chapter 7: rational expressions and equations ssm: elementary...

Chapter 7: Rational Expressions and Equations SSM: Elementary and Intermediate Algebra

212

5. Let w = width, then23

w + 4 = length

area = width ·!length

90 = w23

w + 4Ê Ë Á ˆ

¯ ˜

90 =2w2

3+ 4w

3 90( ) = 32w2

3+ 4w

Ê

Ë Á

ˆ

¯ ˜

270 = 2w2 + 12w2w2 + 12w - 270 = 0w2 + 6w - 135 = 0w +15( ) w - 9( ) = 0

w + 15 = 0 or w - 9 = 0w = -15 w = 9

Since the width cannot be negative, w!= 9l =

23

w + 4

l =23

9( ) + 4l = 6 + 4 = 10The length = 10 inches and the width = 9 inches.

7. Let x = height, then x!+ 5 = basearea =

12

⋅ height · base

42 =12

x x + 5( )

2 42( ) = 2 12

x x + 5( )È Î Í

˘ ˚ ˙

84 = x x + 5( )84 = x 2 + 5x0 = x 2 + 5x - 840 = x - 7( ) x + 12( )

x - 7 = 0 or x + 12 = 0x = 7 x = -12

Since the height cannot be negative, x!=!7.base = x + 5 = 7 + 15 =!12The base is 12 cm and the height is 7 cm.

9. Let b = the base of the triangle, then

†

12

b - 1Ê Ë Á

ˆ ¯ ˜

is

the height.

†

A =12

bh

12 =12

b12

b - 1Ê Ë Á

ˆ ¯ ˜

12 =14

b2 -12

b

4 12( ) = 414

b2 -12

bÊ Ë Á

ˆ ¯ ˜

48 = b2 - 2b0 = b2 - 2b - 480 = b - 8( ) b + 6( )

†

b - 8 = 0 or b + 6 = 0b = 8 or b = -6

Since a length cannot be negative, the base is 8feet.

11. Let one number be x, then the other number is10x.

1x

-1

10x= 3

10x1x

-1

10xÊ Ë Á ˆ

¯ ˜ = 10x 3( )

10 - 1 = 30x9 = 30x

930

= x3

10= x

3 = 10xThe numbers are

310

and 3.

13. Let x = amount by which the numerator wasincreased.

3 + x4

=52

4 3 + x4

Ê Ë Á ˆ

¯ ˜ =

52

Ê Ë Á ˆ

¯ ˜ 4

3 + x = 10x = 7

The numerator was increased by 7.

15. Let r = speed of Creole Queen paddle boat.t =

dr

Time upstream = time downstream

SSM: Elementary and Intermediate Algebra Chapter 7: Rational Expressions and Equations

213

4r - 2

=6

r + 24 r + 2( ) = 6 r - 2( )

4r + 8 = 6r - 1220 = 2r10 = r

The boat’s speed in still water is 10 mph.

17. Let d = distance and

†

t =dr

.

Time going + Time returning =

†

52

.

†

d15

+d15

=52

2d15

=52

4d = 75

d =754

= 18.75

The trolley traveled 18.75 miles in one direction.

19. Let r be the speed of the propeller plane, then 4ris the speed of the jet.dr

= t

time by jet + time by propeller plane = 6 hr1600

4r+

500r

= 6400r

+300

r= 6

900r

= 6

r =9006

= 150

The speed of the propeller plane is 150 mph andthe speed of the jet is 600 mph.

21. Let d = distance traveled by car, then200 – d = distance traveled by train.t =

dr

time traveled by car + time traveled by train= total time

†

d40

+200 - d

120= 2.2

120d40

+200 - d

120Ê Ë Á

ˆ ¯ ˜ = 2.2 120( )

3d + 200 - d = 2642d = 64d = 32

200 - 32 = 168

She travels 32 miles by car and 168 miles bytrain.

23. Let d = the distance flown with the wind, then2800 – d = the time flown against the wind.time with wind =

d600

time against wind =2800 - d

500d

600+

2800 - d500

= 5

3000 d600

+2800 - d

500Ê Ë Á ˆ

¯ ˜ = 5 3000( )

5d + 16,800 - 6d = 15, 000d = 1800

time with wind =1800600

= 3

time against wind =1000500

= 2

It flew 3 hours at 600 mph and 2 hours at500 mph.

25. time at 30 ft/s = d

30

time at 20 ft/s =

†

d20

†

d20

=d30

+ 15

60d20

Ê Ë Á

ˆ ¯ ˜ =

d30

+ 15Ê Ë Á

ˆ ¯ ˜ 60

3d = 2d + 900d = 900

The boat traveled 900 feet in one direction.

27. Felicia’s rate =16

Reynaldo’s rate =18

t6

+t8

= 1

48t6

+t8

Ê Ë Á ˆ

¯ ˜ = 48 1( )

8t + 6t = 4814t = 48

t = 3 37

It will take them 337

hours.


214

29. input rate =12

output rate =13

t2

-t3

= 1

6 t2

-t3

Ê Ë Á ˆ

¯ ˜ = 6 1( )

3t - 2t = 6t = 6

It will take 6 hours.

31. Input rate for small hose

†

=15

Input rat for larger hose =1t

In 2 hours, the smaller hose does

†

25

of the filling

and the larger hose does

†

2t

.

†

25

+2t

= 1

5t25

+2t

Ê Ë Á

ˆ ¯ ˜ = 5t

2t + 10 = 5t10 = 3t

t =103

= 313

It would take the larger hose

†

313

hours

33. Rate for first backhoe =1

12Rate for second backhoe =

115

Work done by first =1

12⋅ 5 =

512

Work done by second =115

⋅ t =t

155

12+

t15

= 1

605

12+

t15

Ê Ë Á

ˆ ¯ ˜ = 1 ⋅ 60

25 + 4t = 604t = 35

t =354

= 8 34

It takes the smaller backhoe 834

days to finish

the trench.

