Title: Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions.
16.3 Simplifying Complex Fractions
Complex Fractions
Defn A rational expression whose numerator,
denominator, or both contain one or more rational
expressions.
26.3 Simplifying Complex Fractions
24 24
LCD 12, 8
LCD 24
2
3
36.3 Simplifying Complex Fractions
LCD y
yy
46.3 Simplifying Complex Fractions
LCD 6xy
6xy 6xy
56.3 Simplifying Complex Fractions
Outers over Inners
LCD
63
66.3 Simplifying Complex Fractions
Outers over Inners
76.5 Solving Equations w/ Rational Expressions
LCD 20
86.5 Solving Equations w/ Rational Expressions
LCD
96.5 Solving Equations w/ Rational Expressions
LCD 6x
106.5 Solving Equations w/ Rational Expressions
LCD x3
116.5 Solving Equations w/ Rational Expressions
LCD
126.5 Solving Equations w/ Rational Expressions
LCD abx
Solve for a
136.6 Rational Equations and Problem Solving
Problems about Numbers
If one more than three times a number is divided
by the number, the result is four thirds. Find
the number.
LCD 3x
146.6 Rational Equations and Problem Solving
Problems about Work
Mike and Ryan work at a recycling plant. Ryan
can sort a batch of recyclables in 2 hours and
Mike can short a batch in 3 hours. If they work
together, how fast can they sort one batch?
2
3
x
156.6 Rational Equations and Problem Solving
Problems about Work
2
3
x
LCD
6x
hrs.
166.6 Rational Equations and Problem Solving
Pippen and Merry assemble Ork action figures. It
takes Merry 2 hours to assemble one figure while
it takes Pippen 8 hours. How long will it take
them to assemble one figure if they work together?
2
8
x
176.6 Rational Equations and Problem Solving
2
8
x
LCD
8x
hrs.
186.6 Rational Equations and Problem Solving
A sump pump can pump water out of a basement in
twelve hours. If a second pump is added, the job
would only take six and two-thirds hours. How
long would it take the second pump to do the job
alone?
12
x
196.6 Rational Equations and Problem Solving
12
x
206.6 Rational Equations and Problem Solving
LCD
60x
hrs.
216.6 Rational Equations and Problem Solving
Distance, Rate and Time Problems
If you drive at a constant speed of 65 miles per
hour and you travel for 2 hours, how far did you
drive?
226.6 Rational Equations and Problem Solving
A car travels six hundred miles in the same time
a motorcycle travels four hundred and fifty
miles. If the cars speed is fifteen miles per
hour faster than the motorcycles, find the speed
of both vehicles.
Rate Time Distance
Motor-cycle
Car
x
t
450 mi
t
x 15
600 mi
236.6 Rational Equations and Problem Solving
Rate Time Distance
Motor-cycle
Car
x
t
450 mi
t
x 15
600 mi
LCD
x(x 15)
x(x 15)
x(x 15)
246.6 Rational Equations and Problem Solving
x(x 15)
x(x 15)
Motorcycle
Car
256.6 Rational Equations and Problem Solving
A boat can travel twenty-two miles upstream in
the same amount of time it can travel forty-two
miles downstream. The speed of the current is
five miles per hour. What is the speed of the
boat in still water?
Rate Time Distance
Up Stream
Down Stream
boat speed x
t
x - 5
22 mi
t
x 5
42 mi
266.6 Rational Equations and Problem Solving
Rate Time Distance
Up Stream
Down Stream
boat speed x
t
x - 5
22 mi
t
x 5
42 mi
LCD
(x 5)(x 5)
(x 5)(x 5)
(x 5)(x 5)
276.6 Rational Equations and Problem Solving
(x 5)(x 5)
(x 5)(x 5)
Boat Speed
286.7 Variation and Problem Solving
Direct Variation y varies directly as x (y is
directly proportional to x), if there is a
nonzero constant k such that
The number k is called the constant of variation
or the constant of proportionality
296.7 Variation and Problem Solving
Direct Variation
Suppose y varies directly as x. If y is 24 when
x is 8, find the constant of variation (k) and
the direct variation equation.
direct variation equation
constant of variation
13
3
5
9
x
y
9
15
27
39
306.7 Variation and Problem Solving
Hookes law states that the distance a spring
stretches is directly proportional to the weight
attached to the spring. If a 56-pound weight
stretches a spring 7 inches, find the distance
that an 85-pound weight stretches the spring.
Round to tenths.
direct variation equation
constant of variation
316.7 Variation and Problem Solving
Inverse Variation y varies inversely as x (y is
inversely proportional to x), if there is a
nonzero constant k such that
The number k is called the constant of variation
or the constant of proportionality.
326.7 Variation and Problem Solving
Inverse Variation
Suppose y varies inversely as x. If y is 6 when
x is 3, find the constant of variation (k) and
the inverse variation equation.
direct variation equation
constant of variation
10
18
3
9
x
y
6
2
1.8
1
336.7 Variation and Problem Solving
The speed r at which one needs to drive in order
to travel a constant distance is inversely
proportional to the time t. A fixed distance can
be driven in 4 hours at a rate of 30 mph. Find
the rate needed to drive the same distance in 5
hours.
direct variation equation
constant of variation
34Additional Problems
356.5 Solving Equations w/ Rational Expressions
LCD 15
366.5 Solving Equations w/ Rational Expressions
LCD x
376.5 Solving Equations w/ Rational Expressions
LCD
Not a solution as equations is undefined at x 1.
386.6 Rational Equations and Problem Solving
Problems about Numbers
The quotient of a number and 2 minus 1/3 is the
quotient of a number and 6. Find the number.
LCD 6