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Rational Expressions: Exercise Supplement

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.

The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.

This module contains the exercise supplement for the chapter "Rational Expressions".

Exercise Supplement

Rational Expressions ((Reference))

For the following problems, find the domain of each rational expression.

Exercise 1

9 x+4 9 x+4

Solution

x4 x4

Exercise 2

10x x+6 10x x+6

Exercise 3

x+1 2x5 x+1 2x5

Solution

x 5 2 x 5 2

Exercise 4

2a+3 7a+5 2a+3 7a+5

Exercise 5

3m 2m( m1 ) 3m 2m( m1 )

Solution

m0,1 m0,1

Exercise 6

5r+6 9r( 2r+1 ) 5r+6 9r( 2r+1 )

Exercise 7

s s( s+8 )( 4s+7 ) s s( s+8 )( 4s+7 )

Solution

s8, 7 4 ,0 s8, 7 4 ,0

Exercise 8

11x x 2 9x+18 11x x 2 9x+18

Exercise 9

y+5 12 y 2 +28y5 y+5 12 y 2 +28y5

Solution

y 1 6 , 5 2 y 1 6 , 5 2

Exercise 10

16 12 a 3 +21 a 2 6a 16 12 a 3 +21 a 2 6a

For the following problems, show that the fractions are equivalent.

Exercise 11

4 5 , 4 5 4 5 , 4 5

Solution

( 4·5 )=20,4( 5 )=20 ( 4·5 )=20,4( 5 )=20

Exercise 12

3 8 , 3 8 3 8 , 3 8

Exercise 13

7 10 , 7 10 7 10 , 7 10

Solution

( 7·10 )=70,7( 10 )=70 ( 7·10 )=70,7( 10 )=70

For the following problems, fill in the missing term.

Exercise 14

3 y5 = y5 3 y5 = y5

Exercise 15

6a 2a+1 = 2a+1 6a 2a+1 = 2a+1

Solution

6a 6a

Exercise 16

x+1 x3 = x3 x+1 x3 = x3

Exercise 17

9 a+4 = a4 9 a+4 = a4

Solution

9

Exercise 18

y+3 y5 = y+5 y+3 y5 = y+5

Exercise 19

6m7 5m1 = 6m+7 6m7 5m1 = 6m+7

Solution

5m+1 5m+1

Exercise 20

2r5 7r+1 = 2r5 2r5 7r+1 = 2r5

Reducing Rational Expressions ((Reference))

For the following problems, reduce the rational expressions to lowest terms.

Exercise 21

12 6x+24 12 6x+24

Solution

2 x+4 2 x+4

Exercise 22

16 4y16 16 4y16

Exercise 23

5m+25 10 m 2 +15m 5m+25 10 m 2 +15m

Solution

m+5 m( 2m+3 ) m+5 m( 2m+3 )

Exercise 24

7+21r 7 r 2 +28r 7+21r 7 r 2 +28r

Exercise 25

3 a 2 +4a 5 a 3 +6 a 2 3 a 2 +4a 5 a 3 +6 a 2

Solution

3a+4 a( 5a+6 ) 3a+4 a( 5a+6 )

Exercise 26

4x4 x 2 +2x3 4x4 x 2 +2x3

Exercise 27

5y+20 y 2 16 5y+20 y 2 16

Solution

5 y4 5 y4

Exercise 28

4 y 3 12 y 4 2 y 2 3 4 y 3 12 y 4 2 y 2 3

Exercise 29

6 a 9 12 a 7 2 a 7 14 a 5 6 a 9 12 a 7 2 a 7 14 a 5

Solution

3 a 2 ( a 2 2 ) a 2 7 3 a 2 ( a 2 2 ) a 2 7

Exercise 30

8 x 4 y 8 +24 x 3 y 9 4 x 2 y 5 12 x 3 y 6 8 x 4 y 8 +24 x 3 y 9 4 x 2 y 5 12 x 3 y 6

