
1 2 1
x  ( x  0) 2
2
2
x
y
-2
2
0
0
2
2
4
8
f-1(x) =  2x
Page 171
Find the inverse of each function. Is the inverse a function?
7.
y = 3x2  2
x  3y2  2
x  2  3y
2
x2
 y2
3

x2
y
3

x2 3
y
3 3

3x  6
 y  f 1 ( x )
3
8. y = (x + 4)2  4
x  ( y  4)2  4
x  4  ( y  4) 2
 x4  y4
4  x  4  y  f 1 ( x)
No, the inverse is a sideways parabola,
so it is not a function
For each function f, find f–1 and the domain and range of f and f–1. Determine whether f–1 is a
function.
9.
1
f(x) =  x + 2 {both will be lines}
5
10. ƒ(x) = x2  2 {vertical parabola}
Domain and Range: All real numbers
Domain: All real numbers
Inverse: f-1(x)= -5 x +10
Range: [2, )
Domain and Range: All real numbers
Inverse: f 1 ( x)   x  2
Domain: [2, )
Range: All real numbers
11.
ƒ(x) = x2 + 4
12. f(x) =
x 1
{vertical parabola: Domain: all real numbers}
Domain: [1, )
Range: [4, )
Range: [0, )
Inverse: f 1 ( x)   x  4
Inverse: f 1 ( x)  x 2  1 where x> 0
Domain: [4, )
Domain: [0, )
Range: All real numbers
Range: [1, )
Algebra Trigonometry Chapter 9
Page 172
Find the inverse of each relation. Graph the given relation and its inverse.
13.
x
-3
-2
-1
0
f-1(x)
-2
-1
0
1
Let f(x) = 2x + 5. Find each value.
14. (f ◦ f1)(3)
f-1(x) =
 35
f

 2 
2( ) + 5
x 5
2
Since we are taking the composition of the function, we should have
known that the answer will be the original input (=3).
2  1  5
2  5
3
Determine whether each pair of functions are inverses.
15.
f ( x) 
x 1
2
Show: f( h(x) ) = x
h( x )  2 x  1
Algebra Trigonometry Chapter 9
f  2 x  1
(
) -1
2
(2x+1) -1
2
2x +1 -1
2
2x
x
x
AND
h( f(x) ) = x
 x 1 
h

 2 
2
 1
 x 1 
2
 1
 2 
2  x 1 

 1
1 2 
2 ( x  1)
1
2
x 11  x
Page 173
16. The equation ƒ(x) = 198,900x + 635,600 can be used to model the number of utility trucks
under 6000 pounds that are sold each year in the U.S. with x = 0 representing the year 1992.
Find the inverse of the function. Use the inverse to estimate in which year the number of
utility trucks under 6000 pounds sold in the U.S. will be 4,000,000.
Original function: f(x) = # utility cars sold.
G(x) = x  635, 600
198,900
Algebra Trigonometry Chapter 9
x is number of year after 1992
Switch domain and range…. Input, x, = # sold, g(x) = year
4, 000, 000  635, 600
 16.9
198,900
Add 17 years to 1992 to get 2009
g ( x) 
Page 174
Algebra Trigonometry Chapter 9
Page 175
Chapter 9
Notes
Algebra Trigonometry Chapter 9
Page 176
Using the graphing calculator…. Let’s talk restrictions.
1) Graph: y =
x4
x2
a) sketch:
b) Where are the asymptotes?
2) Graph: y =
 x  4  ( x  3)
 x  2  ( x  4)
a) sketch:
b) Where are the asymptotes?
c) Are there any other points where the graph is undefined?
This is called a: ___________________ __________________
Algebra Trigonometry Chapter 9
Page 177
State the Restrictions for the variable(s):
( What makes the denominator =0, at any time in the problem…..)
3)
( x  7)
5 x( x  8)
x  0, x  8
5)
( x  7)( x  4)
5 x 2 ( x  1)( x  4)
4)
x8
6)
x  0, x  1, x  4
7)
( x  2)( x  4)
6 x( x  2)( x  4)
x  0, x  2, x  4
Algebra Trigonometry Chapter 9
( x  7)
5( x  8)
( x  2)
5
None
8)
5x2 y7
2 x6 y
y  0, x  0
Page 178
Algebra Trigonometry Chapter 9
Page 179
9.4-9.6 Notes
Warm-up: # 1 – 6.
Factor: { These should be all review for you…. }
1) x2 – 4
3) y3 – 27
2) x2 + 4x + 4
(y-3)(y2 +3y + 9)
(x+2)(x-2)
(x+2)(x+2)
4) y2 – 9
5) z2 – 12z + 27
(y-3)(y+3)
(z - 9 )(z – 3 )
6) 4z2 – 35z – 9
(4z +1 )(z – 9)
Simplify each fractional expression. { Hint use warm-up}
7)
( x  2) ( x  2)
x2  4