35. Ken’s rate =14

Bettina’s rate =16

t6

+t + 3

4= 1

12t6

+t + 3

4Ê Ë Á ˆ

¯ ˜ = 12

2t + 3 t + 3( ) = 122t + 3t + 9 = 12

5t = 3

t =35

It will take them 35

hour or 36 minutes longer.

37. Rate of first skimmer =1

60Rate of second skimmer =

150

Rate of transfer =1

30t

60+

t50

-t

30= 1

300 t60

+t

50-

t30

Ê Ë Á ˆ

¯ ˜ = 300

5t + 6t - 10t = 300t = 300

It will take 300 hours to fill the tank.

39. Let x be the number.4x

+ 5x = 12

x4x

+ 5xÊ Ë Á ˆ

¯ ˜ = 12x

4 + 5x2 = 12x5x2 - 12x + 4 = 05x - 2( ) x - 2( ) = 0

5x - 2 = 0 or x - 2 = 0x =

25

x = 2

The numbers are 2 or 25


215

41. Ed’s rate =18

Samantha’s rate =14

p4

- 1 =p8

8p4

-1Ê Ë Á ˆ

¯ ˜ = 8

p8

Ê Ë Á ˆ

¯ ˜

2p - 8 = pp = 8

Each must pick 8 pints.

43. 12

x + 3( ) - 2x + 6( ) =12

x +32

- 2x - 6

=x2

-4x2

+32

-122

= -3x2

-92

44.

45. x2 - 14x + 48x2 - 5x - 24

÷2x2 - 13x + 62x2 + 5x - 3

=x2 - 14x + 48x2 - 5x - 24

⋅2x2 + 5x - 32x 2 - 13x + 6

=x - 6( ) x - 8( )x + 3( ) x - 8( )

⋅2x - 1( ) x + 3( )2x - 1( ) x - 6( )

= 1

46.x

6x2 - x - 15-

59x2 -12x - 5

=x

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

53x + 1( ) 3x - 5( )

=x 3x + 1( )

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

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

=3x 2 + x - (10x +15)

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

=3x2 + x -10x - 15

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

=3x2 - 9x - 15

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

Exercise Set 7.8

1. a. As one quantity increases, the otherincreases.

b. Answers will vary.

c. Answers will vary.

3. One quantity varies as a product of two or morequantities.

5. a. y decreases

b. inverse variation

b. direct variation

7. Direct

9. Inverse

11. Direct

13. Direct


216

15. Direct

17. Inverse

19. Direct

21. Inverse

23. Inverse

25. a. x = ky

b. Substitute 12 for y and 6 for k.x = 6(12)x = 72

27. a. y = kR

b. Substitute 180 for R and 1.7 for k.y = 1.7(180)y = 306

29. a. R =k

W

b. Substitute 160 for W and 8 for k.R =

8160

R =1

20= 0.05

31. a. A =kBC

b. Substitute 12 for B, 4 for C, and 3 for k.A =

3(12)4

=364

= 9

33. a. x = ky

b. To find k, substitute 12 for x and 3 for y.

†

12= k(3)123 = k

4 = kThus,

†

x = 4y .

Now substitute 5 for y.

†

x = 4(5) = 20

35. a. y = kR2

b. To find k, substitute 5 for y and 5 for R.5 = k(5)25 = k(25)

525

= k15

= k

Thus y =15

R2 .

Now substitute 10 for R.y =

15

(10)2 =15

(100) = 20

37. a. C =kJ

b. To find k, substitute 7 for C and 0.7 for J.7 =

k0.7

7(0.7) = k

4.9 = kThus C =

4.9J

.

Now substitute 12 for J.C =

4.912

ª 0.41

39. a. F =kM1M2

d

b. To find k, substitute 5 for M1, 10 for M2, 0.2for d, and 20 for F.

20 =k(5)(10)

0.220 = k(250 )

20250

= k0.08 = kThus F =

0.08M1M2d .

Now substitute 10 for M1, 20 for M2, and0.4 for d.F =

0.08(10)(20)0. 4 = 16

0. 4 = 40

41. a = kbk(2b) = 2(kb) = 2aIf b is doubled, a is doubled.

43. y =kx

k2x

=12

kx

Ê Ë Á ˆ

¯ ˜ =

12

y

If x is doubled, y is halved.


217

45.

†

F =km1m2

d 2

k 2m1( )m2

d2 =2km1m2

d2 = 2 ⋅km1m2

d2 = 2F

If

†

m1 is doubled, F is doubled.

47.

†

F =km1m2

d 2

k 2m1( ) 12 m2( )

d 2 =2 ⋅ 1

2 km1m2

d2 =1 ⋅km1m2

d2 = F

If

†

m1 is doubled and

†

m2 is halved, F isunchanged.

49.

†

F =km1m2

d 2

k 12 m1( ) 4m2( )

d2 =12 ⋅4 km1m2

d2 =

2 ⋅km1m2d 2 = 2 ⋅

km1m2d2 = 2F

If

†

m1 is halved and

†

m2 is quadrupled, F isdoubled.

51. Notice that as x gets bigger, y gets smaller.This suggests that the variation is inverserather than direct. Therefore use the equation

†

y = kx . To determine the value of k, choose

one of the ordered pairs and substitute thevalues into the equation

†

y = kx and solve for

k. We’ll use the ordered pair (5, 1).

†

y = kx

1= k5 fi k = 5

53. The equation is

†

p= kl To find k substitute 150for l and 2542.50 for p.

†

2542.50 = k(150)

k = 2542.50150

k =16.95Thus p = 16.95l.Now substitute 520 for l.

†

p= 16.95(520)p= 8814

The profit would be $8814.

55. The equation is d = kw. To find k, substitute 2376for d and 132 for w.2376 = k132

k =2376132

k = 18Thus d = 18w. Now substitute 172 for w.d = 18(172)d = 3096The recommended dosage for Bill is 3096 mg.

