Chapter 7: Rational Equations, Expressions, and Functions

1
Chapter 7Rational Equations, Expressions, and
Functions

• 7.1 Multiplication and Division
• 7.2 Addition and Subtraction
• 7.3 Division of Polynomials
• 7.4 Complex Rational Expressions
• 7.5 Solving Rational Equations
• 7.6 Applications and Proportions
• 7.7 Formulas and Applications
• 7.8 Variation and Applications

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7.1
Multiplication Division of Rational Expressions
3
Simplify a Rational Expression

• Factor each polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

1
7.1
4
How about Opposite Factors?

• Factor each polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

-1
7.1
5
How about another one?

• Factor each Polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

1
7.1
6
Okay, one more to be sure!

• Factor each Polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

-1
7.1
7
Oh, dont forget ones like this!

• Factor each Polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

1
Double
7.1
8
Multiply Straight Across

• Factor each Polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

7.1
9
Polynomials TOO!

• Factor each Polynomial
• Like Factors reduce to 1
• Opposite Factors reduce to -1

7.1
10
Divide Multiply by the Reciprocal
7.1
11
Polynomials too!
1
-1
1
7.1
12
7.2
Addition Subtraction of Rational Expressions
7.2
13
Addition Subtraction
Higher Terms
LCD
or -
Reduce
7.2
14
Addition Subtraction Ex 1
Higher Terms
Reduce
or -
LCD
7.2
15
Addition Subtraction Ex 2
Higher Terms
Reduce
or -
LCD
7.2
16
Addition Subtraction Ex 3
Higher Terms
Reduce
or -
LCD
7.2
17
Addition Subtraction Ex 4
Higher Terms
Reduce
or -
LCD
7.2
18
7.3
Division of Polynomials
7.3
19
Short Division
Distribute the monomial divisor
Simplify each term.
7.3
20
Long Division
-
17
3
2
r

-
5
Dividend
Division Symbol
Divisor
Equal Symbol
Quotient
Remainder
7.3
21
Long Division
2
3
-
5
5
17
Divisor
Division Symbol
Dividend
Quotient
Fractional Remainder
7.3
22
Long Division
3
3 4 4
12 4 1 7
3 6
5 7
4 8
9
7.3
23
Try Polynomials?
7.3
X-11
24
Just a little Harder
7.3
4x 7
25
Tricky?
2x2 9x 10
7.3
26
Trinomial Divisor?
x2 3x 4 (6x6)/(x2 x 1)
7.3
27
Synthetic Division
7.3
28
Synthetic Division Ex 1
3x 2
7.3
29
Synthetic Division Ex 2
7.3
30
Synthetic Division Ex 3
7.3
31
7.4
Complex Rational Expressions
7.4
32
Use the LCM or LCD
7.4
33
LCD is EZ to use!
7.4
34
Work Smart!
7.4
35
Do this in your HEAD!
7.4
36
You CAN do this!
7.4
37
You Have the Power!
7.4
38
Careful with the signs!
7.4
39
Calculus Anyone?
7.4
40
7.5
Solving Rational Equations
7.5
41
Strategy
Rational Equations
LCD
Domain
Cancel Denominators
Solve
Check your answer
7.5
42
Start with an EZ one?
7.5
25/14
43
A little harder?
7.5
1,4
44
Whats the LCD?
7.5
7/2
45
7.6
Applications and Proportions
Proportions and Work Problems
7.6
46
Work it out!
Distance rate ? time
Part of job completed