x 2  4 x  4 ( x  2) ( x  2)
8)
( x  2)
( x  2)
( y 2  3 y  9)
( y  3)
{you go… we go… can’t cancel!!!}
9)
z 2  12 z  27
4 z 2  35 z  9
( z  3) ( z  9)
(4 z  1) ( z  9)
( z  3)
(4 z  1)
Algebra Trigonometry Chapter 9
2
y 3  27 ( y  3) ( y  3 y  9)

y2  9
( y  3) ( y  3)
10)
3 x 2  3 xy  7 x  7 y
9 x 2  49
3 x( x  y )  7( x  y )
(3 x  7)(3 x  7)
( x  y ) (3 x  7)
(3 x  7) (3 x  7)
( x  y)
(3 x  7)
Page 180
Practice 9-4
Rational Expressions
Simplify each rational expression. State any restrictions on the variable.
1.
2.
3.
2
4.
x x
2
x  2x
x ( x  1)
x ( x  2)
x 1
x2
2
x  0, x  2
Algebra Trigonometry Chapter 9
3y  3
4x  6
2x  3
2 (2 x  3)
20  40 x
20 x
20 (1  2 x)
20 x
1 2x x  0
x
y2 1
3 ( y 2  1)
2x  3
2
5.
y2 1
3
x
2
3x  6
5 x  10
3 ( x  2)
3
6.
x  1
x 2  3 x  18
x 2  36
( x  3) ( x  6)
5 ( x  2)
( x  6) ( x  6)
3
5
x3
x6
x  2
x  6
Page 181
7.
x  13 x  40
2
x  2 x  35
( x  5)( x  8)
( x  7)( x  5)
x8
x7
8.
x 2  10 x  21
3( x 2  4)
( x  3)( x  2)
4( x 2  9)
( x  7)( x  3)
3 ( x  2) ( x  2)
4( x  3) ( x  3)
( x  3) ( x  2)
( x  7) ( x  3)
3( x  2) 3 x  6

x 3
x 3
4( x  3) 4 x  12

x7
x7
x  7, x  3
x  3, x  2
10.
 x 2  11x  5
3 x  17 x  10
(2 x  1) ( x  5)
2
(3 x  2) ( x  5)
2x 1
3x  2
2
x
, x  5
3
Algebra Trigonometry Chapter 9
4 x 2  36
x2  x  6
2
x  7, x  5
9.
3 x 2  12
11.
6 x2  5x  6
12.
3x  5 x  2
(3x  2) (2 x  3)
2
(3x  2) ( x  1)
{look for an 'easy' factor 1st}
{I did the denominator 1st,
then used it to help with the
7 x  28
x 2  16
7 ( x  4)
( x  4) ( x  4)
7
x4
x  4
numerator}
2x  3
x 1
2
x  ,x 1
3
Page 182
Section 9.4 Multiplying and Dividing Rational Expressions
Multiplication:
1. Factor numerators and denominators completely (if possible).
2. Cancel out any factors that appear in both the numerator and denominator.
3. Multiply straight across leaving everything in factored form.
4. Rewrite the simplified final answer.
Multiply:
6(x 2  16)
x 3
6 x 2  96
( x  3)(6 x 2  96)

{Can also write as a rational expression:
}
2x  8
x2  9
(2 x  8)( x 2  9)
2( x  4)
(x-3)(x+3)
3
( x  3) (6) ( x  4) ( x  4)
2 ( x  4) ( x  3) ( x  3)

3( x  4) 3x  12

x3
x3
Now you try: Simplify each, showing all work. Don’t forget your restrictions for the variables.
1)
3
3
7
16
x5
x6
y7
16 x
5 y9
x y