57. The equation is t =kn

. To find k, substitute

8 for t and 5 for n.8 =

k5

k = 8(5)k = 40Thus t =

40n

.

Now substitute 4 for n.

†

t = 404

t =10It will take 4 bricklayers 10 hours.

59. The equation is V = kP . To find k, substitute 800

for V and 200 for P.800 =

k200

800(200) = k160,000 = k

Thus V =160,000

P.

Now substitute 25 for P.V =

160,00025

= 6400

The volume is 6400 cc.

61. The equation is t =ks

. To find k, substitute 6 for

s and 2.6 for t.


218

†

2.6 = k6

k = 6(2.6)k =15.6

Thus

†

t = 15.6s

.

Now substitute 5 for s.

†

t = 15.65

t = 3.12Leif will take 3.12 hours.

63. The equation is t =ks

. To find k, substitute 122

for s and 0.21 for t.0.21 =

k122

k = 0.21(122)k = 25.62

Thus t =25.62

s.

Now substitute 80 for s.t =

25.6280

ª 0.32

It takes about 0.32 seconds for the ball to hit theground in the service box.

65. The equation is

†

d = ks2 . To find k, substitute40 for s and 60 for d.

†

d = ks2

60= k 40( )2

60= 1600k

k = 601600

= 0.0375

Thus

†

d = 0.0375s2 . Now substitute 56 for s.

†

d = 0.0375s2

d = 0.0375 56( )2

d = 117.6The stopping distance is 117.6 feet.

67. The equation is P = krm. To find k, substitute50,000 for m and 0.07 for r and 332.5 for P.332.5 = k(0.07)(50,000)332.5 = 3500k

k =332.53500

= 0.095

Thus P = 0.095rm.Now substitute 66,000 for m and 0.07 for r.P = 0.095(0.07)(66, 000)P = 438.9The payment is $438.90.

69. The equation is R =kAP

. To find k, substitute

400 for A, 2 for P, and 4600 for R.4600 =

k(400)2

4600 = 200k

k =4600200

= 23

Thus R =23A

P. Now substitute 500 for A and

2.50 for P.R =

23(500)2.50

R =11,5002.50

R = 4600 tapesThey would rent 4600 tapes per week.

71. The equation is R = kLA . To find k, substitute

0.2 for R, 200 for L, and 0.05 for A.0.02 =

k(200)0.05

0.2 = k(4000)0.2

4000= k

0.00005 = kThus R = 0.00005L

A . Now substitute 5000 for Land 0.01 for A.R =

0.00005(5000)0.01

R =0.250.01

R = 25The resistance is 25 ohms.

73. The equation is W = kTA FR

To find k, substitute 78 for T, 5.6 for R, 4 for F,1000 for A, and 68 for W.

68 =k(78)(1000) 4

5.668 =

156, 000k5.6

(68)(5.6)156, 000

= k

0.002441 ª k

Thus W =0.002441TA E

RNow, substitute 78 for T, 5.6 for R, 6 for E, and1500 for A.

W =0.002441(78)(1500) 6

5.6ª 124.92

The water bill is about $124.92.


219

75. a. The equation is F =km1m2

d2 .

b. If m1 becomes 2m1, m2 becomes 3m2 andd becomes d

2 , then

F =k(2m1)(3m2)

d2

Ê Ë Á ˆ

¯ ˜

2

=6km1m2

d2

4= 24 km1m2

d2

The new force is 24 times the original force.

77.

†

4x + 9 8x2 + 6x - 252x - 3

) 8x2 + 18x - 12x - 25 -12x - 27 2

†

8x2 + 6x - 254x + 9

= 2x - 3 +2

4x + 9

78.

†

y z - 2( ) + 3 z - 2( ) z - 2( ) y + 3( )

79.

†

3x2 - 24 = -6x3x2 + 6x - 24 = 0

3 x2 + 2x - 8( ) = 03 x + 4( ) x - 2( ) = 0

†

x + 4 = 0 or x - 2 = 0 x = -4 x = 2

80.

†

x + 3x - 3

⋅x 3 - 27

x2 + 3x + 9=

x + 3x - 3

⋅x - 3( ) x2 + 3x + 9( )

x2 + 3x + 9= x + 3

Review Exercises

1.

†

52x - 12

†

2x -12 = 02 x - 6( ) = 0

x - 6 = 0The expression is defined for all real numbersexcept x = 6.

2. 2x2 - 8x + 15x2 - 8x + 15 = 0x - 3( ) x - 5( ) = 0

x – 3 = 0, x – 5 = 0The expression is defined for all real numbersexcept x!= 3, x!= 5.

3. 25x2 + 4x -15x2 + 4x - 1 = 0

5x - 1( ) x + 1( ) = 05x -1 = 0, x + 1 = 0The expression is defined for all real numbersexcept x =

15

, x = -1 .

†

4. yxy - 3y

=y

y x - 3( )

=1

x - 3

5. x3 + 4x 2 + 12xx

=x x2 + 4x + 12( )

x= x 2 + 4x + 12

6. 9x2 + 6xy3x

=3x 3x + 2y( )

3x= 3x + 2y

7. x2 + 2x - 8x - 2

=x - 2( ) x + 4( )

x - 2= x + 4

†

8. a2 - 36a - 6

=a - 6( ) a + 6( )

a - 6= a + 6


220

†

9. -2x2 + 7x + 4x - 4

=-1 2x2 + 7x + 4( )

x - 4( )

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

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

†

10. b2 - 8b + 15b2 - 3b - 10

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

=b - 3b + 2

11. 4x2 -11x - 34x2 - 7x - 2

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

=x - 3x - 2

12. 2x2 - 21x + 404x2 - 4x -15

=x - 8( ) 2x - 5( )

