Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions. 6.3 – Simplifying Complex Fractions Complex

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


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


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


3759

27

35

)5)(7(

)9)(3(

Outers over Inners

512

56

x

xx

512

56

x

xx

52

5

x

xx


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


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


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


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


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


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


Problems about Work

Time to sort one batch (hours)


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


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

1

21

8

1

x


Time to Assemble one unit (hours)


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


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)


1st pump

2nd pump

Together

x

12

20

3


1

12

1

x

26

3

1203

Time to pump one basement (hours)


1st pump

2nd pump

Together

x

12

1

121 1

12 x

20

3

1

x 1

203

3

20


LCD:

601

12x

5x

60 4xhrs. 15x

1 1 3

12 20x 60x

160

xx

060

3

2x

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

65miles

hour 2 hours 130 miles

dt

r

dr

t


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


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)


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


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


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)


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


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


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


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


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

.x

ky

Inverse Variation

x

ky

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



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



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


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


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.


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


Documents

Defn: A rational expression whose numerator, denominator, or both contain one or more rational expressions. 6.3 – Simplifying Complex Fractions Complex