80 xy 2 5( 80 5 ) x y11 4
y
x5
25 y 4
x  0,y  0
2)
4 x( x  1)
(x-2)(x+3)
4 x2  4 x
x2  2 x  3
(x+3)(x-1)
x2  x  6
12 x 2
4 x ( x  1) ( x  2) ( x  3)
( x  3) ( x  1) (12 x 2 x )
x2
3x
Algebra Trigonometry Chapter 9
x  -3,1,0
Page 183
Division:
1. Change to a multiplication problem by making it the first fraction times the
reciprocal of the second fraction.
2. Factor numerators and denominators completely, if possible.
3. Cancel out any factors that appear in both the numerator and denominator.
5. Multiply straight across leaving everything in factored form.
6. Rewrite the simplified final answer.
Divide:
x 3
3 x 2  27
 2
2x  8
x  5x  4
(x-4)(x-1)
x 3
2x  8
( x  3) ( x  4) ( x  1)
x2  5x  4

2
3 x  27
6 ( x  4) ( x  3) ( x  3)
2(x-4)
3(x-3)(x+3)
x 1
x 1

; x  4,1,3,-3
6( x  3) 6 x  18
Now you try: Simplify each, showing all work.
3)
14c 2 d 35cd 3

9m3 n 2
24mn
2
14 c 2 d
3 2
3 9m n
4)
2x  3 
4 x2  9
x5
8
24 mn 16c 2 dmn

35 5cd 3 15cd 3m3n 2
16c
15d 2 m 2 n
5)
x5
f ( x) 
, g(x)= ( x 2  25). Find f(x)  g(x).
x 5
Algebra Trigonometry Chapter 9
x  5 ( x 2  25)

x 5
1
x5
1
1

x  5 ( x  5)( x  5) ( x  5) 2
Page 184
A rational expression where both the numerator and denominator are rational expressions is
called a COMPLEX FRACTION.
x  2y
x
6y
5
1
x2  4
2
x
x
2
Examples:
2

x
3
6
4
8
x
y
{Hint: Make dividing line longer and put parentheses around numerator and denominator}
To simplify, just rewrite the expression as a division problem and use the rules for division!
Simplify:
x3 y 2 z
a 2b 2
6)
a3 x2 y
b2
x3 y 2 z
a 2b 2
a3 x 2 y
b2
x2  4
2
7)
2 x
8
 x3 y 2 z 
 2 2 
 ab 
 a3 x 2 y 
 2 
 b 
x3 y 2 z
a 2 b2
xy
a5
b2
a3 x 2 y
Algebra Trigonometry Chapter 9
Page 185
More Multiplying and Dividing Rational Expressions
1.
2 x 2  9 x  35 6 x 2  13 x  5
 2
2
2 x  17 x  21 6 x  13 x  6
2 x 2  9 x  35 6 x 2  13 x  6
 2
2
2 x  17 x  21 6 x  13 x  5
(2 x  5) ( x  7) (2 x  3) (3 x  2)
( x  7) (2 x  3) (2 x  5)(3 x  1)
(2 x  5)(3 x  2) 6 x 2  11x  10

(2 x  5)(3 x  1) 6 x 2  13 x  5
2.
4 x 2  49 x 2  8 x  15 4 x 2  6 x  70

x 2  x  12 2 x 2  3x  14 2 x 2  12 x  16
Algebra Trigonometry Chapter 9
Page 186
Section 9.5: Addition and Subtraction of Rational Expressions
Parallel Problems, Part I
4 2
 
7 7
7
x


x2 x2
15 3
 
16 16
x
3x


x 1 x 1
Rule: To add (or subtract) two rational
numbers with like denominators, add
(or subtract) their numerators and place
the result over the denominator.
Rule: To add (or subtract) two rational
expressions with like denominators, add
(or subtract) their numerators and place
the result over the denominator.
Parallel Problems, Part II
4 7
 
18 12
3y2
x


2 x 10 y
6 2
 
7 15
3
4


x  5 x 1
Rule: To add or subtract rational numbers
with unlike denominators, find the least
common denominator (LCD). Then rewrite
each number as an equivalent rational
number using the LCD and proceed as in
the case with like denominators.
Rule: To add/subtract rational expressions
with unlike denominators, find the least
common denominator (LCD). Then rewrite
each expression as an equivalent rational
expression using the LCD and proceed as in
the case with like denominators.
Algebra Trigonometry Chapter 9
Page 187
Examples: Perform the indicated operation and simplify.
1) 5(3x)
x