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

=x - 8

2x + 3

†

13. 5a2

6b⋅

24a2b

=5 ⋅ 26 ⋅ 4

⋅a2

a2 ⋅1

b ⋅ b

=10

24b2

=5

12b2

14. 15x2y3

3z⋅

6z3

5xy3 =15 ⋅6

15x2

xÊ

Ë Á

ˆ

¯ ˜

y3

y3Ê

Ë Á

ˆ

¯ ˜

z3

zÊ

Ë Á

ˆ

¯ ˜

= 6xz2

15.40a3b4

7c3 ⋅14c5

5a5b=

405

⋅147

⋅a3

a5b4

bc5

c3

= 161a2 b3c2

=16b3c2

a2

16.1

x - 4⋅4 - x

3=

1x - 4

⋅-1 x - 4( )

3= -

13

†

17. -m + 415m

⋅10mm - 4

=-1 m - 4( )

3 ⋅ 5m⋅2 ⋅ 5mm - 4

=-23

18. a - 2a + 3

⋅a2 + 4a + 3a2 - a - 2

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

= 1

†

19. 16x6

y 2 ÷x 4

4y=

16x 6

y2 ⋅4yx 4

=64x6 yx4 y 2

=64x2

y

20. 8xy2

z÷

x 4y2

4z2 =8xy2

z⋅

4z2

x4y2

=32zx3

21. 5a + 5ba2 ÷

a2 - b2

a2 =5 a + b( )

a2 ⋅a2

a + b( ) a - b( )

=5

a - b

22.1

a2 + 8a +15÷

3a + 5

=1

a + 5( ) a + 3( )⋅a + 5

3

=1

3 a + 3( )

†

23. t + 8( ) ÷t 2 + 5t - 24

t - 3= t + 8( ) ⋅

t - 3t - 3( ) t + 8( )

= 1

24. x2 + xy - 2y2

4y÷

x + 2y12y2 =

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

⋅12y2

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

†

25. nn + 5

-5

n + 5=

n - 5n + 5

26.3x

x + 7+

21x + 7

=3x + 21

x + 7

=3 x + 7( )

x + 7= 3


221

27. 9x - 4x + 8

+76

x + 8=

9x - 4 + 76x + 8

=9x + 72

x + 8=

9 x + 8( )x + 8

= 9

28.7x - 3

x2 + 7x - 30-

3x + 9x2 + 7x - 30

=7x - 3- (3x + 9)

x2 + 7x - 30=

7x - 3- 3x - 9x 2 + 7x - 30

=4x - 12

x 2 + 7x - 30=

4 x - 3( )x + 10( ) x - 3( )

=4

x + 10

†

29. 5h2 + 12h -1h + 5

-h2 - 5h + 14

h + 5=

5h2 + 12h - 1- h2 - 5h + 14( )h + 5

=5h2 + 12h - 1- h2 + 5h -14

h + 5

=4h2 + 17h - 15

h + 54h - 3( ) h + 5( )

h + 5= 4h - 3

30. 6x2 - 4x2x - 3

--3x + 122x - 3

=6x 2 + 4x - (-3x + 12)

2x - 3

=6x 2 - 4x + 3x - 12

2x - 3

=6x 2 - x - 12

2x - 3

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

2x - 3= 3x + 4

31.

†

a10

+4a3

Least common denominator = 2(5)(3) = 30

32. 8x + 3

+4

x + 3Least common denominator = x + 3

33. 54xy3 -

710x2y

Least common denominator = 20x 2y3

34. 6x +1

-3xx

Least common denominator = x x + 1( )

35.

†

4n - 5

+2n - 3n - 4

Least common denominator

†

= n - 5( ) n - 4( )

36. 7x -12x2 + x

-4

x + 1=

7x - 12x(x + 1)

-4

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

37.

†

2r - 9r - s

-6

r 2 - s 2 =2r - 9r - s

-6

(r + s)(r - s)Least common denominator = (r + s)(r – s)

38. 4x2

x - 7+ 8x2 =

4x2

x - y+

8x2

1Least common denominator = x!– 7

39. 19x - 5x2 + 2x - 35

+3x - 2

x2 + 9x + 14=

19x - 5(x + 7)(x - 5)

+3x - 2

(x + 7)(x + 2)Least common denominator= x + 7( ) x - 5( ) x + 2( )


222

†

40. 43y2 +

y2y

=4

3y 2 +12

=4

3y 2 ⋅22

+12

⋅3y2

3y2

=8

6y2 +3y 2

6y2

=8 + 3y2

6y2

=3y 2 + 8

6y2

41.2xxy

+1

5x=

2xxy

⋅55

+15x

⋅yy

=10x5xy

+y

5xy

=10x + y

5xy

42. 5x3xy

-4x2 =

5x3xy

⋅xx

-4x2 ⋅

3y3y

=5x2

3x2y-

12y3x 2y

=5x2 - 12y

3x2y

43. 6 -2

x + 2= 6 x + 2

x + 2Ê Ë Á ˆ

¯ ˜ -

2x + 2

=6x + 12 - 2

x + 2=

6x + 10x + 2

44.x - y

y-

x + yx

=x - y

y⋅

xx

-x + y

x⋅yy

=x x - y( )

xy-

y x + y( )xy

=x 2 - xy - xy - y2

xy

=x 2 - 2xy - y2

xy

45. 7x + 4

+4x

=7

x + 4⋅xx

+4x

⋅x + 4x + 4

=7x

x x + 4( )+

4 x + 4( )x x + 4( )

=7x + 4x + 16

x x + 4( )

=11x + 16x x + 4( )

46. 23x

-3

3x - 6=

23x

-3

3 x - 2( )

=2

3x⋅x - 2x - 2

-3

3(x - 2)⋅xx

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

-3x

3x x - 2( )

=2x - 4 - 3x3x x - 2( )

=- x - 4

3x x - 2( )

