BASIC ALGEBRA

BASIC ALGEBRA - page 1

Basic algebra problems ask you to solve equations in which one or
more elements are unknown. The unknown quantities are represented by variables, which are letters of the alphabet, such as x or y. The questions in this chapter give you practice in writing algebraic equations and using these expressions to solve problems.

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* 1. Assume that the number of hours Katie spent practicing soccer is represented by x. Michael practiced 4 hours more than 2 times the number of hours that Katie practiced. How long did Michael

practice?

2x + 4

2x − 4

2x + 8

4x + 4

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* 2. Patrick gets paid three dollars less than four times what Kevin gets paid. If the number of dollars that Kevin gets paid is represented by x, what does Patrick get paid?

3 − 4x

3x − 4

4x − 3

4-3x

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* 3. If the expression 9y − 5 represents a certain number, which of the following could NOT be the translation?

five less than nine times y

five less than the sum of 9 and y

the difference between 9y and 5

the product of nine and y, decreased by 5

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* 4. Susan starts work at 4:00 and Dee starts at 5:00. They both finish at the same time. If Susan works x hours, how many hours does Dee work?

x + 1

x − 1

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* 5. Frederick bought six books that cost d dollars each. What is the total cost of the books?

d + 6

d + d

d/6

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* 6. There are m months in a year, w weeks in a month and d days in a week. How many days are there in a year?

mwd

m + w + d

mw/d

d + w/d

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* 7. Carlie received x dollars each hour she spent babysitting. She babysat a total of b hours. She then gave half of the money to a friend who had stopped by to help her. How much money did Carlie have after she had paid her friend?

bx / 2

x/2 + b

b/2 + x

2bx

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* 8. A long distance call costs x cents for the first minute and y cents for each additional minute. How much would a 5-minute call cost?

5xy

x + 5y

xy / 5

x + 4y

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* 9. Melissa is four times as old as Jim. Pat is 5 years older than Melissa. If Jim is y years old, how old is Pat?

4y + 5

5y + 4

4 × 5y

y + 5

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* 10. Sally gets paid x dollars per hour for a 40-hour work week and y dollars for each hour she works over 40 hours. How much did Sally earn if she worked 48 hours?

48xy

40y + 8x

40x + 8y

48x + 48y

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* 11. Eduardo is combining two 6-inch pieces of wood with a piece that measures 4 inches. How many total inches of wood does he have?

10 inches

16 inches

8 inches

12 inches

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* 12. Mary has $2 in her pocket. She does yard work for four different neighbors and earns $3 per yard. She then spends $2 on a soda. How much money does she have left?

$18

$10

$12

$14

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* 13. Ten is decreased by four times the quantity of eight minus three. One is then added to that result. What is the final answer?

−5

−9

−8

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* 14. The area of a square whose side measures four units is added to the difference of eleven and nine divided by two. What is the total value?

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* 15. Four is added to the quantity two minus the sum of negative seven and six. This answer is then multiplied by three. What is the result?

−21

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* 16. John and Charlie have a total of 80 dollars. John has x dollars. How much money does Charlie have?

80 + x

80 − x

x − 80

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* 17. The temperature in Hillsville was 20° Celsius. What is the equivalent of this temperature in degrees Fahrenheit?

4°

43.1°

68°

132°

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* 18. Peggy’s town has an average temperature of 23° Fahrenheit in the winter. What is the average temperature on the Celsius scale?

−16.2°

16.2°

5°

−5°

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* 19. Celine deposited $505 into her savings account. If the interest rate of the account is 5% per year, how much interest will she have made after 4 years?

$252.50

$606

$10,100

$101

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* 20. A certain bank pays 3.4% interest per year for a certificate of deposit, or CD. What is the total balance of an account after 18 months with an initial deposit of $1,250?

$765

$2,015

$63.75

$1,313.75

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* 21. Joe took out a car loan for $12,000. He paid $4,800 in interest at a rate of 8% per year. How many years will it take him to pay off the loan?

2.5

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* 22. What is the annual interest rate on an account that earns $948 in simple interest over 36 months with an initial deposit of $7,900?

