Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions. 6.3 – Simplifying Complex Fractions Complex.
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Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions.
7
5
yx
xyyx
873
4 3
1
1
xx
x
6.3 – Simplifying Complex FractionsComplex Fractions
3 24 31 32 8
3 24 31 32 8
3 43 4
4
3 24 3
1 384 2
24 24
2
3 24 31
43
2 824
9 812 12
4 38 8
6 3 8 2
12 1 3 3
11278
18 16
12 9
2
212
21
LCD: 12, 8 LCD: 24
6.3 – Simplifying Complex Fractions
24
24
8
7
12
1
7
8
12
12
3
LCD: y1
2 1
xy
xy
1
2 1
y y
y
xy
xy
2 1
y x
x
yx
yx
12
1y–y
6.3 – Simplifying Complex Fractions
LCD: 6xy
56
3
yy xyx
2
2 2
5 6
2 6
x y
xy x y
56
3
6 6
6 6
yy x
xy xy
yy
xx xy
25 6x y2xy 3y x
xy
xy
y
3
65
6xy
6xy
6.3 – Simplifying Complex Fractions
LCD:
3759
3759
3 5
7 9
6337
6359
3 9
7 5
9 3
7 5
27
35
27
35
63
Outers over Inners
6.3 – Simplifying Complex Fractions
3759
27
35
)5)(7(
)9)(3(
Outers over Inners
512
56
x
xx
512
56
x
xx
52
5
x
xx
6.3 – Simplifying Complex Fractions
6.5 – Solving Equations w/ Rational Expressions
5 16 1x 4 1
4 5 20
x
4 1
4 520 20
2020
x
5x
LCD: 20
5 16 1x
5 15x
3x
44 11
LCD:
2 3 3 3 2x x 2
2 3 2
3 3 9x x x
2 3 2
3 3 3 3x x x x
3 3x x
3 32
3xx x
2 3 3 3 2x x
2 6 3 9 2x x
5 3 2x
5 5x 1x
33
33
xx x
33 3
32
x xx x
6.5 – Solving Equations w/ Rational Expressions
LCD: 6x5 3 3
3 2 2x
65
3x
2 5 3 3 3 3x x 10 9 9x x
9x
36
2xx
26
3x
10 9 9x x
6.5 – Solving Equations w/ Rational Expressions
LCD: x+36 2
23 3
xxx x
3x x
2 3x x
2 12 0x x 3x
3 0 4 0x x 3x
36
3xx
2
33
xx
x
3 2x
6 2x 2 6x 2 3x x 6 4 6x
0 4x 4x
6.5 – Solving Equations w/ Rational Expressions
LCD:
2
5 11 1 12
2 7 10 5
x
x x x x
5 11 1 12
2 2 5 5
x
x x x x
2 5x x
2 55
2xx x
5 5 11 1 12 2x x x
5 25 11 1 12 24x x x
5 25 23x x
6 48x 8x
1
2 52
1 15
x
xx
xx
2 5
12
5x
xx
6.5 – Solving Equations w/ Rational Expressions
LCD: abx1 1 1
a b x
1abx
a
bx
bx ab ax
bx
bx
b x
Solve for a
1
babx 1
xabx
ax ab
a b x
a
6.5 – Solving Equations w/ Rational Expressions
Problems about NumbersIf one more than three times a number is divided by the number, the result is four thirds. Find the number.
3x
33 1
xx
x
3 3 1 4x x
9 3 4x x
5 3x 3
5x
LCD = 3x1
x 4
3
3
34
x
9 4 3x x
6.6 – 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?
Time to sort one batch (hours)
Fraction of the job completed in one hour
Ryan
Mike
Together
1
2
1
31
x
2
3
x
6.6 – Rational Equations and Problem Solving
Problems about Work
Time to sort one batch (hours)
Fraction of the job completed in one hour
Ryan
Mike
Together
1
2
1
31
x
2
3
x
1
2 61
2x
3x 5 6x 6
5x hrs.
LCD =1
3
1
x 6x
36
1x 1
6x
x
2x 61
15
6.6 – 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?
Time to Assemble one unit (hours)
Fraction of the job completed in one hour
Merry
Pippen
Together
2
8
x
1
21
8
1
x
6.6 – Rational Equations and Problem Solving
Time to Assemble one unit (hours)
Fraction of the job completed in one hour
Merry
Pippen
Together
2
8
x
1
21
8
1
x
LCD:1
2 8
1
2x
4x 5 8x 8
5x
hrs.