†

47. 3z + 5

+7

z + 5( )2 =3

z + 5⋅z + 5z + 5

+7

(z + 5)2

=3 z + 5( )z + 5( )2 +

7z + 5( )2

=3z + 15+ 7

z + 5( )2

=3z + 22z + 5( )2


223

48. x + 2x2 - x - 6

+x - 3

x2 - 8x + 15=

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

+x - 3

x - 3( ) x - 5( )

=1

x - 3+

1x - 5

=1

x - 3⋅

x - 5x - 5

+1

x - 5⋅x - 3x - 3

=x - 5

x - 3( ) x - 5( )+

x - 3x - 5( ) x - 3( )

=x - 5 + x - 3

(x - 5)( x - 3)

=2x - 8

x - 5( ) x - 3( )

†

49. x + 4x + 6

-x - 5x + 2

=x + 4( ) x + 2( )x + 6( ) x + 2( )

-x - 5( ) x + 6( )x + 2( ) x + 6( )

=x2 + 6x + 8 - (x 2 + x - 30)

(x + 6)(x + 2)

=x2 + 6x + 8 - x2 + x + 30

x + 6( ) x + 2( )

=5x + 38

x + 6( ) x + 2( )

50. 3 +x

x - 4=

3 x - 4( )x - 4

+x

x - 4=

3x - 12 + xx - 4

=4x - 12

x - 4

51. a + 2b

÷a - 24b2 =

a + 2b

⋅4b2

a - 2

=4b a + 2( )

a - 2=

4ab + 8ba - 2

52. x + 3x2 - 9

+2

x + 3=

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

+2

x + 3

=x + 3

x - 3( ) x + 3( )+

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

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

=3x - 3

x - 3( ) x + 3( )

†

53. 5p + 10qp2q

⋅p4

p + 2q=

5 p + 2q( ) p4

p + 2q( )p2q

=5p2

q


224

54.4

x + 2( ) x - 3( )-

4x - 2( ) x + 2( )

=4(x - 2)

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

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

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

=4x - 8 - 4x +12

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

=4

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

55.x + 7

x2 + 9x + 14-

x -10x 2 - 49

=x + 7

(x + 7)(x + 2)-

x -10(x + 7)(x - 7)

=x + 7

x + 7( ) x + 2( )⋅

x - 7( )x - 7( )

-x -10

x + 7( ) x - 7( )⋅

x + 2( )x + 2( )

=x2 - 49 - x 2 - 8x - 20( )

x + 7( ) x - 7( ) x + 2( )

=8x - 29

x + 7( ) x - 7( ) x + 2( )

56. x - yx + y

⋅xy + x2

x 2 - y2 =x - yx + y

⋅x y + x( )

x + y( ) x - y( )

=x

x + y

†

57. 3x2 - 27y 2

20÷

x - 3y( )2

4=

3 x2 - 9y 2( )20

⋅4

x - 3y( )2

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

4 ⋅ 5⋅

4(x - 3y)2

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

58. a2 - 9a + 20a - 4

⋅a2 - 8a + 15

a2 - 10a + 25=

a - 4( ) a - 5( )a - 4

⋅a - 5( ) a - 3( )

a - 5( )2

= a - 3

59. aa2 - 1

-2

3a2 - 2a - 5=

aa -1( ) a +1( )

-2

3a - 5( ) a +1( )

=a(3a - 5)

(a -1)(a + 1)(3a - 5)-

2(a - 1)(a -1)(a + 1)(3a - 5)

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

=3a2 - 5a - 2a + 2a -1( ) a +1( ) 3a - 5( )

=3a2 - 7a + 2

a -1( ) a +1( ) 3a - 5( )


225

60. 2x2 + 6x - 20x2 - 2x

÷x 2 + 7x + 10

4x 2 -16=

2 x2 + 3x -10( )x x - 2( )

⋅4 x2 - 4( )

x + 5( ) x + 2( )

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

x x - 2( )⋅4 x + 2( ) x - 2( )

x + 5( ) x + 2( )

=8 x - 2( )

x=

8x -16x

†

61. 3+ 2

335

=15 3 + 2

3( )15 3

5( )

=45 + 10

9

=559

62.1+ 5

84 - 9

16=

16 1+ 58( )

16 4 - 916( )

=16 + 1064 - 9

=2655

63.12ab9c4ac2

=12ab

9c⋅c2

4a

=bc3

64. 36x4y2

9xy 5

4z 2

=36x 4y2 ⋅ 4z2

9xy 5

4 z 2 ⋅ 4z2

=144x4y2z2

9xy5

=16x3z2

y3

65.a - a

b1+a

b=

a - ab( )b

1+ab( )b

=ab - a1+ a

†

66. r 2 + 1

s

s 2 =r 2 + 1

s( )ss2( )s

=r 2s + 1

s3

67.4x + 2

x2

6 - 1x

=

4x + 2

x2( )x 2

6 - 1x( )x 2

=4x + 2

6x2 - x=

4x + 2x 6x - 1( )

68.x

x +yx2

2x +2y

=x

x + y⋅2x + 2y

x2

=1

x + y⋅2 x + y( )

x

=2x

69.3x3x2

=3x

⋅x 2

3= x

70.1a + 21a + 1

a=

1a + 2

2a

=1a + 2( )a

2a( )a

=1 + 2a

2


226

†

71. 1

x 2 - 1x

1x 2 + 1

x=

x2 1x 2 - 1

x( )x2 1

x 2 + 1x( )

=1 - x1 + x

=-x + 1x + 1

72.3xy - xyx -1

=

3xy - x( )xyyx - 1( )xy

=3x 2 - x2y

y2 - xy

=3x 2 - x2yy y - x( )

