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7.8 Review Exercises and Sample Exam

Review Exercises

Simplifying Rational Expressions

Evaluate for the given set of x-values.

1. 252x2; {−5, 0, 5}

2. x42x1; {1/2, 2, 4}

3. 1x2+9; {−3, 0, 3}

4. x+3x29; {−3, 0, 3}

State the restrictions to the domain.

5. 5x

6. 1x(3x+1)

7. x+2x225

8. x1(x1)(2x3)

State the restrictions and simplify.

9. x8x264

10. 3x2+9x2x318x

11. x25x24x23x40

12. 2x2+9x54x21

13. x214412x

14. 8x210x394x2

15. Given f(x)=x3x2+9, find f(3), f(0), and f(3).

16. Simplify g(x)=x22x242x29x18 and state the restrictions.

Multiplying and Dividing Rational Expressions

Multiply. (Assume all denominators are nonzero.)

17. 3x5x3x39x2

18. 12y2y3(2y1)(2y1)3y

19. 3x2x2x24x+45x3

20. x28x+159x512x2x3

21. x236x2x302x2+10xx2+5x6

22. 9x2+11x+2481x29x2(x+1)2

Divide. (Assume all denominators are nonzero.)

23. 9x2255x3÷3x+515x4

24. 4x24x21÷2x2x1

25. 3x213x10x2x20÷9x2+12x+4x2+8x+16

26. 2x2+xyy2x2+2xy+y2÷4x2y23x2+2xyy2

27. 2x26x208x2+17x+2÷(8x239x5)

28. 12x227x415x4+10x3÷(3x2+x2)

29. 25y215y4(y2)15y1÷10y2(y2)2

30. 10x4136x2÷5x26x27x+1x12x

31. Given f(x)=16x29x+5 and g(x)=x2+3x104x2+5x6, calculate (fg)(x) and state the restrictions.

32. Given f(x)=x+75x1 and g(x)=x24925x25x, calculate (f/g)(x) and state the restrictions.

Adding and Subtracting Rational Expressions

Simplify. (Assume all denominators are nonzero.)

33. 5xy3y

34. xx2x63x2x6

35. 2x2x+1+1x5

36. 3x7+12xx2

37. 7x4x29x+22x2

38. 5x5+209x2x215x+25

39. xx52x35(x3)x28x+15

40. 3x2x1x4x+4+12(2x)2x2+7x4

41. 1x2+8x91x2+11x+18

42. 4x2+13x+36+3x2+6x27

43. y+1y+212y+2yy24

44. 1y11y2y21

45. Given f(x)=x+12x5 and g(x)=xx+1, calculate (f+g)(x) and state the restrictions.

46. Given f(x)=x+13x and g(x)=2x8, calculate (fg)(x) and state the restrictions.

Complex Fractions

Simplify.

47. 42x2x13x

48. 1313y1515y

49. 16+1x1361x2

50. 11001x21101x

51. xx+32x+1xx+4+1x+3

52. 3x1x55x+22x

53. 112x+35x2125x2

54. 215x+25x22x5

Solving Rational Equations

Solve.

55. 6x6=22x1

56. xx6=x+2x2

57. 13x29=1x

58. 2x5+35=1x5

59. xx5+4x+5=10x225

60. 2x122x+3=23x22x2+3x

61. x+12(x2)+x6x=1

62. 5x+2x+1xx+4=4

63. xx+5+1x4=4x7x2+x20

64. 23x1+x2x+1=2(34x)6x2+x1

65. xx1+1x+1=2xx21

66. 2xx+512x3=47x2x2+7x15

67. Solve for a:   1a=1b+1c.

68. Solve for y:   x=2y13y.

Applications of Rational Equations

Use algebra to solve the following applications.

69. A positive integer is twice another. The sum of the reciprocals of the two positive integers is 1/4. Find the two integers.

70. If the reciprocal of the smaller of two consecutive integers is subtracted from three times the reciprocal of the larger, the result is 3/10. Find the integers.

71. Mary can jog, on average, 2 miles per hour faster than her husband, James. James can jog 6.6 miles in the same amount of time it takes Mary to jog 9 miles. How fast, on average, can Mary jog?

72. Billy traveled 140 miles to visit his grandmother on the bus and then drove the 140 miles back in a rental car. The bus averages 14 miles per hour slower than the car. If the total time spent traveling was 4.5 hours, then what was the average speed of the bus?

