Pearson Educationwps.prenhall.com/wps/media/objects/708/725216/ssm/ch8_1.pdf223 Chapter 8 Exercise Set 8.1 1. Answers will vary. 3. The value of the variable does not make the denominator

223

Chapter 8Exercise Set 8.1

1. Answers will vary.

3. The value of the variable does not make thedenominator equal to 0.

5. There is no factor common to both the

numerator and denominator of

†

2 + 3x4

.

7. The denominator cannot be 0.

9. x!– 2 = 0x ≠ 2

†

11. - x + 55 - x

=x + 5

-(5- x)

=x + 5

-5+ x

=x + 5x - 5

†

x + 5x - 5

≠ -1

No

13. The expression is defined for all real numbersexcept x = 0.

†

15. 4n - 12 = 04n = 12

n = 3The expression is defined for all real numbersexcept n = 3.

17. x2 - 4 = 0x - 2( ) x + 2( ) = 0

The expression is defined for all real numbersexcept x = 2, x = –2.

†

19. 2 x 2 - 9x + 9 = 02x - 3( ) x - 3( ) = 0

The expression is defined for all real numbers

†

except x =32

, x = 3.

21. All real numbers because x2 + 16 ≠ 0.

†

23. 4 p2 - 25 = 04 p2 = 25

p2 =254

p = ±52

The expression is defined for all real numbers

†

except p = ±52

.

†

25. 7x3y

21x2 y5=

721

⋅ x3- 2 ⋅ y1- 5

=13

xy -4

=x

3y4

†

27. 2a4b5( )32a12b20

=23 ⋅ a4( 3) ⋅b 5(3)

2a12b20

=8a12b15

2a12b20

= 4a12 -12b15- 20

= 4a0b-5

=4

b5

29.x

x + xy=

xx 1 + y( )

=1

1+ y

31. 5x +15x + 3

=5 x + 3( )

x + 3= 5

33. x3 + 6x2 + 3x

2x=

x x2 + 6x + 3( )2x

=x 2 + 6x + 3

2

†

35. r2 - r - 2r + 1

=r - 2( ) r + 1( )

r + 1= r - 2

Chapter 8: Rational Expressions and Equations SSM: Elementary Algebra/Early Graphing

224

37. x2 + 2x

x2 + 4x + 4=

x x + 2( )x + 2( )2

=x

x + 2

†

39. k2 - 6k + 9k2 - 9

=k - 3( ) k - 3( )k - 3( ) k + 3( )

=k - 3k + 3

41. x2 - 2x - 3x2 - x - 6

=(x + 1)(x - 3)(x + 2)(x - 3)

=x + 1x + 2

43. 2x - 33 - 2x

=2x - 3

- 2x - 3( )= -1

45. x2 - 2x - 8

4 - x=

x - 4( ) x + 2( )- x - 4( )

= - x + 2( )

47. x2 + 3x - 18-2x 2 + 6x

=x + 6( ) x - 3( )-2x x - 3( )

= -x + 62x

49. 2x2 + 5x - 31- 2x

=2x - 1( ) x + 3( )

- 2x - 1( )= - x + 3( )

†

51. m - 24m2 -13m + 10

=m - 2

4m - 5( ) m - 2( )=

14m - 5

53. x2 - 25

x + 5( )2=

x - 5( ) x + 5( )x + 5( )2

=x - 5x + 5

55. 6x2 - 13x + 63x - 2

=3x - 2( ) 2x - 3( )

3x - 2= 2x - 3

57.x2 - 3x + 4x - 12

x - 3=

x x - 3( ) + 4 x - 3( )x - 3

=x + 4( ) x - 3( )

x - 3= x + 4

59. 2x2 - 8x + 3x - 12

2x2 + 8x + 3x + 12=

2x x - 4( ) + 3 x - 4( )2x x + 4( ) + 3 x + 4( )

=x - 4( ) 2x + 3( )x + 4( ) 2x + 3( )

=x - 4x + 4

†

61. a3 - 8

a - 2=

a - 2( ) a2 + 2a + 4( )a - 2

= a2 + 2a + 4

†

63. 9s2 -16 t 2

3s - 4 t=

3s + 4t( ) 3s - 4 t( )3s - 4t

= 3s + 4 t

†

65. 4x + 6y2x 2 + xy - 3y 2

=2(2x + 3y)

2x + 3y( ) x - y( ) =2

x - y

67.

