Intermediate Algebra
Chapter 6 - Gay
•Rational Expressions
Intermediate Algebra 6.1
• Introduction
•to
•Rational Expressions
Definition: Rational Expression
• Can be written as
P( x)
Q( x)
• Where P and Q are polynomials and Q(x) is
not 0.
Determine Domain of rational
function.
Solve the equation Q(x) = 0
• 2. Any solution of that equation
is a restricted value and must be
excluded from the domain of
the function.
• 1.
Graph
• Determine domain, range, intercepts
• Asymptotes
1
f ( x) 
x
Graph
• Determine domain, range, intercepts
• Asymptotes
1
g ( x)  2
x
Calculator Notes:
• [MODE][dot] useful
• Friendly window useful
• Asymptotes sometimes occur that are not
part of the graph.
• Be sure numerator and denominator are
enclosed in parentheses.
Fundamental Principle of
Rational Expressions
ac a

bc b
Simplifying Rational Expressions
to Lowest Terms
• 1. Write the numerator and
denominator in factored form.
• 2. Divide out all common
factors in the numerator and
denominator.
Negative sign rule
p p
p
 

q
q
q
Problem
y  4 (1)  y  4 

4  y  1 4  y 
1 y  4 


 1
y4
Objective:
• Simplify a Rational
Expression.
Denise Levertov – U. S. poet
• “Nothing is ever enough.
Images split the truth in
fractions.”
Robert H. Schuller
• “It takes but one positive
thought when given a chance
to survive and thrive to
overpower an entire army of
negative thoughts.”
Intermediate Algebra 6.1
•Multiplication
•and
•Division
Multiplication of Rational
Expressions
• If a,b,c, and d represent algebraic
expressions, where b and d are not 0.
a c ac

b d bd
Procedure
• 1. Factor each numerator
and each denominator
completely.
• 2. Divide out common
factors.
Definition of Division of
Rational Expressions
• If a,b,c,and d represent algebraic
expressions, where b,c,and d are not 0
a c a d ad
 

b d b c bc
Procedure for Division
• Write down problem
• Invert and multiply
• Reduce
Objective:
•Multiply and divide
rational expressions.
John F. Kennedy – American
President
• “Don’t ask ‘why’, ask
instead, why not.”
Intermediate Algebra 6.2
• Addition
•and
• Subtraction
Objective
• Add and Subtract
• rational expressions with
the same denominator.
Procedure adding rational
expressions with same
denominator
Add or subtract the
numerators
• 2. Keep the same denominator.
• 3. Simplify to lowest terms.
• 1.
Algebraic Definition
a b ab
 
c c
c
a b a b
 
c c
c
LCMLCD
• The LCM – least common
multiple of denominators is
called LCD – least common
denominator.
Objective
• Find the lest common denominator (LCD)
Determine LCM of polynomials
• 1. Factor each polynomial completely
– write the result in exponential form.
• 2. Include in the LCM each factor that
appears in at least one polynomial.
• 3. For each factor, use the largest
exponent that appears on that factor in
any polynomial.
Procedure: Add or subtract rational
expressions with different denominators.
• 1. Find the LCD and write down
• 2. “Build” each rational expression so
the LCD is the denominator.
• 3. Add or subtract the numerators and
keep the LCD as the denominator.
• 4. Simplify
Elementary Example
• LCD = 2 x 3
1 2 13 2 2
 


2 3 23 32
3 4 3 4 7
 

6 6
6
6
Objective
• Add and Subtract
• rational expressions with
unlike denominator.
Martin Luther
• “Even if I knew that
tomorrow the world
would go to pieces, I
would still plant my apple
tree.”
Intermediate Algebra 6.3
•Complex Fractions
Definition: Complex rational
expression
• Is a rational expression
that contains rational
expressions in the
numerator and
denominator.
Procedure 1
• 1. Simplify the numerator and
denominator if needed.
• 2. Rewrite as a horizontal division
problem.
• 3. Invert and multiply
• Note – works best when fraction
over fraction.
Procedure 2
• 1. Multiply the numerator and
denominator of the complex rational
expression by the LCD of the
secondary denominators.
• 2. Simplify
• Note: Best with more complicated
expressions.
• Be careful using parentheses where
needed.
Objective
• Simplify a complex
rational expression.
Paul J. Meyer
• “Enter every activity without
giving mental recognition to the
possibility of defeat.
Concentrate on your strengths,
instead of your weaknesses…on
your powers, instead of your
problems.”
Intermediate Algebra 6.4
•Division
Long division Problems
x  5x  7
x2
2
Long division Problems
x  5x  7
x2
2
Maya Angelou - poet
• “Since time is the one
immaterial object which we
cannot influence – neither
speed up nor slow down, add
to nor diminish – it is an
imponderably valuable gift.”
Intermediate Algebra 6.5
•Equations
•with
•Rational Expressions
Extraneous Solution
• An apparent solution that is a
restricted value.
Procedure to solve equations containing
rational expressions
• 1. Determine and write LCD
• 2. Eliminate the denominators of the
rational expressions by multiplying
both sides of the equation by the LCD.
• 3. Solve the resulting equation
• 4. Check all solutions in original
equation being careful of extraneous
solutions.
Graphical solution
• 1. Set = 0 , graph and look for x intercepts.
• Or
• 2. Graph left and right sides and look for
intersection of both graphs.
• Useful to check for extraneous solutions
and decimal approximations.
Proportions and Cross Products
• If
a c
 where b, d  0
b d
then ad  bc
Thomas Carlyle
• “Ever noble work is at
first impossible.”
Intermediate Algebra 6.6
•Applications
Objective
• Use Problem Solving
methods including charts,
and table to solve problems
with two unknowns
involving rational
expressions.
Problems involving work
• (person’s rate of work) x
(person's time at work) =
amount of the task
completed by that person.
Work problems continued
• (amount completed by
one person) + (amount
completed by the other
person) = whole task
Section 6.7 – Gay
Variation and Problem Solving
•
•
•
•
Direct Variation
Inverse Variation
Joint Variation
Applications
Def: Direct Variation
• The value of y varies
directly with the value of
x if there is a constant k
such that y = kx.
Objective
• Solve Direct Variation
Problems
• Determine constant of
proportionality.
Procedure:Solving Variation Problems
• 1. Write the equation
• Example y = kx
• 2. Substitute the initial values and
find k.
• 3. Substitute for k in the original
equation
• 4. Solve for unknown using new
equation.
Example: Direct Variation
• y varies directly as x. If y =
18 when x = 5, find y when x
=8
• Answer: y = 28.8
Helen Keller – advocate for he
blind
• “Alone we can do so
little, together we can
do so much.”
Definition: Inverse Variation
• A quantity y varies inversely with x if there
is a constant k such that
k
y
x
• y is inversely proportional to x.
• k is called the constant of variation.
Procedure: Solving inverse
variation problems
• 1. Write the equation
• 2. Substitute the initial values
and find k
• 3. Substitute for k in the
equation found in step 1.
• 4. Solve for the unknown.
Joint Variation
• Three variables y,x,z are
in joint variation if y =
kxz where k is a constant.
Leonardo Da Vinci - scientist,
inventor, and artist
• “Time stays long
enough for those who
use it.”