Download Wetzel - Wsimg.com

Wetzel
Math 102
7.1 – Simplifying Rational Expressions
What is a rational expression?
The quotient of two polynomials. (a fraction)
How to Simplify a Rational Expressions Steps to Simplify:
1.) Completely factor the numerator AND denominator
2.) Cancel any common factors between the numerator and denominator.
Ex.: Simplify
1.)
10 x 5
25 x 3
2.)
( x  9)( x  3)
( x  3)( x  3)
3.)
x2
2
x  4x  4
4.)
x 2  7 x  12
x 2  5 x  24
5.)
3x 2  6 x  105
x 2  11x  28
6.)
x4
4x  9x  9
2
Determine where the following functions are undefined.
7.) f ( x) 
x
x5
9.) r ( x) 
x2  9
3
8.) g ( x) 
x 2  5x  4
x 3  12 x 2  20 x
Wetzel
Math 102
7.2 – Multiplying and Dividing Rational Expressions
Multiplying Rationals –
Steps:
1.) Completely factor all numerators and denominators
2.) Cross-Cancel
3.) Multiply the rational expressions leaving the answer in factored form.
Ex.:
1.)
x 2  144
x 2  7 x  18

x 2  16 x  63 x 2  8 x  48
2.)
x 2  6 x  55 x 2  4 x  12
 2
36  x 2
x  15 x  50
Dividing Rational Expressions Steps:
1.) Change the division sign to multiplication by reciprocal of the rational that
follows the division sign.
2.) Now that it is a multiplication problem, follow the Multiplication steps.
Ex.:
x 2  2 x  80 x 2  17 x  70
3.) 2
 2
x  3x  18
x  4x  3
x 2  6 x  27 x 2  18 x  81
4.)
 2
x 2  3x  4
x  8x  7
Wetzel
Math 102
7.3 – Adding and Subtracting Rational Expressions with Like Denominators
Addition / Subtraction with like denominators –
Steps:
1.) Combine like terms in the numerators & write in standard form.
3.) Simplify the remaining rational expression.
Ex.:
1.)
x  14
3 x  10
 2
x  7x  6 x  7x  6
2
x 2  20 x  45
3x  27
3.) 2
 2
x  15 x  56 x  15 x  56
2.)
6x  1
3 x  22
 2
x  16 x  63 x  16 x  63
2
x 2  3x  7 x 2  5 x  9 x 2  6 x  14
4.) 2


x  2x  8 x 2  2x  8 x 2  2x  8
Wetzel
Math 102
7.4 – Adding and Subtracting Rational Expressions with Unlike Denominators
Addition / Subtraction with unlike denominators –
Steps:
1.) Find the Least Common Denominator (LCD). First, factor all of the
denominators. The LCD is then the product of each different or distinct factor from each
denominator.
2.) Rewrite each rational to have this LCD.
3.) Add / Subtract the rational expressions.
4.) Simplify.
Ex.:
1.)
3 11

8 6
2.)
7
2

x 5 x 8
3.)
2
4
 2
x  x  6 x  4 x  21
4.)
x5
4
 2
x  2 x  24 x  16 x  60
2
2
5.)
x4
14
 2
x  20 x  96 x  6 x  16
6.)
2
x3
 2
x  4 x  12 x  36
2
2
Wetzel
Math 102
7.5 – Complex Fractions
Complex Fractions –
A fraction within a fraction
3 10

x x2
2
1
x
1
Not considered simplified!!
To Simplify:
1.) Multiply the num. and den. by the LCD.
2.) It should reduce to ONE rational/fraction.
3.) Simplify the remaining rational.
Ex.:
8
x
1.)
20
25 
x
18
x
2.)
1 42

1
x x2
x 2  19 x  88
2
3.) x2  4 x  5
x  5 x  66
x 2  x  30
10
6

4.) x  4 x  6
3 x  63
x6
10 
3
Wetzel
Math 102
7.6 – Rational Equations
Solving Rational Equations –
Steps :
1.) Find the LCD.
2.) Multiply the equation by the LCD. (This will clear all of the fractions)
3.) Solve the remaining equation.
4.) Check for extraneous roots. (solutions that make a denominator equal to zero)
Ex.:
1.)
6x 9 x 3
  
7 2 4 14
2.)
6 x  13 4 x  15

x 1
x 1
3.)
2
3
7x  8

 2
x x  2 x  2x
4.)
x
x  4 9x  5


x  3 x  3 x2  9
5.)
x
3
36

 2
x  5 x  7 x  2 x  35
6.)
x2
x5
x3
 2
 2
x  13x  40 x  7 x  8 x  4 x  5
2
Wetzel
Math 102
7.7 – Applications
To Solve:
1.) Understand the problem. Organizing the given information by sketching a
diagram, making a table, or just rewriting it and know exactly what the problem is asking
you to find.
2.) Derive a plan to solve the problem. This may not even require any writing.
3.) Carry your plan out. Solve the problem!
4.) Completely answer the question making sure your answer makes
sense!. There may be more than one answer or an answer may not make sense so you
would cancel it out. Include any units in the answer.
Ex.:
Numbers
1.) One positive number is nine larger than another positive number. If the reciprocal of
the smaller number is added to four times the reciprocal of the larger number, the sum is
2
equal to . Find the two numbers.
3
Hourly Jobs
2.) Martha can rake the leaves in her yard in 6 hours. Her brother can do the job in 7
hours. How long will it take them to do the job working together?
Distance
3.) Selma is kayaking in a river that flows downstream at a rate of 1 mph. Selma paddles
5 miles downstream and then turns around and paddles 6 miles upstream, and the trip
takes 3 hours.
a.) How fast can Selma paddle in still water?
b.) Selma is now 1 mile upstream of her starting point. How many minutes will
it take her to paddle back to her starting point?
EXTRA PRACTICE:
1.) One positive number is six less than another positive number. If two times the
reciprocal of the smaller number is added to five times the reciprocal of the larger
number, the sum is 1. Find the two numbers.
2.) A small pipe takes three times longer to fill a tank than a larger pipe takes. If both
pipes are used at the same time, it takes 12 minutes to fill the tank. How long does it take
the smaller pipe alone to fill the tank?
3.) Kim ran 7 miles this morning. After running the first 3 miles, she increased her speed
by 2 miles per hour. If it took her exactly 1 hour to finish her run, find the speed at which
she was running for the last 4 miles.
4.) A company has printed out 500 surveys to be put into preaddressed envelopes and
mailed out. Dusty can fill all of the envelopes in 5 hours, Felipe can do the job in 6
hours, and Grady can do it in nine hours. If all three work together, how long will it take
them to stuff the 500 envelopes?