†

3J12

=3J3 ⋅ 4

=J4

69. 7D14D + 21

=7D

7 2D + 3( )=

D2D + 3

71. 3D - 22 - 3D

=- 2 - 3D( )

2 - 3D( )= -1

73. x2 - x - 6 = x - 3( ) x + 2( )Denominator = x!+ 2

75. x + 3( ) x + 4( ) = x 2 + 7x + 12Numerator = x2 + 7x + 12

77. a.x + 3

x2 - 2x + 3x - 6=

x + 3x x - 2( ) + 3 x - 2( )

=x + 3

x + 3( ) x - 2( )x!≠ –3, x!≠ 2

b. x + 3x + 3( ) x - 2( )

=1

x - 2

SSM: Elementary Algebra/Early Graphing Chapter 8: Rational Expressions and Equations

225

79. a.x + 5

2x3 + 7x2 -15 x=

x + 5x 2x2 + 7x - 15( )

=x + 5

x 2x - 3( ) x + 5( )

x ≠ 0, x ≠32

, x ≠ -5

b. x + 5x 2x - 3( ) x + 5( )

=1

x 2x - 3( )

81.15 x

5 - 23 x4

15 x

5 - 23 x4 = 1

84. z = x - y2

2z = x - y2z - x = -y

y = x - 2z

85. Let x = measure of the smallest angle. Then thesecond angle = x + 30 and third angle = 3x!+ 10.angle 1 + angle 2 + angle 3 = 180°x + x + 30( ) + 3x + 10( ) = 180

5x + 40 = 1805x = 140x = 28

x + 30 = 28 + 30 = 583x + 10 = 3 28( ) +10 = 84 + 10 = 94 . The threeangles are 28°, 58°, and 94°.

†

86. 4x2y 2

9x4 y 3Ê

Ë Á

ˆ

¯ ˜

2

=4

9x 2yÊ

Ë Á

ˆ

¯ ˜

2

=16

81x 4 y2

†

87. 3x2 - 4x - 8 - -3x2 + 6x + 9( ) = 3x 2 - 4x - 8 + 3x 2 - 6x - 9 = 6x2 -10x - 17

†

88. 3a2 - 30a + 72 = 3(a2 -10a + 24)= 3(a - 4)(a - 6)

†

89. a2 + b2 = c2

52 + 122 = c2

25+ 144 = c2

169 = c2

169 = c 2

13 = cThe hypotenuse is 13 inches long.

Exercise Set 8.2


3. x + 3x - 4

⋅x + 3

= x + 2

Numerator must be x + 2( ) x - 4( ) = x2 - 2x - 8

5. x - 4x + 5

⋅x + 5

=1

x + 3Denominator must bex + 3( ) x - 4( ) = x 2 - x - 12

†

7. 5x4y

⋅y2

10=

5x4 y

⋅y2

5 ⋅ 2

=xy8

9. 16x2

y4⋅5x 2

y2=

80x4

y6

11. 6x5y3

5z3⋅

6x 4

5yz4=

36x9y2

25z7

13.3x - 23x + 2

⋅4x - 11- 4x

=3x - 23x + 2

⋅4x - 1

- 4x - 1( )

=-3x + 23x + 2


226

15. x2 + 7x + 12

x + 4⋅

1x + 3

=x + 4( ) x + 3( )x + 4( ) x + 3( )

= 1

17. aa2 - b2

⋅a + b

a2 + ab=

a(a + b)(a + b)(a - b) ⋅ a(a + b)

=1

a - b( ) a + b( )

=1

a2 - b2

19. 6x2 - 14x -12

6x + 4⋅

x + 32x2 - 2x - 12

=2 3x2 - 7x - 6( )

2 3x + 2( )⋅

x + 32 x2 - x - 6( )

=2 3x + 2( ) x - 3( ) x + 3( )

2 3x + 2( ) ⋅ 2 x - 3( ) x + 2( )

=x + 3

2 x + 2( )

†

21. 3x2 -13x -10

x 2 - 2x -15⋅

x 2 + x - 23x2 - x - 2

=3x + 2( ) x - 5( )x + 3( ) x - 5( )

⋅x + 2( ) x - 1( )

3x + 2( ) x - 1( )

