6.3 – Simplifying Complex Fractions

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 24 314 32 8

24

9 812 12

4 38 8

6 3 8 212 1 3 3

11278

18 1612 9

221

221

LCD: 12, 8 LCD: 24

6.3 – Simplifying Complex Fractions

24

24

87

121

78

121

2

3

LCD: y1

2 1

xy

xy

1

2 1

y y

y

xy

xy

2 1y xx

yxyx

12

1y–y


LCD: 6xy5

6

3

yy xy x

2

2 2

5 62 6x yxy x y

56

3

6 6

6 6

yy x

xy xy

y y xx xy

25 6x y2xy 3y x

xyxy

y

3

656xy

6xy


LCD:

3759

3759

3 57 9

63 37

63 59

3 97 5

9 37 5

2735

2735

63

Outers over Inners6.3 – Simplifying Complex Fractions

3759

2735

)5)(7()9)(3(

Outers over Inners

512

56

x

xx

512

56xxx

52

5xxx


6.5 – Solving Equations w/ Rational Expressions

5 16 1x 4 14 5 20x

4 14 5

20 2020

20x

5x

LCD: 20

5 16 1x

5 15x

3x

44 11

LCD:

2 3 3 3 2x x 2

2 3 23 3 9x x x

2 3 2

3 3 3 3x x x x

3 3x x

3 3 23x

x x

2 3 3 3 2x x

2 6 3 9 2x x

5 3 2x

5 5x 1x

3 33

3x

x x

3

3 33 2x x

x x


LCD: 6x5 3 33 2 2x

6 53

x

2 5 3 3 3 3x x 10 9 9x x

9x

362x

x 2

6 3x

10 9 9x x


LCD: x+36 2 23 3

xxx x

3x x

2 3x x

2 12 0x x 3x

3 0 4 0x x 3x

3 63x

x

233

x xx

3 2x

6 2x 2 6x 2 3x x 6 4 6x

0 4x 4x


LCD:

2

5 11 1 122 7 10 5

xx x x x

5 11 1 12

2 2 5 5x

x x x x

2 5x x

2 5 52x

x x

5 5 11 1 12 2x x x

5 25 11 1 12 24x x x

5 25 23x x

6 48x 8x

12 5

2 1 15 xx

xx

x

2 5 12

5x

xx


LCD: abx1 1 1a b x

1abxa

bx

bx ab ax

bx

bxb x

Solve for a

1b

abx 1x

abx

ax ab

a b x

a


Problems about NumbersIf one more than three times a number is divided by the number, the result is four thirds. Find the number.

3x

3 3 1x xx

3 3 1 4x x

9 3 4x x

5 3x 35

x

LCD = 3x1

x 4

3

3

3 4x

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

12131x

2

3

x


Problems about Work

Time to sort one batch (hours)


Ryan

Mike

Together

12131x

2

3

x

12 6 1

2x

3x 5 6x 65

x hrs.

LCD =13

1x

6x 3

6 1x 16x

x

2x 6115


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)


Merry

Pippen

Together

2

8

x

12181x


Time to Assemble one unit (hours)


Merry

Pippen

Together

2

8

x

12181x

LCD:12

8 12

x

4x 5 8x 85

x

hrs.

18

1x

8x 8

8 1x 18x

x

x 8315


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?

112

1x

263

1203

Time to pump one basement (hours)


1st pump

2nd pump

Together

x

12

203


112

1x

263

1203

Time to pump one basement (hours)


1st pump

2nd pump

Together

x

12

112

1 112 x

203

1x

1203

320


LCD:

60 112

x

5x

60 4xhrs. 15x

1 1 312 20x

60x

160x

x 0

60 32

x

60 9x

5x60 9x


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

65mileshour

2 hours 130 miles

d tr

d rt


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

rdt

x450

15600x


Rate Time Distance

Motor-cycle

Car

x

x + 15

450 mi

600 mi

t

t

rdt

x450

15600x

x450

15

600x

LCD: x(x + 15)

15600450

xx

x(x + 15) x(x + 15)


15600450

xx

x(x + 15) x(x + 15)

xx 60045015

xx 60015450450

x15015450

x

15015450

x45

45x mphMotorcycle

6015 x mphCar


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

rdt

522x

542x


boat speedx

Rate Time Distance

UpStream

DownStream

x - 5

x + 5

22 mi

42 mi

t

t

rdt

522x

542x

522x

5

42x

LCD: (x – 5)(x + 5)

542

522

xx(x – 5)(x + 5) (x – 5)(x + 5)


542225 xx

2104211022 xx

xx 2242210110

x20

320

x1616 mph

Boat Speed

542

522

xx(x – 5)(x + 5) (x – 5)(x + 5)

x20320


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

k824


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

xy

39

515

927

1339

kwd 567 k

k567

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.

81

k

xy31

direct variation equation


8531

y

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


The number k is called the constant of variation or the constant of proportionality.

.xk

y

Inverse Variation

xk

y

36

k

k18


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



xy

36

92

101.8

181

tk

r

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



5120

r

24r mph

Additional Problems

LCD: 15

5 2 3 1 1x x 2 1 13 5 15x x

15 152 1 13 5 15

15x x

5 2x

5 10 3 3 1x x

2 13 1x

2 12x

6x 13 x 11


LCD: x

22 6 7x x x 62 7xx

62 7xx

x x x x

2x

20 5 6x x

1 0 6 0x x

1 6x x

0 1 6x x

6 2x 7x


LCD:5 5 3

1 1x

x x

51

1x xx

5 5 3 3x x

2 2x 1x

1x

1 51x

x

1 3x

5 3 5 3x x

Not a solution as equations is undefined at x = 1.


Problems about Numbers

The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number.

2x

62x

3 2x x 3 2x x

1x

LCD = 613

6x

3

6 1 6

6x

2 2x


Documents

6.3 – Simplifying Complex Fractions