Exercise 31

21 y 8 z 10 w 2 7 y 7 w 2 21 y 8 z 10 w 2 7 y 7 w 2

Solution

3y z 10 3y z 10

Exercise 32

35 a 5 b 2 c 4 d 8 5ab c 3 d 6 35 a 5 b 2 c 4 d 8 5ab c 3 d 6

Exercise 33

x 2 +9x+18 x 3 +3 x 2 x 2 +9x+18 x 3 +3 x 2

Solution

x+6 x 2 x+6 x 2

Exercise 34

a 2 -12a+35 2 a 4 14 a 3 a 2 -12a+35 2 a 4 14 a 3

Exercise 35

y 2 7y+12 y 2 4y+3 y 2 7y+12 y 2 4y+3

Solution

y4 y1 y4 y1

Exercise 36

m 2 6m16 m 2 9m22 m 2 6m16 m 2 9m22

Exercise 37

12 r 2 7r10 4 r 2 13r+10 12 r 2 7r10 4 r 2 13r+10

Solution

3r+2 r2 3r+2 r2

Exercise 38

14 a 2 5a1 6 a 2 +9a6 14 a 2 5a1 6 a 2 +9a6

Exercise 39

4 a 4 8 a 3 4 a 2 4 a 4 8 a 3 4 a 2

Solution

a( a2 ) a( a2 )

Exercise 40

5 m 2 10 m 3 +5 m 2 5 m 2 10 m 3 +5 m 2

Exercise 41

6a1 5a2 6a1 5a2

Solution

6a+1 5a+2 6a+1 5a+2

Exercise 42

r 5r1 r 5r1

Multiplying and Dividing Rational Expressions ((Reference)) - Adding and Subtracting Rational Expressions ((Reference))

For the following problems, perform the indicated operations.

Exercise 43

x 2 18 · 3 x 3 x 2 18 · 3 x 3

Solution

1 6x 1 6x

Exercise 44

4 a 2 b 3 15 x 4 y 5 · 10 x 6 y 3 a b 2 4 a 2 b 3 15 x 4 y 5 · 10 x 6 y 3 a b 2

Exercise 45

x+6 x1 · x+7 x+6 x+6 x1 · x+7 x+6

Solution

x+7 x1 x+7 x1

Exercise 46

8a12 3a+3 ÷ ( a+1 ) 2 4a6 8a12 3a+3 ÷ ( a+1 ) 2 4a6

Exercise 47

10 m 4 5 m 2 4 r 7 +20 r 3 ÷ m 16 r 8 +80 r 4 10 m 4 5 m 2 4 r 7 +20 r 3 ÷ m 16 r 8 +80 r 4

Solution

20mr( 2 m 2 1 ) 20mr( 2 m 2 1 )

Exercise 48

5 r+7 3 r+7 5 r+7 3 r+7

Exercise 49

2a 3a1 9a 3a1 2a 3a1 9a 3a1

Solution

7a 3a1 7a 3a1

Exercise 50

9x+7 4x6 + 3x+2 4x6 9x+7 4x6 + 3x+2 4x6

Exercise 51

15y4 8y+1 2y+1 8y+1 15y4 8y+1 2y+1 8y+1

Solution

13y5 8y+1 13y5 8y+1

Exercise 52

4 a+3 + 6 a5 4 a+3 + 6 a5

Exercise 53

7a a+6 + 5a a8 7a a+6 + 5a a8

Solution

2a( 6a13 ) ( a+6 )( a8 ) 2a( 6a13 ) ( a+6 )( a8 )

Exercise 54

x+4 x2 + x+7 x1 x+4 x2 + x+7 x1

Exercise 55

2y+1 y+4 y+6 y+1 2y+1 y+4 y+6 y+1

Solution

y 2 7y23 ( y+4 )( y+1 ) y 2 7y23 ( y+4 )( y+1 )