=x + 2x + 3

23. x + 3x - 3

⋅x3 - 27

x2 + 3x + 9=

x + 3( ) x - 3( ) x2 + 3x + 9( )x - 3( ) x2 + 3x + 9( )

= x + 3

25. 9x3

y2÷

3xy3

=9x3

y2⋅

y3

3x

=3 ⋅ 3x3

y2⋅y3

3x= 3x 2y

†

27. 15xy2

4z÷

5x2 y2

12z 2=

3 ⋅ 5xy 2

4z⋅

12z 2

5x2 y2

=3 ⋅ 5xy 2

4z⋅3 ⋅ 4z 2

5x2 y2

=9zx

†

29. 5xy7ab2

÷6xy7

=5xy7ab2

⋅7

6xy

=5

6ab2

†

31. 10r + 5r

÷2r + 1

r 2=

5 2r + 1( )r

⋅r 2

2r + 1= 5r

†

33. x2 + 5x - 14

x÷

x - 2x

=x - 2( ) x + 7( )

x⋅

xx - 2( )

= x + 7


227

35. x2 - 12x + 32

x 2 - 6x - 16÷

x 2 - x -12x 2 - 5x - 24

=x2 - 12x + 32x2 - 6x -16

⋅x2 - 5x - 24x2 - x - 12

=x - 8( ) x - 4( )x - 8( ) x + 2( )

⋅x - 8( ) x + 3( )x - 4( ) x + 3( )

=x - 8x + 2

37. 2x2 + 9x + 4

x 2 + 7x + 12÷

2x2 - x - 1x + 3( )2

=2x + 1( ) x + 4( )x + 3( ) x + 4( )

⋅x + 3( )2

2x + 1( ) x -1( )

=x + 3x - 1

39. x2 - y2

x2 - 2xy + y2÷

x + yy - x

=x - y( ) x + y( )

x - y( )2⋅-(x - y)

x + y= -1

†

41. 5x2 - 4x -1

5x2 + 6x + 1÷

x2 - 5x + 4x 2 + 2x + 1

=5x + 1( ) x - 1( )5x + 1( ) x + 1( )

⋅x + 1( ) x + 1( )x - 4( ) x - 1( )

=x + 1x - 4

43. 9x6y2

⋅24x2y4

9x= 4x2y2

†

45. 63a2b3

16c3⋅

4c 4

9a3b5=

4 ⋅ 7 ⋅ 9a2b3c4

9 ⋅ 4 ⋅ 4a3b5c 3=

7c4ab2

47.- xy

a÷

-2ax6y

=-xya

⋅6y

-2ax

=3y2

a2

†

49. 100m6

21x5 y7⋅14 x12y 5

25m5=

5 ⋅ 5 ⋅ 4m6

3 ⋅ 7x5 y7⋅2 ⋅ 7x12y 5

5 ⋅ 5m5

=8mx 7

3y 2

51. 3x + 5( ) ⋅1

6x + 10= 3x + 5( ) ⋅

12 3x + 5( )

=12

†

53. 14x2 y2

÷1

28x3 y=

14x 2y 2

⋅4 ⋅ 7x3 y

1

=7xy

†

55. 4m( )2

8n3÷

m6n8

4=

16m2

8n3⋅

4m6n8

=8 ⋅ 2m2

8n3⋅

4m6n8

=8

m4 n11

†

57. r2 + 5r + 6

r 2 + 9r + 18⋅r 2 + 4r - 12r 2 - 5r + 6

=r + 2( ) r + 3( )r + 6( ) r + 3( )

⋅r + 6( ) r - 2( )r - 3( ) r - 2( )

=r + 2r - 3


228

†

59. x2 - 10x + 24

x 2 - 8x + 12÷

x 2 - 7x + 12x2 - 6x + 8

=x2 -10x + 24x2 - 8x + 12

⋅x2 - 6x + 8

x 2 - 7x + 12

=x - 6( ) x - 4( )x - 6( ) x - 2( )

⋅x - 4( ) x - 2( )x - 4( ) x - 3( )

=x - 4x - 3

†

61. 3z2 - 4z - 4z 2 - 4

⋅2z 2 + 5z + 22z 2 - 3z - 2

=3z + 2( ) z - 2( )z + 2( ) z - 2( )

⋅2z + 1( ) z + 2( )2z + 1( ) z - 2( )