1
8
1
x 8x
88
1x 1
8x
x
x 83
15
6.6 – 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?
1
12
1
x
26
3
1203
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together
x
12
20
3
6.6 – Rational Equations and Problem Solving
1
12
1
x
26
3
1203
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together
x
12
1
121 1
12 x
20
3
1
x 1
203
3
20
6.6 – Rational Equations and Problem Solving
LCD:
601
12x
5x
60 4xhrs. 15x
1 1 3
12 20x 60x
160
xx
060
3
2x
60 9x
5x60 9x
6.6 – 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?
d r t
65miles
hour 2 hours 130 miles
dt
r
dr
t
6.6 – 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 car’s speed is fifteen miles per hour faster than the motorcycle’s, find the speed of both vehicles.
Rate Time Distance
Motor-cycle
Car
x
x + 15
450 mi
600 mi
t
t
r
dt
x
450
15
600
x
6.6 – Rational Equations and Problem Solving
Rate Time Distance
Motor-cycle
Car
x
x + 15
450 mi
600 mi
t
t
r
dt
x
450
15
600
x
x
450
15
600
xLCD: x(x + 15)
15
600450
xx
x(x + 15) x(x + 15)
6.6 – Rational Equations and Problem Solving
15
600450
xx
x(x + 15) x(x + 15)
xx 60045015
xx 60015450450
x15015450
x
150
15450
x45
45x mph
Motorcycle
6015 x mph
Car
6.6 – 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?
boat speedx
Rate Time Distance
UpStream
DownStream
x - 5
x + 5
22 mi
42 mi
t
t
r
dt
5
22
x
5
42
x
6.6 – Rational Equations and Problem Solving
boat speedx
Rate Time Distance
UpStream
DownStream
x - 5
x + 5
22 mi
42 mi
t
t
r
dt
5
22
x
5
42
x
5
22
x
5
42
xLCD: (x – 5)(x + 5)
5
42
5
22
xx(x – 5)(x + 5) (x – 5)(x + 5)
6.6 – Rational Equations and Problem Solving
542225 xx
2104211022 xx
xx 2242210110
x20
320
x1616 mph
Boat Speed
5
42
5
22
xx(x – 5)(x + 5) (x – 5)(x + 5)
x20320
6.6 – Rational Equations 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
6.7 – Variation and Problem Solving
The number k is called the constant of variation or the constant of proportionality
.kxy
Direct Variation
kxy 824 k
k8
24
6.7 – Variation and Problem Solving
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.
3k
xy 3direct variation equation
constant of variation
x
y
3
9
5
15
9
27
13
39
kwd 567 k
k56
7
6.7 – Variation and Problem SolvingHooke’s 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.
8
1k
xy3
1
direct variation equation
constant of variation
853
1y
6.10y inches
Inverse Variation: y varies inversely as x (y is inversely proportional to x), if there is a nonzero constant k such that
6.7 – Variation and Problem Solving
The number k is called the constant of variation or the constant of proportionality.
.x
ky
Inverse Variation
x
ky
36
k
k18
6.7 – Variation and Problem Solving
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.
xy
18
direct variation equation
constant of variation
x
y
3
6
9
2
10
1.8
18
1
t
kr
430
k
k120
6.7 – Variation and Problem SolvingThe 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.
xr
120
direct variation equation
constant of variation
5
120r
24r mph
Additional Problems
LCD: 15
5 2 3 1 1x x 2 1 1
3 5 15
x x
15 152 1 1
3 5 1515
x x
5 2x
5 10 3 3 1x x
2 13 1x
2 12x
6x 13 x 11
6.5 – Solving Equations w/ Rational Expressions
LCD: x
22 6 7x x x 62 7xx
62 7x
xx x x x
2x
20 5 6x x
1 0 6 0x x
1 6x x
0 1 6x x
6 2x 7x
6.5 – Solving Equations w/ Rational Expressions
LCD:5 5
31 1
x
x x
5
11x
x
x
5 5 3 3x x
2 2x 1x
1x
15
1xx
1 3x
5 3 5 3x x
Not a solution as equations is undefined at x = 1.
6.5 – Solving Equations w/ Rational Expressions
Problems about Numbers
The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number.
2
x
62
x
3 2x x 3 2x x
1x
LCD = 61
3
6
x
3
61
66
x
2 2x
6.6 – Rational Equations and Problem Solving
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