Exercise 56

x3 ( x+2 )( x+4 ) + 2x1 x+4 x3 ( x+2 )( x+4 ) + 2x1 x+4

Exercise 57

6a+5 ( 2a+1 )( 4a3 ) + 4a+1 2a+1 6a+5 ( 2a+1 )( 4a3 ) + 4a+1 2a+1

Solution

2( 8 a 2 a+1 ) ( 2a+1 )( 4a3 ) 2( 8 a 2 a+1 ) ( 2a+1 )( 4a3 )

Exercise 58

4 x 2 +3x+2 + 9 x 2 +6x+8 4 x 2 +3x+2 + 9 x 2 +6x+8

Exercise 59

6r r 2 +7r18 3r r 2 3r+2 6r r 2 +7r18 3r r 2 3r+2

Solution

3r( 3r+7 ) ( r1 )( r2 )( r+9 ) 3r( 3r+7 ) ( r1 )( r2 )( r+9 )

Exercise 60

y+3 y 2 11y+10 y+1 y 2 +3y4 y+3 y 2 11y+10 y+1 y 2 +3y4

Exercise 61

2a+5 16 a 2 1 6a+7 16 a 2 12a+2 2a+5 16 a 2 1 6a+7 16 a 2 12a+2

Solution

16 a 2 18a17 2( 4a1 )( 4a+1 )( 2a1 ) 16 a 2 18a17 2( 4a1 )( 4a+1 )( 2a1 )

Exercise 62

7y+4 6 y 2 32y+32 + 6y10 2 y 2 18y+40 7y+4 6 y 2 32y+32 + 6y10 2 y 2 18y+40

Exercise 63

x 2 x12 x 2 3x+2 · x 2 +3x4 x 2 3x18 x 2 x12 x 2 3x+2 · x 2 +3x4 x 2 3x18

Solution

( x+4 )( x4 ) ( x2 )( x6 ) ( x+4 )( x4 ) ( x2 )( x6 )

Exercise 64

y 2 1 y 2 +9y+20 ÷ y 2 +5y6 y 2 16 y 2 1 y 2 +9y+20 ÷ y 2 +5y6 y 2 16

Exercise 65

( r+3 ) 4 · r+4 ( r+3 ) 3 ( r+3 ) 4 · r+4 ( r+3 ) 3

Solution

( r+3 )( r+4 ) ( r+3 )( r+4 )

Exercise 66

( b+5 ) 3 · ( b+1 ) 2 ( b+5 ) 2 ( b+5 ) 3 · ( b+1 ) 2 ( b+5 ) 2

Exercise 67

( x7 ) 4 ÷ ( x7 ) 3 x+1 ( x7 ) 4 ÷ ( x7 ) 3 x+1

Solution

( x7 )( x+1 ) ( x7 )( x+1 )

Exercise 68

( 4x+9 ) 6 ÷ ( 4x+9 ) 2 ( 3x+1 ) 4 ( 4x+9 ) 6 ÷ ( 4x+9 ) 2 ( 3x+1 ) 4

Exercise 69

5x+ 2 x 2 +1 x4 5x+ 2 x 2 +1 x4

Solution

7 x 2 20x+1 ( x4 ) 7 x 2 20x+1 ( x4 )

Exercise 70

2y+ 4 y 2 +5 y1 2y+ 4 y 2 +5 y1

Exercise 71

y 2 +4y+4 y 2 +10y+21 ÷( y+2 ) y 2 +4y+4 y 2 +10y+21 ÷( y+2 )

Solution

( y+2 ) ( y+3 )( y+7 ) ( y+2 ) ( y+3 )( y+7 )

Exercise 72

2x3+ 4 x 2 +x1 x1 2x3+ 4 x 2 +x1 x1

Exercise 73

3x+1 x 2 +3x+2 + 5x+6 x 2 +6x+5 3x7 x 2 2x35 3x+1 x 2 +3x+2 + 5x+6 x 2 +6x+5 3x7 x 2 2x35