=3z + 2z - 2

†

63. 2x2 - 19x + 24

x2 -12x + 32÷

2x 2 + x - 6x2 + 7x + 10

=2x2 -19x + 24x 2 - 12x + 32

⋅x 2 + 7x + 102x2 + x - 6

=2x - 3( ) x - 8( )x - 4( ) x - 8( )

⋅x + 5( ) x + 2( )

2x - 3( ) x + 2( )=

x + 5x - 4

†

65. 4n2 - 9

9n2 - 1⋅

3n2 - 2n - 12n2 - 5n + 3

=2n + 3( ) 2n - 3( )3n + 1( ) 3n - 1( )

⋅3n + 1( ) n - 1( )2n - 3( ) n -1( )

=2n + 33n -1

67. 6D2

12⋅

1236D5

=6D2

12⋅

126 ⋅ 6D5

=1

6D3

71. x + 2( ) x +1( ) = x2 + x + 2x + 2= x2 + 3x + 2

Numerator is x2 + 3x + 2.

73. x - 5( ) x + 2( ) = x 2 + 2x - 5x - 10= x 2 - 3x - 10

Numerator is x2 - 3x - 10 .

75. x2 - 4( ) x - 1( ) ÷ x + 2( )x - 2( ) x + 2( ) x - 1( )⋅ 1

x + 2= x 2 - x - 2x + 2

= x 2 - 3x + 2Numerator is x2 - 3x + 2 .

77. x + 2x2 - 4x - 12

⋅x2 - 9x + 18

x - 2Ê

Ë Á

ˆ

¯ ˜ ÷

x2 + 5x + 6x2 - 4

=x + 2

x - 6( ) x + 2( )⋅

x - 3( ) x - 6( )x - 2

⋅x + 2( ) x - 2( )x + 2( ) x + 3( )

=x - 3x + 3

79. x2 + 4x + 3x2 - 6x - 16

Ê

Ë Á

ˆ

¯ ˜ ÷

x 2 + 5x + 6x2 - 9x + 8

⋅x 2 - 1

x 2 + 4x + 4Ê

Ë Á

ˆ

¯ ˜ =

x2 + 4x + 32

x2 - 6x - 16⋅x 2 - 9x + 8x 2 + 5x + 6

⋅x2 + 4x + 4

x 2 - 1

=x + 1( ) x + 3( )x - 8( ) x + 2( )

⋅x - 8( ) x - 1( )x + 2( ) x + 3( )

x + 2( )2

x - 1( ) x + 1( )= 1


229

†

81. x - 3( ) x - 2( )x + 4( ) x - 5( )

⋅x + 4( ) x -1( )x - 3( ) x -1( )

=x - 2x - 5

The numerator is x - 3( ) x - 2( ) = x2 - 5x + 6. The denominator is x + 4( ) x - 5( ) = x2 - x - 20.

84. Let x = the time it takes the tug boat to reachthe barge.Then x + 2 = the time it takes the tug boat toreturn to the dock.

Rate Time Distance

Trip Out 15 x 15(x)

Return Trip 5 x + 2 5(x + 2)

Distance to barge = Distance back to dock15(x) = 5(x + 2) 15x = 5x + 10 10x = 10 x = 1It took 1 hour for the tug boat to reach the barge.

85. 4x 3y2z4( ) 5xy3z7( ) = 4 ⋅5 ⋅ x3xy2y3z4z7= 20x4y5z11

86. 2x - 1 4x3 + 0x2 - 5x + 04x3 – 2x2

2x2 - 5x2x2 – x

- 4x + 0-4x + 2

- 2

2x2 + x - 2)

4x3 - 5x2x - 1

= 2x 2 + x - 2 -2

2x - 1

87. 3x 2 - 9x - 30 = 3 x 2 - 3x -10( )= 3 x - 5( ) x + 2( )

88. 3x2 - 9x - 30 = 03 x2 - 3x -10( ) = 03 x - 5( ) x + 2( ) = 0x - 5 = 0 or x + 2 = 0

x = 5 x = -2

Exercise Set 8.3



†

5. 5x + 6 -

2x

The only factor (other than 1) of the firstdenominator is x + 6 . The only factor (otherthan 1) of the second denominator is x . The

†

LCD is therefore x(x + 6).