2

3x 2
15x
x(3 x)

2
15 x  2

x(3 x)
3x 2
2)
3
4
3x  3
4 x  20



x  5 x  1 ( x  5)( x  1) ( x  5)( x  1)
 x  17
( x  5)( x  1)
4)
4x

2 x
1(2  x)
1( x  2)
x (3x) x(3x)
3)
x
7
x3
x
7

1 x3
x( x  3)
7

x3
x3
4 x
3

x2 x2
x 2  3x  7
x3
4 x  3
x2
Algebra Trigonometry Chapter 9
3
x2
Page 188
9.5 Notes and Practice – Denominator is a Binomial or Trinomial
Simplify:
1)
1
4x

3 x  21x  30 3 x  15
3( x 2  7 x  10) 3(x+5)
2
3(x+5)(x+2)
1
4 x( x  2)

3(x+5)( x  2) 3(x+5)( x  2)
Steps:
1) Factor denominators
a) GCF first
b) Factor quadratic
2) Find Least common denominator
a) Find common factor and write it
b) Write remaining factors
3) Change numerators according to the ‘new’
denominator.
4) Write as one fraction combining like terms
4x2  8x  1
3(x+5)( x  2)
2)
2x
3

Common Denominator: (x+1) 4 (x-3)
x  2x  3 4x  4
( x  3)( x  1) 4(x+1)
2
2 x(4)
3( x  3)

4( x  1)( x  3) 4( x  1)( x  3)
8 x  3x  9
5x  9

4( x  1)( x  3) 4( x  1)( x  3)
3)
1
3x

x  4 x  12 4 x  8
( x  6)( x  2) 4(x+2)
2
4(1)
3 x( x  6)

4( x  6)( x  2) 4( x  6)( x  2)
4  3 x 2  18 x
3 x 2  18 x  4

4( x  6)( x  2) 4( x  6)( x  2)
Algebra Trigonometry Chapter 9
Page 189
4)
1 1
x y


2 1
 y  x


Before “Multiplying by the reciprocal”, we need
to have ONE fraction divided by ONE fraction.
1 1
x y


 2 1 
y x 


 yx
1 y 1x

 xy 
xy yx


 
2 x 1y  2 x  y 

 xy 
xy xy


y  x xy
yx

xy 2 x  y 2 x  y
5)
2
3 x

3 x  36 x  105 6 x  30
3( x 2  12 x  35) 6(x+5)
2
3(x+7)(x+5)
2 3(x+5)
2(2)
3 x( x  7)
+
3( x  5)(2)( x  7) 3( x  5)(2)( x  7)
4  3 x 2  21x
3 x 2  21x  4

3( x  5)(2)( x  7) 6( x  5)( x  7)
6)
x
1(2 x  1)
 2
3x  9 x  6 3x  3x  6
3( x 2  3 x  2) 3(x 2 +x - 2)
2
3( x  2)( x  1) 3(x+2)(x-1)
x( x  2)
(2 x  1)( x  2)
+
3( x  1)( x  2)( x  2) 3( x  1)( x  2)( x  2)
x2  2x  2x2  4x  x  2
 x2  5x  2

3( x  1)( x  2)( x  2)
3( x  1)( x  2)( x  2)
Algebra Trigonometry Chapter 9
Page 190
7)
1
x 
y
8)
3
1
1
 
x
 y
 
1

1 1 1

x y xy
1
2y
3
3
 3
 
 
 
1
1 
1
  

1   2 y 1   2 y  1 
1  2 y   2 y  2 y   2 y 

 
 

3 2y

1 2 y 1
9)
6y
2 y 1
2 
 x2



x 1 
 x
1 
 3



 x 1 x  1 
 ( x  1)  x  2 
2x 



 ( x  1)( x  1) x( x  1) 
 x2  x  2  2x 


x( x  1)



 3( x  1)



3
x

3

x

1
1( x  1)