73.59

=5

x + 35 x + 3( ) = 5 9( )5x +15 = 45

5x = 30x = 6

74. x6

=x - 4

22x = 6(x - 4)2x = 6x - 2424 = 4x6 = x

†

75. n5

+ 12 =n2

10n5

+ 12Ê Ë Á

ˆ ¯ ˜ = 10

n2

Ê Ë Á

ˆ ¯ ˜

2n + 120 = 5n120 = 3n40 = n

76.3x

-16

=1x

6x3x

-16

Ê Ë Á ˆ

¯ ˜ = 6x

1x

Ê Ë Á ˆ

¯ ˜

18 - x = 6-x = -12

x = 12

†

77. -4d

=32

+4 - d

d

2d-4d

Ê Ë Á

ˆ ¯ ˜ = 2d

32

+4 - d

dÊ Ë Á

ˆ ¯ ˜

-8 = 3d + 8 - 2d-8 = d + 8

-16 = d

78. 1x - 7

+1

x + 7=

1x2 - 49

1x - 7

+1

x + 7=

1(x - 7)(x + 7)

(x - 7)(x + 7) 1x - 7

+1

x + 7È Î Í

˘ ˚ ˙ =

1(x - 7)(x + 7)

È

Î Í ˘

˚ ˙ (x - 7)(x + 7)

x + 7 + x - 7 = 12x = 1

x =12


227

79. x - 3x - 2

+x + 1x + 3

=2x 2 + x + 1x2 + x - 6

x - 3x - 2

+x + 1x + 3

=2x2 + x + 1

(x - 2)(x + 3)

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

+x +1x + 3

Ê Ë Á ˆ

¯ ˜ = x - 2( ) x + 3( ) 2x2 + x + 1

(x - 2)(x + 3)È

Î Í

˘

˚ ˙

(x + 3)( x - 3) + ( x - 2)( x + 1) = 2x 2 + x + 1x 2 - 9 + x2 - x - 2 = 2x 2 + x + 1

2x 2 - x - 11= 2x 2 + x + 1-12 = 2x-6 = x

80.a

a2 - 64+

4a + 8

=3

a - 8a

(a + 8)(a - 8)+

4a + 8

=3

a - 8

a + 8( ) a - 8( )a

a + 8( ) a - 8( )+

4a + 8

È

Î Í

˘

˚ ˙ = a + 8( ) a - 8( )

3a - 8

Ê Ë Á ˆ

¯ ˜

a + 4 a - 8( ) = 3 a + 8( )a + 4a - 32 = 3a + 24

5a - 32 = 3a + 242a = 56a = 28

†

81. dd - 2

- 2 =2

d - 2

d - 2( )d

d - 2- 2Ê

Ë Á ˆ ¯ ˜ = d - 2( )

2d - 2

Ê Ë Á

ˆ ¯ ˜

d - 2 d - 2( ) = 2d - 2d + 4 = 2

-d + 4 = 2-d = -2

d = 2Since

†

20

is not a real number, there is no

solution.

82. John and Amy rate =16

Paul and Cindy rate =15

t6

+t5

= 1

30t6

+t5

Ê Ë Á ˆ

¯ ˜ = 30 1( )

5t + 6t = 3011t = 30

t =3011

= 2 811

It will take the 4 people 2811

hours.

83. 34

-inch hose’s rate =17

516

-inch hose’s rate =1

12t7

-t

12= 1

12 t - 7t84

= 15t = 84

t =845

= 1645

It will take 1645

hours to fill the pool.


228

84. Let x be one number then, 5x!is the othernumber.

1x

+1

5x= 6

5x1x

+1

5xÊ Ë Á ˆ

¯ ˜ = 5x 6( )

5 + 1 = 30x6 = 30x

630

= x15

= x

5x = 515

Ê Ë Á ˆ

¯ ˜ = 1

The numbers are 15

and 1.

85. Let x!= Robert’s speed, then3.5 + x = Tran’s speedt =

dr

Robert’s time = Tran’s time3x

=8

3.5 + x3 3.5 + x( ) = 8x10.5 + 3x = 8x

10.5 = 5x2.1 = x

x!+ 3.5 = 2.1 + 3.5 = 5.6Robert’s speed is 2.1 mph and Tran’s speed is5.6 mph.

86. The equation is

†

x = ky 2 . To find k, substitute 45for x and 3 for y.

†

45= k(3)2

45= 9k459

= k

5= kThus

†

x = 5y 2 . Now substitute 2 for y.

†

A = 5(2)2 = 5(4) = 20

87. The equation is

†

W = kL2

A. To find k, substitute 4

for W, 2 for L, and 10 for A.

†

4 =k(2)2

104 = 4k

1040= 4k10= k

Thus

†

W = 10L2

A. Now substitute 5 for L and 20

for A.

†

W =10 ⋅ (5)

202

= 25020 = 25

2

88. The equation is z =kxyr2 . To find k, substitute 12

for z, 20 for x, 8 for y, and 8 for r.12 =

k(20 )(8)(8)2

12 =160k

6412(64) = 160k

768 = 160k768160

= k

k = 4.8Thus z =

4.8xyr2 . Now substitute 10 for x, 80 for

y, and 3 for r.z =

4.8(10)(80)32 ª 426.7

89. Let s represent the surcharge and let Erepresent the energy used in kilowatt-hours.The equation iss = kE .To find k, substitute 7.20 for s and 3600 forE.

†

s = kE

7.20 = k 3600( ) fi k =7.203600

= 0.002

Thus s = 0.002E. Now substitute 4200 for E.

†

s = 0.002Es = 0.002(4200)s = 8.40The surcharge is $8.40.

90. The equation is d = kt2 . To find k, substitute16 for d and 1 for t.16 = k(1)216 = k(1)16 = kThus d = 16t2 . Now substitute 10 for t.

†

d = 16(10)2 =16(100) =1600An object will fall 1600 feet.


229

91. The equation is A = kr2 . To find k, substitute78.5 for A and 5 for r.78.5 = k(5)2

78.5 = k(25)78.525

= k3.14 = k

Thus A = 3.14r2 . Now substitute 8 for r.A = 3.14(8)2

A = 3.14(64)A = 200.96The area is 200.96 square units.