40%

3.3%

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* 23. Marty used the following mathematical statement to show he could change an expression and still get the same answer on both sides: 10 × (6 × 5) = (10 × 6) × 5 Which mathematical property did Marty use?

Identity Property of Multiplication

Commutative Property of Multiplication

Distributive Property of Multiplication over Addition

Associative Property of Multiplication

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* 24. Tori was asked to give an example of the commutative property of addition. Which of the following choices would be correct?

3 + (4 + 6) = (3 + 4) + 6

3(4 + 6) = 3(4) + 3(6)

3 + 4 = 4 + 3

3 + 0 = 3

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* 25. Jake needed to find the perimeter of an equilateral triangle whose sides measure x + 4 cm each. Jake realized that he could multiply 3 (x + 4) = 3x + 12 to find the total perimeter in terms of x. Which property did he use to multiply?

Associative Property of Addition

Distributive Property of Multiplication over Addition

Commutative Property of Multiplication

Inverse Property of Addition

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* 26. The product of −5 and a number is 30. What is the number?

−6

−35

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* 27. When ten is subtracted from the opposite of a number, the difference between them is five. What is the number?

−15

−5

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* 28. The sum of −4 and a number is equal to −48. What is the number?

−12

−44

−52

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* 29. Twice a number increased by 11 is equal to 32 less than three times the number. Find the number.

−21

21/5

43/2

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* 30. If one is added to the difference when 10x is subtracted from −18x, the result is 57. What is the value of x?

−2

−7

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* 31. If 0.3 is added to 0.2 times the quantity x − 3, the result is 2.5. What is the value of x?

1.7

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* 32. If twice the quantity x + 6 is divided by negative four, the result is 5. Find the number.

−18

−16

−13

−0.5

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* 33. The difference between six times the quantity 6x + 1 and three times the quantity x − 1 is 108. What is the value of x?

22/11

35/11

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* 34. Negative four is multiplied by the quantity x + 8. If 6x is then added to this, the result is 2x + 32. What is the value of x?

No solution

Identity

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* 35. Patrice has worked a certain amount of hours so far this week. Tomorrow she will work four more hours to finish out the week with a total of 10 hours. How many hours has she worked so far?

2.5

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* 36. Michael has 16 CDs. This is four more than twice the amount that Kathleen has. How many CDs does Kathleen have?

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* 37. The perimeter of a square can be expressed as x + 4. If one side of the square is 24, what is the value of x?

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* 38. The perimeter of a rectangle is 21 inches. What is the measure of its width if its length is 3 inches greater than its width?

3.75

4.5

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* 39. The sum of two consecutive integers is 41. What are the integers?

20, 21

21, 22

20, 22

10.5, 10.5

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* 40. The sum of two consecutive even integers is 126. What are the integers?

62, 64

62, 63

60, 66

2, 63

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* 41. The sum of two consecutive odd integers is −112. What is the larger integer?

−55

−57

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* 42. The sum of three consecutive even integers is 102. What is the value of the largest consecutive integer?

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* 43. Two commuters leave the same city at the same time but travel in opposite directions. One car is traveling at an average speed of 63 miles per hour, and the other car is traveling at an average speed of 59 miles per hour. How many hours will it take before the cars are 610 miles apart?

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* 44. Two trains leave the same city at the same time, one going east and the other going west. If one train is traveling at 65 mph and the other at 72 mph, how many hours will it take for them to be 822 miles apart?

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* 45. Two trains leave two different cities 1,029 miles apart and head directly toward each other on parallel tracks. If one train is traveling at 45 miles per hour and the other at 53 miles per hour, how many hours will it take before the trains pass?

9.5

11.5

10.5

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* 46. Nine minus five times a number, x, is no less than 39. Which of the following expressions represents all the possible values of the number?

x ≤ 6

x ≥ −6

x ≤ −6

x ≥ 6

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* 47. Will has a bag of gumdrops. If he eats 2 of his gumdrops, he will have between 2 and 6 of them left. Which of the following represents how many gumdrops, x, were originally in his bag?

4 < x < 8

0 < x < 4

0 > x > 4

4 > x > 8

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* 48. The value of y is between negative three and positive eight inclusive. Which of the following represents y?