73. Jerry takes twice as long as Manny to assemble a skateboard. If they work together, they can assemble a skateboard in 6 minutes. How long would it take Manny to assemble the skateboard without Jerry’s help?

74. Working alone, Joe completes the yard work in 30 minutes. It takes Mike 45 minutes to complete work on the same yard. How long would it take them working together?

Variation

Construct a mathematical model given the following.

75. y varies directly with x, and y = 12 when x = 4.

76. y varies inversely as x, and y = 2 when x = 5.

77. y is jointly proportional to x and z, where y = 36 when x = 3 and z = 4.

78. y is directly proportional to the square of x and inversely proportional to z, where y = 20 when x = 2 and z = 5.

79. The distance an object in free fall drops varies directly with the square of the time that it has been falling. It is observed that an object falls 16 feet in 1 second. Find an equation that models the distance an object will fall and use it to determine how far it will fall in 2 seconds.

80. The weight of an object varies inversely as the square of its distance from the center of earth. If an object weighs 180 pounds on the surface of earth (approximately 4,000 miles from the center), then how much will it weigh at 2,000 miles above earth’s surface?

Sample Exam

Simplify and state the restrictions.

1. 15x3(3x1)23x(3x1)

2. x2144x2+12x

3. x2+x122x2+7x4

4. 9x2(x3)2

Simplify. (Assume all variables in the denominator are positive.)

5. 5xx225x525x2

6. x2+x6x24x+43x25x2x29

7. x24x1212x2÷x66x

8. 2x27x46x224x÷2x2+7x+310x2+30x

9. 1x5+1x+5

10. xx+182x12xx2x2

11. 1y+1x1y21x2

12. 16x+9x225x3x2

13. Given f(x)=x281(4x3)2 and g(x)=4x3x9, calculate (fg)(x) and state the restrictions.

14. Given f(x)=xx5 and g(x)=13x5, calculate (fg)(x) and state the restrictions.

Solve.

15. 13+1x=2

16. 1x5=32x3

17. 19x+20x2=0

18. x+2x2+1x+2=4(x+1)x24

19. xx21x3=3x10x25x+6

20. 5x+4x4x=9x4x216

21. Solve for r:   P=1201+3r.

Set up an algebraic equation and then solve.

22. An integer is three times another. The sum of the reciprocals of the two integers is 1/3. Find the two integers.

23. Working alone, Joe can paint the room in 6 hours. If Manny helps, then together they can paint the room in 2 hours. How long would it take Manny to paint the room by himself?

24. A river tour boat averages 6 miles per hour in still water. With the current, the boat can travel 17 miles in the same time it can travel 7 miles against the current. What is the speed of the current?

25. The breaking distance of an automobile is directly proportional to the square of its speed. Under optimal conditions, a certain automobile moving at 35 miles per hour can break to a stop in 25 feet. Find an equation that models the breaking distance under optimal conditions and use it to determine the breaking distance if the automobile is moving 28 miles per hour.

Review Exercises Answers

1: 1/2, undefined, 1/2

3: 1/18, 1/9, 1/18

5: x0

7: x±5

9: 1x+8; x±8

11: x+3x+5; x5,8

13: (x+12); x12

15: f(3)=13, f(0)=13, f(3)=0

17: x33

19: 3(x2)5x

21: 2xx1

23: 3x(3x5)

25: x+43x+2

27: 2(8x+1)2

29: (5y+1)(y2)50y6

31: (fg)(x)=(4x+3)(x2)x+2; x5,2,34

33: 5x3y

35: 2x28x+1(2x+1)(x5)

37: 14x1

39: x5x3

41: 3(x1)(x+2)(x+9)

43: yy2

45: (f+g)(x)=3x23x+1(2x5)(x+1); x1,52

47: 6

49: 6xx6

51: (x3)(x+4)(x+1)(x+2)

53: x7x+5

55: −3/5

57: −3

59: −10, 1

61: 3, 8

63: 3

65: Ø

67: a=bcb+c

69: 6, 12

71: 7.5 miles per hour

73: 9 minutes

75: y=3x

77: y=3xz

79: d=16t2; 64 feet

Sample Exam Answers

1: 5x2(3x1); x0,13

3: x32x1; x4,12

5: 15x(x+5)

7: x+22x

9: 2x(x5)(x+5)

11: xyxy

13: (fg)(x)=x+94x3; x34,9

15: 3/5

17: 4, 5

19: 4

21: r=40P13

23: 3 hours

25: y=149x2; 16 feet