92. The equation is t =kw

where t is time and w

is water temperature. To find k, substitute 1.7for t and 70 for w.

1.7 =k70

(1.7)(70) = k119 = k

Thus t =119w

. Now substitute 50 for w.

t =11950

= 2.38

It takes the ice cube 2.38 minutes to melt.

Practice Test

1.

†

-6 + xx - 6

=x - 6x - 6

= 1

2. x3 - 1x2 - 1

=x -1( ) x2 + x + 1( )

x - 1( ) x + 1( )

=x 2 + x + 1

x +1

†

3. 15x2 y3

4z 2 ⋅8xz 3

5xy 4 =3 ⋅ 5x 2y 3

4z 2 ⋅2 ⋅ 4 xz3

5xy 4

=6x3 y3z 3

xy 4z 2

=6x2z

y

4. a2 - 9a + 14a - 2

⋅a2 - 4a - 21

a - 7( )2 =a - 7( ) a - 2( )

a - 2⋅

a - 7( ) a + 3( )a - 7( )2

= a + 3

5. x2 - x - 6x 2 - 9

⋅x2 - 6x + 9x2 + 4x + 4

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

⋅x - 3( ) x - 3( )x + 2( ) x + 2( )

=x - 3( )2

x + 3( ) x + 2( )

=x 2 - 6x + 9

(x + 3)(x + 2)

6. x2 - 1x + 2

⋅2x + 42 - 2x2 =

x - 1( ) x + 1( )x + 2

⋅2 x + 2( )

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

†

7. x 2 - 4y2

3x + 12y÷

x + 2yx + 4 y

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

3 x + 4 y( ) ⋅x + 4yx + 2y

=x - 2y

3


230

8.15

y2 + 2y - 15÷

3y - 3

=15

y - 3( ) y + 5( )⋅y - 3

3

=5

y + 5

†

9. m2 + 3m - 18m - 3

÷m2 - 8m + 15

3 - m=

m + 6( ) m - 3( )m - 3

⋅-1 m - 3( )

(m - 5)(m - 3)

=-(m + 6)

m - 5

= -m + 6m - 5

10. 4x + 32y

+2x - 5

2y=

4x + 3+ 2x - 52y

=6x - 2

2y

=2 3x - 1( )

2y

=3x - 1

y

11. 7x2 - 4x + 3

-6x + 7x + 3

=7x2 - 4 - (6x + 7)

x + 3

=7x2 - 4 - 6x - 7

x + 3

=7x2 - 6x -11

x + 3

12.4xy

-3

xy3 =4xy

⋅y2

y2 -3

xy3

=4y2

xy3 -3

xy3

=4y2 - 3

xy3

†

13. 4 -5z

z - 5= 4

z - 5z - 5

Ê Ë Á

ˆ ¯ ˜ -

5zz - 5

=4z - 20z - 5

-5z

z - 5

=4z - 20 - 5z

z - 5

=-z - 20z - 5

=-1 z + 20( )

z - 5

= -z + 20z - 5


231

14. x - 5x2 - 16

-x - 2

x2 + 2x - 8=

x - 5x - 4( ) x + 4( )

-x - 2

x - 2( ) x + 4( )

=x - 5

x - 4( ) x + 4( )-

1x + 4

=x - 5

( x - 4)( x + 4)-

1x + 4

⋅x - 4x - 4

=x - 5- x - 4( )x - 4( ) x + 4( )

=x - 5 - x + 4

( x - 4)( x + 4)

=-1

x - 4( ) x + 4( )

†

15. 5 + 1

23 - 1

5=

10 5 + 12( )

10 3 - 15( )

=50 + 530 - 2

=5528

16.x + x

y1x

= x +xy

Ê

Ë Á

ˆ

¯ ˜

x1

=xy + x

yÊ

Ë Á

ˆ

¯ ˜

x1

Ê Ë Á ˆ

¯ ˜

=yx2 + x2

y

17.2 + 3

x2x - 5

=x 2 + 3

x( )x 2

x - 5( )=

2x + 32 - 5x

18. 6 +2x

= 7

x 6 +2x

Ê Ë Á ˆ

¯ ˜ = 7x

6x + 2 = 7x2 = x

19. 2x3

-x4

= x + 1

12 2x3

-x4

Ê Ë Á ˆ

¯ ˜ = 12 x + 1( )

8x - 3x = 12x + 125x = 12x + 12

-7x = 12

x = -127

20. xx - 8

+6

x - 2=

x2

x2 -10 x + 16x

x - 8+

6x - 2

=x2

(x - 8)(x - 2)

x - 8( ) x - 2( )x

x - 8+

6x - 2

Ê Ë Á ˆ

¯ ˜ = x - 8( ) x - 2( )

x2

x - 8( ) x - 2( )È

Î Í

˘

˚ ˙

x x - 2( ) + 6 x - 8( ) = x2

x2 - 2x + 6x - 48 = x2

4x - 48 = 04x = 48

x = 12


232

21.

t8

+t5

= 15t + 8t

40= 1

13t = 40

t =4013

= 31

13It will take them 3

113

hours to level one acre

together.

22. Let x be the number.x +

1x

= 2

x x +1x

Ê Ë Á ˆ

¯ ˜ = x 2( )

x2 + 1 = 2xx2 - 2x + 1 = 0

x - 1( ) x - 1( ) = 0x - 1 = 0

x = 1The number is 1.

23. Let d = the distance she rollerblades, then 12 – dis the distance she bicycles.

t =dr

d4

+12 - d

10= 1.5

20 d4

+12 - d

10Ê Ë Á ˆ

¯ ˜ = 20(1.5)

5d + 2(12 - d) = 305d + 24 - 2d = 30

3d = 6d = 2

She rollerblades for 2 miles.

24.

†

w =kf

4.3 =k

2631130.9 = k

Now substitute 1000 for f and 1130.9 for k.