7. 2x + 3

+1x

+13

The only factor (other than 1) of the firstdenominator is x!+ 3. The only factor (other than1) of the second denominator is x. The onlyfactor (other than 1) of the third denominator is3. The LCD is therefore 3x x + 3( ) .

9. a. The negative sign in –(2x – 7) was notdistributed.

b. 4x - 35x + 4

-2x - 75x + 4

=4x - 3 - 2x - 7( )

5x + 4=

4x - 3 - 2x + 75x + 4

≠4x - 3 - 2x - 7

5x + 4

†

10. a. The negative sign in - -3x - 7( ) was not distributed.


230

†

b. 5x2x - 3

--3x - 72x - 3

=5x - -3x - 7( )

2x - 3

=5x + 3x + 7

2x - 3

≠5x + 3x - 7

2x - 3

11. a. The negative sign in - 3x2 - 4x + 5( ) was not distributed.

b. 6x - 2x2 - 4x + 3

-3x2 - 4x + 5x 2 - 4x + 3

=6x - 2 - 3x2 - 4x + 5( )

x2 - 4x + 3

=6x - 2 - 3x 2 + 4x - 5

x2 - 4x + 3

≠6x - 2 - 3x 2 - 4x + 5

x2 - 4x + 3

†

13. x - 27

+2x7

=x - 2 + 2x

7

=3x - 2

7

†

15. 3r + 24

-34

=3r + 2 - 3

4

=3r - 1

4

17. 2x

+x + 4

x=

2 + x + 4x

=x + 6

x

†

19. n - 5n

-n + 7

n=

n - 5 - n - 7n

= -12n

21. xx -1

+4x + 7x - 1

=x + 4x + 7

x - 1=

5x + 7x - 1

†

23. 4t + 75t 2

-3t + 4

5t 2=

4t + 7 - 3t + 4( )5t 2

=4t + 7 - 3t - 4

5t 2

=t + 35t 2

25.5x + 4

x2 - x - 12+

-4x - 1x 2 - x - 12

=5x + 4 - 4x - 1

x2 - x - 12=

x + 3x + 3( ) x - 4( )

=1

x - 4

27. x + 43x + 2

-x + 4

3x + 2=

x + 4 - x + 4( )3x + 2

= 0

†

29. 2p- 5p - 5 -

p + 5p - 5

=2p - 5- p + 5( )

p - 5

=2p - 5- p - 5

p - 5

=p -10p - 5

31. x2 + 4x + 3

x + 2-

5x + 9x + 2

=x 2 + 4x + 3- 5x + 9( )

x + 2

=x 2 + 4x + 3- 5x - 9

x + 2

=x 2 - x - 6

x + 2

=x - 3( ) x + 2( )

x + 2= x - 3


231

†

33. 3x + 112x + 10

-2 x + 3( )2x + 10

=3x + 11 - 2 x + 3( )

2x + 10

=3x + 11 - 2x - 6

2x + 10

=x + 5

2 x + 5( )

=12

†

35. b2 - 2b - 3

b2 - b - 6+

b - 3b2 - b - 6

=b2 - 2b - 3 + b - 3

b2 - b - 6

=b2 - b - 6b2 - b - 6

= 1

†

37. t - 3t + 3

--3t -15

t + 3=

t - 3 - -3t - 15( )t + 3

=t - 3 + 3t + 15

t + 3

=4 t + 12t + 3

=4 t + 3( )

t + 3= 4

39. x2 + 2x

x + 6( ) x - 3( )-

15x + 6( ) x - 3( )

=x2 + 2x - 15x + 6( ) x - 3( )

=(x + 5)(x - 3)(x + 6)(x - 3)

=x + 5x + 6

41.3x2 - 7x4x2 - 8x

+x

4x2 - 8x=

3x2 - 7x + x4x2 - 8x

=3x2 - 6x4x2 - 8x

=3x x - 2( )4x x - 2( )

=34

43.3x 2 - 4x + 43x2 + 7x + 2

-10x + 9

3x2 + 7x + 2=

3x2 - 4x + 4 - 10x + 9( )3x2 + 7x + 2

=3x2 - 4x + 4 -10 x - 9

3x2 + 7x + 2

=3x2 - 14x - 53x2 + 7x + 2

=3x + 1( ) x - 5( )3x + 1( ) x + 2( )

