Definition: A rational expression is a quotient of
polynomials .
What are some examples of rational expressions?
What is a rational function?
Recall the concept of the domain of a function and how it
is related
to rational functions.
Simplify rational expressions.
To simplify a rational expression, factor the numerator
and
denominator and cross out any common factors .
Simplify.
Remember: Factoring for the sake of factoring is not
simplification.
Simplification only occurs if factors are crossed out.
Simplify.
Simplify.
Multiplying and Dividing Rational Expressions.
Recall the rules for multiplying and dividing fractions.
Since simplification is expected any time you work with
rational
expressions, multiplication and division problems almost always
become simplification problems which focus on factoring!
Multiply.
Divide.
Adding and Subtracting Rational Expressions
Recall the rules for adding and subtracting fractions .
If the ex pressions have like denominators , adding and
subtracting
will be easy but simplification concerns may loom on the horizon.
If the expressions have unlike denominators, you will have to find a
least common denominator which again focuses on factoring.
Add.
Subtract.
Subtract.
Add.
§7.5 Rational Equations
To solve a rational equation, find the LCD of the
denominators.
Then multiply the LCD through the equation to clear all fractions.
Solve the resulting equation and check to make sure the proposed
solution doesn’t make a fraction in the original equation undefined.
Solve.
Solve.
Solve.
Solve.
Solve.
Solve.
Solve for M.
§7.6 Applications involving Rational Equations
Part 1: Proportions
Proportions are a special kind of rational equations.
A proportion is an equality of two ratios.
A ratio is a comparison of two quantities, often expressed using
fractions.
To solve a proportion, apply
cross‐multiplication: AD = BC
A machine can process 300 parts in 20 minutes. Find how many
parts can be processed in 45 minutes.
In Quack County with a voting population of 50,000, a
pre‐election
poll of 250 eligible voters was taken. Donald Duck® had the support
of 39 voters in the poll. Estimate the number of voters in Quack
County who would vote for Donald.
Part 2: Shared Work Problems
t1& t2 are the times necessary for individuals to complete a job
T is the time needed working together to complete the job
An experienced bricklayer can construct a long wall in 6
days.
The apprentice can complete the job in 12 days. Find how long
it will take if they work together.
Mr. Dodson can paint a bedroom by himself in 4 hours. With
his
son helping him, the bedroom can be painted in 3 hours. How long
would Mr. Dodson’s son need to paint the bedroom by himself?
One pipe fills a storage pool in 20 minutes. A second pipe
fills the
same pool in 15 minutes. When a third pipe is added, and all three
are used to fill the pool, it only takes 5 minutes. How long does it
take the third pipe to do the job alone?
Part 3: distance = rate X time
The current on a portion of the Mississippi River is 3
miles per hour.
A barge can go 6 miles upstream in the same amount of time as it
takes to go 10 miles downstream. Find the speed of the barge in
still water.
The use of a chart will make it easier to set up the
problem.
In the time it takes Bob to ride his bicycle 150 miles
over the flat
countryside he can climb 90 miles of hills and small mountains. If
Bob cycles 10 mph s lower on the hills than he does on level terrain,
how fast can Bob cycle on the flat countryside?
A marketing manager travels 1080 miles in a corporate jet
and then
an additional 240 miles by car. If the car ride takes one hour longer
than the jet ride takes, and if the rate of the jet is 6 times the rate of
the car, find the time the manager travels by jet and find the time
the manager travels by car.
§7.7 Complex Fractions
A complex fraction is a fraction with at least one
fraction inside its
numerator or denominator.
When we simplify a complex fraction, we want to rewrite
the
expression so that neither the numerator nor denominator has a
fraction in it.
There are two strategies for simplifying complex fractions
based on
two different families of complex fractions:
Group 1:
Group 2:
Simplifying by treating like a division problem (Group 1)
If a complex fraction is composed of one fraction over one
fraction,
then treat the complex fraction like a division problem.
Process: Rewrite N/D as N ÷D. Then take the reciprocal of
D and
multiply it to N. Simplify the product completely .
Simplify.
Simplify.
While this process works very well for expressions in
group 1, the
problem is that most complex fractions have multiple expressions in
their numerator or denominator. While it is possible to combine
the numerator and denominator expressions so as to create a group
1 example, a better strategy emerges for the group 2 examples .
Simplify by using a LCD (Group 2)
For most complex fractions, finding the LCD of the
fractions in the
numerator and denominator—then multiplying the LCD to each—is
the best method for simplification .
Process: Take all fractions in the numerator and
denominator and
find their LCD. Then multiply the complex fraction by a
unit fraction of the LCD/LCD and simplify completely.
Simplify.
Simplify.
Simplify.
Simplify.
If a fraction has variables /expressions with negative
exponents ,
recognize that the fraction is complex by rewriting the expression
with positive exponents.
Simplify.
Simplify.