Solution

5 x 3 26 x 2 192x105 ( x 2 2x35 )( x+1 )( x+2 ) 5 x 3 26 x 2 192x105 ( x 2 2x35 )( x+1 )( x+2 )

Exercise 74

5a+3b 8 a 2 +2ab b 2 3ab 4 a 2 9ab+2 b 2 a+5b 4 a 2 +3ab b 2 5a+3b 8 a 2 +2ab b 2 3ab 4 a 2 9ab+2 b 2 a+5b 4 a 2 +3ab b 2

Exercise 75

3 x 2 +6x+10 10 x 2 +11x6 + 2 x 2 4x+15 2 x 2 11x21 3 x 2 +6x+10 10 x 2 +11x6 + 2 x 2 4x+15 2 x 2 11x21

Solution

13 x 3 39 x 2 +51x100 ( 2x+3 )( x7 )( 5x2 ) 13 x 3 39 x 2 +51x100 ( 2x+3 )( x7 )( 5x2 )

Rational Equations ((Reference))

For the following problems, solve the rational equations.

Exercise 76

4x 5 + 3x1 15 = 29 25 4x 5 + 3x1 15 = 29 25

Exercise 77

6a 7 + 2a3 21 = 77 21 6a 7 + 2a3 21 = 77 21

Solution

a=4 a=4

Exercise 78

5x1 6 + 3x+4 9 = 8 9 5x1 6 + 3x+4 9 = 8 9

Exercise 79

4y5 4 + 8y+1 6 = 69 12 4y5 4 + 8y+1 6 = 69 12

Solution

y=2 y=2

Exercise 80

4 x1 + 7 x+2 = 43 x 2 +x2 4 x1 + 7 x+2 = 43 x 2 +x2

Exercise 81

5 a+3 + 6 a4 = 9 a 2 a12 5 a+3 + 6 a4 = 9 a 2 a12

Solution

a=1 a=1

Exercise 82

5 y3 + 2 y3 = 3 y3 5 y3 + 2 y3 = 3 y3

Exercise 83

2m+5 m8 + 9 m8 = 30 m8 2m+5 m8 + 9 m8 = 30 m8

Solution

No solution; m=8 m=8 is excluded.

Exercise 84

r+6 r1 3r+2 r1 = 6 r1 r+6 r1 3r+2 r1 = 6 r1

Exercise 85

8b+1 b7 b+5 b7 = 45 b7 8b+1 b7 b+5 b7 = 45 b7

Solution

No solution; b=7 b=7 is excluded.

Exercise 86

Solve z= x x ¯ x  for s. z= x x ¯ x  for s.

Exercise 87

Solve A=P( 1+rt ) for t. A=P( 1+rt ) for t.

Solution

t= AP Pr t= AP Pr

Exercise 88

Solve 1 R = 1 E + 1 F  for E. 1 R = 1 E + 1 F  for E.

Exercise 89

Solve Q= 2mn s+t  for t. Q= 2mn s+t  for t.

Solution

t= 2mnQs Q t= 2mnQs Q

Exercise 90

Solve I= E R+r  for r. I= E R+r  for r.

Applications ((Reference))

For the following problems, find the solution.

Exercise 91

When the same number is subtracted from both terms of the fraction 7 12 , 7 12 , the result is 1 2 . 1 2 . What is the number?

Solution

2

Exercise 92

When the same number is added to both terms of the fraction 13 15 , 13 15 , the result is 8 9 . 8 9 . What is the number?

Exercise 93

When three fourths of a number is added to the reciprocal of the number, the result is 173 16 . 173 16 . What is the number?

Solution

No rational solution.

Exercise 94

When one third of a number is added to the reciprocal of the number, the result is 127 90 . 127 90 . What is the number?