†

w =kf

w =1130.91000

w = 1.1309The wavelength would be about 1.13 feet.

Cumulative Review Test

†

1. 3 x2 - 5xy 2 + 3 = 3 -4( )2 - 5 -4( ) -2( )2 + 3= 3 16( ) - 5 -4( ) 4( ) + 3= 48 + 80 + 3= 131

2. [7 - [3(8÷ 4)]2 + 9 ⋅ 4]2 = [7 - [3(2)]2 + 9 ⋅ 4]2

= [7 - (6)2 + 9 ⋅ 4]2

= [7 - 36 + 9 ⋅ 4]2

= [7 - 36 + 36]2

= [7]2

= 49

3. 5z + 4 = -3 z - 7( )5z + 4 = -3z + 21

5z + 3z = 21- 48z = 17

z =178

4.

†

12 inches1 foot

=162 inches

x feet121

=162

x12x = 162

x = 13.5 feet

5. Let (–7, 8) be

†

x1, y1( ) and (3, 8) be

†

x2, y2( ) .

†

m =y2 - y1x2 - x1

m =8 - 8

3- -7( )

m =0

10= 0

The slope is 0.

6. To put in slope-intercept form, solve for y.

†

3x - 4y = 123x - 3x - 4y = -3x + 12

-4y = -3x + 12-4y-4

=-3x-4

+12-4

y =34

x - 3


233

7. 6x2 - 3x - 5( ) - -2x2 - 8x - 9( ) = 6x2 - 3x - 5 + 2x2 + 8x + 9

= 6x2 + 2x2 - 3x + 8x - 5 + 9= 8x2 + 5x + 4

8.

†

3n2 - 4n + 3( ) 2n - 5( ) = 3n2 2n - 5( ) - 4n 2n - 5( ) + 3 2n - 5( )

= 6n3 -15n2 - 8n2 + 20n + 6n -15= 6n3 - 23n2 + 26n - 15

9.

†

4x - 348

=4 x8

-348

=12

x -174

10. 6a2 - 6a - 5a + 5 = 6a a -1( ) - 5 a - 1( )= 6a - 5( ) a - 1( )

11. 13x2 + 26x - 39 = 13 x2 + 2x - 3( )= 13 x + 3( ) x - 1( )

12. 2x2 = 11x -122x2 - 11x + 12 = 0x - 4( ) 2x - 3( ) = 0

x - 4 = 0 or 2x - 3 = 0x = 4 x = 3

2

13. x 2 - 9x2 - x - 6

⋅x2 - 2x - 8

2x2 - 7x - 4=

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

⋅x - 4( ) x + 2( )

2x + 1( ) x - 4( )

=x + 3

2x +1

†

14. rr + 2

-6

r - 5=

rr + 2

⋅r - 5r - 5

-6

r - 5⋅r + 2r + 2

=r(r - 5)

(r + 2)(r - 5)-

6(r + 2)(r + 2)(r - 5)

=r 2 - 5r - (6r + 12)

(r + 2)(r - 5)

=r 2 - 5r - 6r -12

(r + 2)(r - 5)

=r 2 -11r - 12(r + 2)(r - 5)


234

15. 4x2 - 3x - 10

+2

x2 + 5x + 6=

4x - 5( ) x + 2( )

+2

x + 2( ) x + 3( )

=4(x + 3)

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

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

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

=4x + 12 + 2x -10x - 5( ) x + 2( ) x + 3( )

=6x + 2

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

16.x9

-x6

=1

12

36x9

-x6

Ê Ë Á ˆ

¯ ˜ = 36

112

Ê Ë Á ˆ

¯ ˜

4x - 6x = 3-2x = 3

x = -32

17. 7x + 3

+5

x + 2=

5x2 + 5x + 6

7x + 3

+5

x + 2=

5(x + 3)( x + 2)

x + 3( ) x + 2( )7

x + 3+

5x + 2

Ê Ë Á ˆ

¯ ˜ = x + 3( ) x + 2( )

5x + 3( ) x + 2( )

È

Î Í

˘

˚ ˙

7 x + 2( ) + 5 x + 3( ) = 57x +14 + 5x + 15 = 5

12x + 29 = 512x = -24

x = -2Check:

7x + 3

+5

x + 2=

5x 2 + 5x + 6

7-2 + 3

+5

-2 + 2=

5-2( )2 + 5 -2( ) + 6

71

+50

=50

Since 50

is not a real number, there is no solution.

18. Let x = the total medical bills. The cost underplan 1 is 0.10x, while the cost under plan 2 is100 + 0.05x.0.10 x = 100 + 0.05x0.05x = 100

x = 2000

The cost under both plans is the same for $2000in total medical bills.


235

19. Let x = pounds of sunflower seed and y = pounds of premixed assorted seed mixx + y = 500.50 x + 0.15y = 14.50Solve the first equation for y.y = 50 – xSubstitute 50 – x for y in the second equation.0.50 x + 0.15 50 - x( ) = 14.500.50x + 7.5 - 0.15x = 14.50

0.35x = 7x = 20

y = 50 - x = 50 - 20 = 30He will have to use 20 pounds of sunflower seedand 30 pounds of premixed assorted seed mix.

20. Let d = distance on first leg, then the distance onthe second leg is 12.75 – d.t =

dr

Time for first leg + time for second leg = totaltime

d6.5

+12.75 – d

9.5= 1.5

9.5( ) 6.5( )d

6.5+

12.75 - d9.5

È Î Í

˘ ˚ ˙ = 9.5( ) 6.5( ) 1.5( )

9.5d + 6.5 12.75 - d( ) = 92.6259.5d + 82.875 - 6.5d = 92.625

3d = 9.75d = 3.25

12.75 – d = 12.75 – 3.25 = 9.5The distance traveled during the first leg of therace was 3.25 miles and the distance traveled inthe second leg of the race was 9.5 miles.

chapter 7: rational expressions and equations ssm: elementary...

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