 ( x  1)( x  1) 


  x  1 ( x  1)  x  1 ( x  1) 
x 2  3 x  2 ( x  1) ( x  1)
2x  4
x ( x  1)
( x  1)( x 2  3 x  2)
2 x ( x  2)
Algebra Trigonometry Chapter 9
Page 191
More 9.5 Practice
Simplify.
1.
3.
x2
x2

5
5
12
xy
5.
3

6
2
9
xy

3
5
5 x y 10 xy
2
Algebra Trigonometry Chapter 9
2.
x2  2 x

12
6
4.
6y  4
6.
3
y 5
2
3 3
8x y


3y 1
y2  5
1
4 xy
Page 192
7.
9.
11.
4
x  25
2

6
x  6x  5
2
xy  y
y

x2 x2
3
x  x6
2

8.
2
x  6x  5
Algebra Trigonometry Chapter 9
2
7x y
10.
2
3
12.

4
21xy 2
x2
x  4x  4
2
1

2
x2

1
6 x  11x  3 8x  18
2
2
Page 193
13.
3
1
x 5
2
15. 1  2 x  7
3x  1
17.
4c
4c

c 3 c 3
Algebra Trigonometry Chapter 9
1
14.
5
16.
2a
3a

a2 a2
18.
x  5x  6
2
f  1 f 1

fgh
fgh
Page 194
19.
2t 2t

t 5 t 5
21.
x y y

x y x
23.
2
x
6
4
x
1
Algebra Trigonometry Chapter 9
20.
22.
24.
4r
4r

r2 r2
2
x
3
y
1
x2
1
2
x
Page 195
9.6: Solving Rational Equations:
Warm-up:
1-3: Simplify and include restrictions for each expression:
1)
2) x  3 x  10
2
5
x2  4
x3  25 x
( x  5) ( x  2)
5
5x  5x
3x 2  3
5 x ( x  1)
2
5x

3( x 2  1) 3( x  1)
( x  2) ( x  2) x( x 2  25)
3 ( x 1) ( x 1)
5
x  0, 2, 2, 5, 5
x( x  2)( x  5)
x(x-5) (x+5)
x  1, 1
3)
10 x  10
20 x  5
10( x  1) 12 x 2  17 x  5


3x 2  2 x  5 12 x 2  17 x  5 (3x  5)( x  1)
20 x  5
10 ( x  1)
(3x  5) (4 x  1)
(3x  5) ( x  1)
5 (4 x  1)
2
2
x   5 ,1, 1
3
4
#4-6: Solve each equation
4) Solve: 2x = 5x + 6
5) Solve: x2 = 16
-3x=6
x=4,-4
X=-2
6) Solve: 2x2 - x – 6 = 0
(2x + 3 ) (x – 2 ) = 0
Algebra Trigonometry Chapter 9
3
x   ,2
2
Page 196
Solve each equation. Check each solution.
1.
1
x

x
9
2.
Method 1: Cross multiplication:
1
2

x x3
Method 1: Cross multiplication:
(Means-extremes theorem)
2x = x + 3
x=3
x2 = 9
check :
x= + 3
Check :
1
3

3
9
True!
1
2

3 33
1 2

TRUE
3 6
Method2: Common Denominator of 9x:
Method 2: Common Denominator of x(x+3)
91
xx

9x
9x
1( x  3)

x( x  3)
9x
Restrictions: x  0
2
 9 x
x  3
Since 3 or -3 are not restrictions, this works!
Algebra Trigonometry Chapter 9
x  3  2x
3 x
2( x)
 x  3 ( x)
restrictions: x  0 or -3
Since 3 is not a restriction, it works!
Page 197
6)
8)
3y 1
y
 
5 2 10
6y 5
y


10 10 10
6y  5  y
5  5 y
1  y
2
5
6

 2
x 3 3 x x 9
-1(x-3)
2
5
6

 2
x  3 ( x  3) x  9
2  x  3
5  x  3
6

 2
 x  3 x  3 ( x  3)  x  3 x  9
2 x  6  5 x  15  6 x  3,-3
-3x-21=6
-3x=27
x= -9
There are two methods to solve rational equations:
Cross-multiply, then solve. Check solution against restrictions, or
plug it back into the original equation.
1) _________________________________________________________________________________________________________
2)
Get common denominators on both sides of the equation, write down
restrictions, then drop denominators, setting the numerators = to
each other. (Use when have a common factors in denominators)
_________________________________________________________________________________________________________
Algebra Trigonometry Chapter 9
Page 198
3.
3x 5 x  1