rate of work ? time worked
7.6
47
Think about it!
Maria can paint a room in 8 hours
Her rate is 1/8th of the room per hour
If she works for 6 hours, she can finish 6 x 1/8
or 3/4ths of the room
7.6
48
Teamwork!
Renee can build a wall in 10 hours. Sean can
build the wall in 15 hours. How long will it
take to build the wall if they work together?
7.6
6 hours
49
Higher Math?
Jacob requires 8 hours to shingle a roof by
himself. Jacob and Trisha work on a roof for 2
hours, then Jacob leaves for another job. Trisha
takes 10 more hours to finish the job. How long
would it take Trisha to do the job working alone?
7.6
16 hours
50
Dont Dive In Yet!
A large pump can fill a pool in 6 hours while a
small pump would take 15 hours. How long will it
take if both pumps work together?
4 hours 17 minutes
7.6
51
Oh no, a leak!
A large pipe can fill a tank in 10 hours while a
small pipe could fill it in 14 hours.
Inadvertently, a drain is left open which can
empty the tank in 35 hours. If the tank starts
out empty and both inlets and the drain are
working, when will the tank be full? (just
before it starts to overflow)
7.6
7 hours
52
A RATIO
is the quotient of two quantities that have the
same units.
7.6
53
A Space Shuttle Ratio
On average, the shuttle loses 50 of its 24,000
heat protection tiles during each trip.
7.6
http//www.ed.arizona.edu/ward/Shuttle/shuttle.htm
l
54
A RATE
is the quotient of two quantities that have
different units.
7.6
55
A UNIT RATE
is a rate with a denominator of 1.
7.6
56
Starbucks? Yum!
A 16 ounce White Chocolate Mocha has 470 calories.
This is a UNIT rate
7.6
http//www.starbucks.com
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A PROPORTION
is the equality of two ratios or rates.
7.6
58
Parts of a PROPORTION
a is to b as c is to d ab cd
means
extremes
7.6
59
Solve This?
24
3x - 6

7.6
60
What if ?
3x 6
-10x 20

7.6
61
Similar Figures?
4x
42

7.6
62
OKtry this one!

7.6
63
Trouble Parking?
A theatre that can seat 1500 people has a parking
lot with 600 spaces. At the same rate, how many
parking spaces should a new theatre with 2200
seats create?
7.6
64
Time to Vote?
A survey showed that 5 out of every 8 voters
would vote in a special election.At this rate,
how many people would be expected to vote in a
city of 180,000 voters ?
7.6
65
7.7
Formulas and Applications
7.7
66
Literally, Solve It!
7.7
67
One More!
7.7
68
7.8
Variation
7.8
69
Direct Variation
The harder he hits, the higher it goes!
y varies directly as x y kx k is a constant
7.8
70
Inverse Variation
As the elevation increases, the oxygen
concentration decreases!
y varies inversely as x
7.8
71
Types of Variation
y varies directly as x y kx
y varies inversely as x
z varies jointly as x and y z kxy
k is the constant of proportionality or the
constant of variation
7.8
combined variation
72
DIRECT multiplication
I varies directly as h. If I 256 when h
8, determine I when h 36.
1152
7.8
73
Direct again?
s varies directly as the square of v. If v 30
when s 63, determine s when v 55.
211.75
7.8
74
Inverse Divide
t is inversely proportional to r. If t5 when r
55, determine t when r 65.
4.23077
7.8
75
Force Yourself to Try this one!
The repulsive force, f, between the north poles
of two magnets is inversely proportional to the
square of the distance, d, between them. If
the repulsive force is 20 lbs. when the distance
is 4 inches, find the repulsive force when the
distance is 2 inches.
7.8
80 lbs.
76
Ohms Law
If the voltage, V, in an electric circuit is held constant, the current, I, is inversely proportional to the resistance R. If the current is 40 amperes when the resistance is 270 ohms, find the current when the resistance is 150 ohms.
7.8
10800 k 72 amps
77
Assembly Line
The number of cars manufactured on an assembly line varies jointly as the number of workers and the time they work. 200 workers can produce 60 cars in 2 hours. Determine how many cars 240 workers should be able to produce in 3 hours.
7.8
kv 3/20 108 cars
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Anthropology
The cephalic index, C, varies directly as the skull width, w, and inversely as the length of the skull, n. The cephalic index is 70 for a width of 7 and a length of 10. Find the index for a skull with a width of 6 and a length of 8.
7.8
75
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Thats All For Now!