=x - 5x + 2

45. x2 + 3x - 6

x2 - 5x + 4-

-2x2 + 4x - 4x2 - 5x + 4

=x2 + 3x - 6 - (-2x2 + 4x - 4)

x 2 - 5x + 4

=x2 + 3x - 6 + 2x2 - 4x + 4

x 2 - 5x + 4

=3x2 - x - 2x2 - 5x + 4

=3x + 2( ) x - 1( )x - 4( ) x - 1( )

=3x + 2x - 4


232

47. 5x2 + 40x + 8x 2 - 64

+x2 + 9xx2 - 64

=5x 2 + 40x + 8 + x2 + 9x

x 2 - 64

=6x2 + 49x + 8

x2 - 64

=6x + 1( ) x + 8( )x - 8( ) x + 8( )

=6x +1x - 8

†

49. x5

+x + 4

5Least common denominator = 5

†

51. 1n

+15n

Least common denominator = 5n

53. 35x

+74

Least common denominator = 5x ⋅ 4 = 20x

†

55. 6p

+3p3

Least common denominator = p3

†

57. m + 33m - 4

+ m =m + 33m - 4

+m1

Least common denominator = 3m - 4

†

59. x2x + 3

+4x 2

Least common denominator = x 2 2x + 3( )

61. x + 112x2 y

-7

9x3=

x + 13 ⋅ 4x2y

-7

32 x3Least common denominator= 4 ⋅ 32 ⋅ x3 ⋅ y = 36x3y

†

63. 42r 4s 5

-5

9r 3s 7=

42r 4s 5

-5

3 ⋅ 3r 3s7 Least common denominator = 2 ⋅ 3 ⋅ 3r 4 s7 = 18r 4 s7

65. x2 - 724x

-x + 3

9 x + 5( )=

x2 - 73 ⋅ 8x

-x + 3

32(x + 5)Least common denominator= 8 ⋅ 32 x x + 5( ) = 72x x + 5( )

67. 5x - 2x2 + x

-x 2

x=

5x - 2x(x + 1)

-x 2

xLeast common denominator = x x + 1( )

†

69. n4n - 1 +

n - 21 - 4n

=n

4n -1 +-1( ) n - 2( )-1( ) 1- 4n( )

=n

4n -1 +-n + 24n - 1

†

Least common denominator = 4 n -1

†

71. 64k - 5r

-5

-4k + 5r=

64k - 5r

--1( )5

-1( ) -4k + 5r( )

=6

4k - 5r-

-54k - 5r

Least common denominator = 4k – 5r

†

73. 52q2 + 2q

-53q

=5

2q q + 1( ) -5

3q

†

Least common denominator = 2 ⋅ 3q q + 1( )= 6q q + 1( )

75. 2124x2y

+x + 4

15xy3=

213 ⋅8 x2y

+x + 4

3 ⋅ 5xy3Least common denominator= 8 ⋅ 3 ⋅ 5x2y3 = 120x2y3

77. 33x + 12

+3x + 62x + 4

=3

3(x + 4)+

3x + 62(x + 2)

Least common denominator= 3 ⋅ 2 x + 4( ) x + 2( ) = 6 x + 4( ) x + 2( )

79. 9x + 4x + 6

-3x - 6x + 5

Least common denominator = x + 6( ) x + 5( )


233

81. x - 2x2 - 5x - 24

+3

x2 + 11x + 24=

x - 2(x - 8)( x + 3)

+3

(x + 8)(x + 3)Least common denominator= x - 8( ) x + 3( ) x + 8( )

†

83. 7a - 4( )2

-a + 2

a2 - 7a + 12

=7

a - 4( )2-

a + 2(a - 4)(a - 3)

Least common denominator

†

= a - 4( )2 a - 3( )

85. 2xx2 + 6x + 5

-5x2

x2 + 4x + 3

=2x

(x + 5)(x + 1)-

5x2

(x + 3)(x + 1)Least common denominator= x + 5( ) x +1( ) x + 3( )

87. 3x - 5x2 - 6x + 9

+3

x - 3=

3x + 5(x - 3)2

+3

x - 3Least common denominator = x - 3( )2

89. 8x2

x2 - 7x + 6+ x - 3

=8x2

(x - 6)(x - 1)+

x - 31

Least common denominator = x - 6( ) x - 1( )

†

91. t -13t 2 + 10t - 8

-6

3t 2 + 11t - 4

=t -1

(3t - 2)(t + 4)-

6(3t -1)(t + 4)