4
3
3(3x)  4(5 x  1)
9 x  20 x  4
4.
11x  4
4
x
11
9.
x3
x  3x  4
( x  4)( x  1)
2
x2

x  16
(x-4)(x+4)
2
( x  3)( x  4)
( x  2)( x  1)

( x  4)( x  1)( x  4) ( x  4)( x  1)( x  4)
Before looking at only numerators,
write down restrictions: x  -4, 4, 1
11.
3
2

1 x 1 x
3(1  x)  2(1  x)
3  3x  2  2 x
3  5x  2
5 x  1
1
x
5
7
x2  5x
x( x  5)
2
3

x 2 x  10
2(x-5)

7(2)
2( x  5) x
4 ( x 5 )

2(2( x  5))
x

3x
2( x  5) x
14  4 x  20  3 x
x  x  12  x  x  2
 x  12  x  2
2
2
12  2 x  2
10  2 x
x  5
Algebra Trigonometry Chapter 9
4 x  6  3x
6   x
x6
Page 199
5.
2x  3 2x  5

42 2
62 3
7.
10
x 2  3x
x

x2
x2
x( x  2) 10
x 2  3x


x2
x2
x2
x2
3(2 x  3)
2(2 x  5)

223
223
6 x  9  4 x  10
2 x  1
1
x
2
x 2  2 x  10  x 2  3x
5 x  10
x  2 (because of restriction)
NO SOLUTION!!
10.
2
x
 2
6 x  2 3x  11
1
R:x 
3
2(3x 2  11)  x(6 x  2)
6 x 2  22  6 x 2  2 x
22  2 x
12.
2y 2 y 1
  
5 6 2 6
2 y ( 2 3) 2(5)
y (3 5) 1(5)



235
235
235 235
12 y  10  15 y  5
10  3 y  5
15  3 y
y5
x  11
Algebra Trigonometry Chapter 9
Page 200
13.
2 3x  1 5


3
6
2
32

32

14.
32
2 2 3x  1 5 3


32 32
32
4  3 x  1  15
3  3 x  15
2y
8

y 5 y 5
4( y  5) 2 y
8


y 5
y 5 y 5
4 y  20  2 y  8
6 y  20  8
6 y  28
14
y
3
4
3 x  12
x4
15.
4
2
16


x  3 x  1 x2  2 x  3
(x-3)(x+1)
4( x  1)
2( x  3)
16


\
( x  3)( x  1) ( x  3)( x  1) ( x  3)( x  1)
4 x  4  2 x  6  16
4 x  4  2 x  10
2x  6
x3
Algebra Trigonometry Chapter 9
Page 201
9.4-9.6 Review
Medium Problems.
Simplify each rational expression. State any restrictions on the variable.
1.
2
2
9 x
5x  2 x
x 3
5 x3  17 x 2  6 x

x
-1 1 (x-3) (x+3)
1( x 2  9)
x (5 x  17 x  6)
2
1

x (5 x  2)
x  31
2
,3, 3, 0
5
 1
(5x+2) (x+3)
2.
( x  1)(2 x  4) ( x  1)( x  4)

x4
2x  4
( x  1) (2 x  4) ( x  1) ( x  4)

x4
2x  4
x  2, 4
( x  1)( x  1) or x 2  1
3.
( x  3) ( x  4) ( x  3) ( x  1)

( x  1) ( x  3)
x4
x3
Simplify each rational expression. State any restrictions on the variable.
3y  3
18

Algebra
9
6 y Trigonometry
12 5 y  Chapter
5
3( y  1) 5 y  5
6( y  2)
18
Page 202
4.
y  2, 1
5.
y 2  49
( y  7) 2

5 y  35
y  7, 0, 7
y2  7 y
( y  7) ( y  7) y ( y  7)
( y  7)
Simplify.
6.
3
5 ( y  7)
2

2

y
5
No common factors in the denominator!
x  x6
x  6x  5
( x  3)( x  2) ( x  5)( x  1)
3( x  5)( x  1)
2( x  3)( x  2)

( x  3)( x  2)( x  5)( x  1) ( x  5)( x  1)( x  3)( x  2)
2
2
3( x 2  6 x  5)
2( x 2  x  6)