Exercise 95

Person A working alone can complete a job in 9 hours. Person B working alone can complete the same job in 7 hours. How long will it take both people to complete the job working together?

Solution

3 15 16  hrs 3 15 16  hrs

Exercise 96

Debbie can complete an algebra assignment in 3 4 3 4 of an hour. Sandi, who plays her radio while working, can complete the same assignment in 1 1 4 1 1 4 hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?

Exercise 97

An inlet pipe can fill a tank in 6 hours and an outlet pipe can drain the tank in 8 hours. If both pipes are open, how long will it take to fill the tank?

Solution

24 hrs

Exercise 98

Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?

Exercise 99

The pressure due to surface tension in a spherical bubble is given by P= 4T r , P= 4T r , where T T is the surface tension of the liquid, and r r is the radius of the bubble.
(a) Determine the pressure due to surface tension within a soap bubble of radius 1 2 1 2 inch and surface tension 22.
(b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18.

Solution

(a) 176 units of pressure; (b) 5 4 5 4 units of length

Exercise 100

The equation 1 p + 1 q = 1 f 1 p + 1 q = 1 f relates an objects distance p p from a lens and the image distance q q from the lens to the focal length f f of the lens.
(a) Determine the focal length of a lens in which an object 8 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 10 inches and the image distance is 10 inches.
(c) Determine how far an object will be from a lens that has a focal length of 1 7 8 1 7 8 cm and the object distance is 3 cm away from the lens.

Dividing Polynomials ((Reference))

For the following problems, divide the polynomials.

Exercise 101

a 2 +9a+18 a 2 +9a+18 by a+3 a+3

Solution

a+6 a+6

Exercise 102

c 2 +3c88 c 2 +3c88 by c8 c8

Exercise 103

x 3 +9 x 2 +18x+28 x 3 +9 x 2 +18x+28 by x+7 x+7

Solution

x 2 +2x+4 x 2 +2x+4

Exercise 104

y 3 2 y 2 49y6 y 3 2 y 2 49y6 by y+6 y+6

Exercise 105

m 4 +2 m 3 8 m 2 m+2 m 4 +2 m 3 8 m 2 m+2 by m2 m2

Solution

m 3 +4 m 2 1 m 3 +4 m 2 1

Exercise 106

3 r 2 17r27 3 r 2 17r27 by r7 r7

Exercise 107

a 3 3 a 2 56a+10 a 3 3 a 2 56a+10 by a9 a9

Solution

a 2 +6a2 8 a9 a 2 +6a2 8 a9

Exercise 108

x 3 x+1 x 3 x+1 by x+3 x+3

Exercise 109

y 3 + y 2 y y 3 + y 2 y by y+4 y+4

Solution

y 2 3y+11 44 y+4 y 2 3y+11 44 y+4

Exercise 110

5 x 6 +5 x 5 2 x 4 +5 x 3 7 x 2 8x+6 5 x 6 +5 x 5 2 x 4 +5 x 3 7 x 2 8x+6 by x 2 +x1 x 2 +x1

Exercise 111

y 10 y 7 +3 y 4 3y y 10 y 7 +3 y 4 3y by y 4 y y 4 y

Solution

y 6 +3 y 6 +3

Exercise 112

4 b 7 3 b 6 22 b 5 19 b 4 +12 b 3 6 b 2 +b+4 4 b 7 3 b 6 22 b 5 19 b 4 +12 b 3 6 b 2 +b+4 by b 2 +6 b 2 +6

Exercise 113

x 3 +1 x 3 +1 by x+1 x+1

Solution

x 2 x+1 x 2 x+1

Exercise 114

a 4 +6 a 3 +4 a 2 +12a+8 a 4 +6 a 3 +4 a 2 +12a+8 by a 2 +3a+2 a 2 +3a+2

Exercise 115

y 10 +6 y 5 +9 y 10 +6 y 5 +9 by y 5 +3 y 5 +3

Solution

y 5 +3 y 5 +3

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