Least common denominator

†

= 3t - 2( ) t + 4( ) 3t -1( )

93. 2x - 34x2 + 4x + 1

+x2 - 4

8x2 + 10x + 3

=2x - 3

(2 x + 1)2+

x2 - 4(2x + 1)(4x + 3)

Least common denominator= 2x + 1( )2 4x + 3( )

95. x2 - 6x + 3+ = 2x2 - 5x - 6= 2x2 - 5x - 6 - (x2 - 6x + 3)= 2x2 - 5x - 6 - x2 + 6x - 3= x 2 + x - 9

Sum of numerators must be 2x2 - 5x - 6

97. - x2 - 4x + 3 + = 5x - 7= 5x - 7 - (-x2 - 4x + 3)= 5x - 7 + x2 + 4x - 3= x 2 + 9x - 10

Sum of numerator must be 5x!– 7

98. -3x2 - 9 - = x2 + 3x= -3x2 - 9 - (x 2 + 3x)= -3x2 - 9 - x2 - 3x= -4x 2 - 3x - 9

Difference of numerators must be x2 + 3x

99.

†

3J +

45J

Least common denominator = 5J

101. 8D2 - 9

-2

D + 3=

8(D + 3)(D - 3)

-2

D + 3Least common denominator = D + 3( ) D - 3( )

103.4x - 1

x2 - 25-

3x2 - 8x2 - 25

+8x - 3x2 - 25

=4x - 1- 3x2 - 8( ) + 8x - 3( )

x 2 - 25

=4x - 1- 3x2 + 8+ 8x - 3

x2 - 25

=-3x2 +12 x + 4

x2 - 25


234

105. 76x5y9

-9

2x3y+

45x12y2

=7

2 ⋅3 x5y9-

92x3y

+4

5x12y2

Least common denominator = 2 ⋅ 3 ⋅ 5x12y9 = 30x12y9

107. 4x2 - x - 12

+3

x 2 - 6x + 8+

5x 2 + x - 6

=4

x - 4( ) x + 3( )+

3x - 4( ) x - 2( )

+5

x + 3( ) x - 2( )Least common denominator = x - 4( ) x + 3( ) x - 2( )

109. 4 35

- 259

=235

-239

=20745

-11545

=9245

= 2 245

110. 6x + 4 = - x + 2( ) - 3x + 46x + 4 = -x - 2 - 3x + 46x + 4 = -4x + 2

6x + 4x = 2 - 410x = -2

x = -210

= -15

111. 6128

=x

48128x = 6 ⋅ 48128x = 288

x = 288128

x =94

= 214

You should use 2.25 ounces of concentrate.

112. Let h = the number of hours played.The cost under Plan 1 is C = 125 + 2.5h whilethe cost under Plan 2 is C != 300.

Set the two costs equal125 + 2.5h = 300

2.5h = 175h = 70

If Malcolm plays 70 hours in a year, the cost ofthe two plans is equal.

113. Use the two points given and find the slopebetween the two points. This is the slope of theline.

†

m =y2 - y1x2 - x1

m =5- -2( )4 - -3( )

m = 77

= 1

The slope of the line is 1.

114.

†

4.2 ¥ 108

2.1¥ 10-3= 2.0 ¥ 108- -3( ) = 2.0 ¥ 1011

115. 2x2 - 3 = x2x2 - x - 3 = 0

(2x - 3)(x + 1) = 02x - 3 = 0 or x +1 = 0

2x = 3 x = -1x = 32

Exercise Set 8.4

1. For each fraction, divide the LCD by thedenominator.

3. a. Answers will vary.

b.

xx 2 - x - 6

+3

x2 - 4=

xx - 3( ) x + 2( )

+3

x - 2( ) x + 2( )

=x x - 2( )

x - 3( ) x + 2( ) x - 2( )+

3 x - 3( )x - 2( ) x + 2( ) x - 3( )

=x2 - 2x + 3x - 9

x - 3( ) x + 2( ) x - 2( )

=x 2 + x - 9

x - 3( ) x + 2( ) x - 2( )

Documents

Pearson Educationwps.prenhall.com/wps/media/objects/708/725216/ssm/ch8_1.pdf223 Chapter 8 Exercise Set 8.1 1. Answers will vary. 3. The value of the variable does not make the denominator