( x  3)( x  2)( x  5)( x  1) ( x  5)( x  1)( x  3)( x  2)
3 x 2  18 x  15  2 x 2  2 x  12
5 x 2  16 x  3

( x  5)( x  1)( x  3)( x  2)
( x  5)( x  1)( x  3)( x  2)
(5 x  1)( x  3)
y  5, 1,3, 2
( x  5)( x  1)( x  3)( x  2)
Algebra Trigonometry Chapter 9
Page 203
Simplify.
7.
1
1

6 x  11x  3
2
(2 x  3)(3 x  1)
8 x  18
2
2(4 x 2  9)
2(2 x  3)(2 x  3)
1
1

(2 x  3)(3 x  1)
2(2 x  3)(2 x  3)
2(2 x  3)
(3 x  1)

(2 x  3)(3 x  1)2(2 x  3)
2(2 x  3)(2 x  3)(3 x  1)
y
3 1 3
, ,
2 3 2
4 x  6  3x  1
7x  5

(2 x  3)(3 x  1)2(2 x  3) (2 x  3)(3 x  1)2(2 x  3)
8.
3
x 2  3 x  10
( x  5)( x  2)

1
x2  6 x  5
( x  5)( x  1)
3( x  1)
1( x  2)

( x  5)( x  2)( x  1) ( x  5)( x  1)( x  2)
3x  3  x  2
4x 1

( x  5)( x  1)( x  2) ( x  5)( x  1)( x  2)
9.
10.
1
4 1

4  4 4 
3 10 3
2

5 5 5
1
3
 
4
7
 
5

y  5, 1, 2
3 5 15

4 7 28
3
4x
3
4 x( x  9) 3  4 x 2  36 x 4 x 2  36 x  3





x 9 1
x  9 1( x  9)
x 9
x 9
x9
Algebra Trigonometry Chapter 9
Page 204
FUN Problems
Simplify.
1.
2
x  0,
3
2
 x2
x 2



x
x2
x2
x  x x   x   x2


6
4x 6
4x  6
4 x  6 2(2 x  3)
 4x  6 
x
4



x
x
x
x 

2(2 x  3)
1
2.
 1 
1
1
x  0, 2


x
x
x2  x2   x2  1

1
2x 1
x  2 ( 2 x  1) ( x  2)(2 x  1)
 2x 1 
2



x
x
x
 x 
3.
5
2
1 5
  
2x 3
x 6
23 x

23 x

23 x

x0
23 x
15
6x
6
5



23 x 23 x
23 x 23 x
15  6 x  11
6 x  4
x
4.
2
3
3
x 1
3( x  1)
x 1  3

5
x 1
5
5( x  1)
x 1
Algebra Trigonometry Chapter 9
x  1 or -1
Page 205
Simplify.
5.
2y
y

y 2  4 y  12
y 2  10 y  24
( y  6)( y  2) ( y  6)( y  4)
2 y ( y  4)
y ( y  2)

( y  6)( y  2)( y  4) ( y  6)( y  2)( y  4)
2 y2  8 y  y2  2 y
3y2  6 y

( y  6)( y  2)( y  4) ( y  6)( y  2)( y  4)
3 y ( y  2)
3y  6 y

( y  6)( y  2)( y  4) ( y  6)( y  4)( y  2)
6. 2
x
6
1
y
y  6,-2,4
2 6x
2  6x

2  6 x y 2 y  6 xy
x 
x
 x


1
1
x
1
x
y
y
x  0, y  0
Simplify each rational expression. State any restrictions on the variable.
7. 6 x 2  32 x  10 3 x 2  11x  4

3 x 2  15 x
2 x 2  32
2 ( x  5) (3x  1) 2( x  4) ( x  4)
3x ( x  5)
8.
( 3 x  1) ( x  4)
x2  6 x

x2  2x  8
x6
3x 2  6 x  24
x ( x  6)
( x  4) ( x  2)
4
1
x  0,5, , 4, 4
3x
3

3( x  2 x  8)
2
( x  6)

x
3
x  -4,2,-6
3 ( x  4) ( x  2)
9.
7 x4
24 y 5
7
x3

21x
12 y 4
x4
24 2 y 5 y
12 y 4
x3

213 x 6 y
Algebra Trigonometry Chapter 9
x0 y0
Page 206