What is the Lowest Common Denominator (LCD)? 5.3 – Addition & Subtraction of Rational Expressions

6

5

3

2and

What is the Lowest Common Denominator (LCD)?

2

1

7

2,

4

3and

3 5

5 13and

x x4 2

17 3

4 6and

y y

6 28

5x 12 4y

5.3 – Addition & Subtraction of Rational Expressions

5

7

5

3

a

aand

a

a


123

5

4

72

2

x

xand

x

x

5a

3

27xand 5x

4x 4x

243 x

4x 4x 3 4x 5a


23

3

32

522

yy

yand

yy

y


1y

5y

3y 2y

3y 1y and

3y

1y 2y


73

7

73

3

xx

x

Examples (Like Denominators):

1

3 7x 3x 7


2

64

2

52 2

x

x

x

xx


2x 22 5

2

x x

x

22

2

x

x

2x

32 x22 5x x 4 6x

4x 6

x 6

2x 2 3x


65

72

65

1322

xx

x

xx

x


65

72132

xx

xx

65

132

xx

x x2 7

65

62

xx

x

6x

x 1 x 6

1

1

x

1

1


Examples:

LCD 15


𝑦5−

4 𝑦15

( 33 ) 𝑦5 − 4 𝑦

15

3 𝑦15−

4 𝑦15

3 𝑦−4 𝑦15

−𝑦15

Examples:

210

11

8

5

xx

LCD 40x2

210

11

8

5

xx

x

x

5

5

22 40

44

40

25

xx

x 240

4425x

x

4

4


Examples:

1

2

7

5

xx

LCD

5 2

7 1x x

7 1 7 1x x x x

7 1x x

7x 1x

1

1

x

x

7

7

x

x

5x 5 14x 19x 5


3

5

9

102

xx

xExamples:

10x

10 5

3 3 3

x

x x x

10

3 3

x

x x

3 3x x

3 3x x

3 3x x

3

5

x

3x 3x

LCD

3

3

x

x

3 3x x 5x 15

10x 155x

5x 15

5 3x 3x 3x

5

3x


812

3

23

42

x

x

xx

Examples:

LCD

4

4 3

3 2 4 3 2

x

x x x

x 3 2x 3x

4 3 2x

4 x 3 2x

4

4

x

x


Examples: continued

4 3 2 4 3 2x x x x

4 3 2x x

x

x

x

x

xx 234

3

23

4

4

4

16 23x

216 3x


444

622

x

x

xx

xExamples:

6x x

LCD

6

2 2 2 2

x x

x x x x

2x 2x 2x 2x

2x 2x 2x

2

2

x

x

2

2

x

x


Examples: continued

2

2

2222

6

2

2

x

x

xx

x

xx

x

x

x

2 2 2 2 2 2x x x x x x

222

2126 22

xxx

xxxx

2 2 2x x x 22 2x x

26x 12x2x 2x

27x 10x x 7x 10


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

5.4 – 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 24

8

7

12

1

7

8

12

12

3

24

5.4 – Complex Fractions

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

6xy6xy


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


5.5 – Equations with 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 22

3 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 = 3x1x

4

3

3

34

x

9 4 3x x

5.6 – Applications

Problems about Work

Mike and Ryan work at a recycling plant. Ryan can sort a batch of recyclables in 2 hours and Mike can sort 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

21

31

x

2

3

x


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 6

1

2x

3x 5 6x 6

5x hrs.

LCD =1

3

1

x 6x

36

1x 1

6x

x

2x 61

15


James and Andy mow lawns. It takes James 2 hours to mow an acre while it takes Andy 8 hours. How long will it take them to mow one acre if they work together?

Time to mow one acre (hours)


James

Andy

Together

2

8

x

1

21

81

x


Time to mow one acre (hours)


James

Andy

Together

2

8

x

1

2

1

81

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

121

x

26

3

Time to pump one basement (hours)


1st pump

2nd pump

Together

x

12

20

3


1203

1

12

1

x2

63

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


Rate Time Distance

UpStream

DownStream

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 speed = x

x - 5

x + 5

22 mi

42 mi

t

t

r

dt

5

22

x

5

42

x


Rate Time Distance

UpStream

DownStream

boat speed = x

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

x16 16mph

Boat Speed

5

42

5

22

xx(x – 5)(x + 5) (x – 5)(x + 5)

x20320


Dividing by a Monomial

where 0a b a b

cc c c

3 2

2

25 5

5

x x

x

3

2

25

5

x

x5x

8 6 4

3

21 9 12

3

x x x

x

8

3

21

3

x

x57x

2

2

5

5

x

x 1

6

3

9

3

x

x

4

3

12

3

x

x

33x 4x

5.7 – Division of Polynomials

Dividing by a Monomial

7 2 224 12 4 4x x x x

3 312 18 6

3

x y xy y

xy

7 2

2

24 12 4

4

x x x

x

7

2

24

4

x

x56x

3 312

3

x y

xy

2 24x y

2

2

12

4

x

x 2

4

4

x

x 3

1

x

18

3

xy

xy

6

3

y

xy

62

x


285

5

Review of Long Division

783 4

4 7835 2855

253 5

7

350

43 836

2

1 9

3

5

203


4

3195

4

3195

35125 2 xxx

x

xx 52 x7 35

7

357 x 0

5x 7x 35122 xx

Long Division

5

35122

x

xx


15023 2 xxx

x2

xx 62 2 x6 15

6

186 x 3

3x

3

362x

x 152 2 x

3

3

x

Long Division

3

152 2

x

x


72812 2 xxx

x4

xx 48 2 x6 7

3

36 x 4

12 x

12

434

xx 728 2 xx

12

4

x

Long Division

12

728 2

x

xx


Documents

What is the Lowest Common Denominator (LCD)? 5.3 – Addition & Subtraction of Rational Expressions