−3 ≤ y ≤ 8

−3 < y ≤ 8

−3 ≤ y < 8

−3 ≥ y ≥ 8

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* 49. Five more than the quotient of a number and 2 is at least that number. What is the greatest value of the number?

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* 50. Carl worked three more than twice as many hours as Cindy did. What is the maxim sum amount of hours Cindy worked if together they worked 48 hours at most?

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* 51. The cost of renting a bike at the local bike shop can be represented by the equation y = 2x + 2, where y is the total cost and x is the number of hours the bike is rented. Which of the following ordered pairs would be a possible number of hours rented, x, and the corresponding total cost, y?

(0, −2)

(2, 6)

(6, 2)

(−2, −6)

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* 52. A telephone company charges $.35 for the first minute of a phone call and $.15 for each additional minute of the call. Which of the following represents the cost y of a phone call lasting x minutes?

y = 0.15(x − 1) + 0.35

x = 0.15(y − 1) + 0.35

y = 0.15x + 0.35

x = 0.15y + 0.35

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* 53. A ride in a taxicab costs $1.25 for the first mile and $1.15 for each additional mile. Which of the following could be used to calculate the total cost y of a ride that was x miles?

x = 1.25(y − 1) + 1.15

x = 1.15(y − 1) + 1.25

y = 1.25(x − 1) + 1.15

y = 1.15(x − 1) + 1.25

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* 54. The cost of shipping a package through Shipping Express is $4.85 plus $2 per ounce of the weight of the package. Sally only has $10 to spend on shipping costs. Which of the following could Sally use to find the maximum number of ounces she can ship for $10?

4.85x + 2 ≤ 10

4.85x + 2 ≥ 10

2x + 4.85 ≤ 10

2x + 4.85 ≥ 10

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* 55. Green Bank charges a monthly fee of $3 for a checking account and $.10 per check. Savings-R-Us bank charges a $4.50 monthly fee and $.05 per check. How many checks need to be used for the monthly costs to be the same for both banks?

100

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* 56. Easy Rider taxi service charges a pick-up fee of $2 and $1.25 for each mile. Luxury Limo taxi service charges a pick-up fee of $3.25 and $1 per mile. How many miles need to be driven for both services to cost the same amount?

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* 57. The sum of two integers is 36, and the difference is 6. What is the smaller of the two numbers?

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* 58. One integer is two more than another. The sum of the lesser integer and twice the greater is 7. What is the greater integer?

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* 59. One integer is four times another. The sum of the integers is 5. What is the value of the lesser integer?

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* 60. The sum of three times a greater integer and 5 times a lesser integer is 9. Three less than the greater equals the lesser. What is the value of the lesser integer?

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* 61. The perimeter of a rectangle is 104 inches. The width is 6 inches less than 3 times the length. Find the width of the rectangle.

13.5 inches

37.5 inches

14.5 inches

15 inches

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* 62. The perimeter of a parallelogram is 50 cm. The length of the parallelogram is 5 cm more than the width. Find the length of the parallelogram.

15 cm

11 cm

5 cm

10 cm

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* 63. Jackie invested money in two different accounts, one of which earned 12% interest per year and another that earned 15% interest per year. The amount invested at 15% was 100 more than twice the amount at 12%. How much was invested at 12% if the total annual interest earned was $855?

$4,100

$2,100

$2,000

$4,000

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* 64. Kevin invested $4,000 in an account that earns 6% interest per year and $x in a different account that earns 8% interest per year. How much is invested at 8% if the total amount of interest earned annually is $405.50?

$2,075.00

$4,000.00

$2,068.75

$2,075.68

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* 65. Megan bought x pounds of coffee that cost $3 per pound and 18 pounds of coffee at $2.50 per pound for the company picnic. Find the total number of pounds of coffee purchased if the average cost per pound of both types together is $2.85.

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* 66. The student council bought two different types of candy for the school fair. They purchased 40 pounds of candy at $2.15 per pound and x pounds at $1.90 per pound. What is the total number of pounds they bought if the total amount of money spent on candy was $158.20?

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* 67. The manager of a garden store ordered two different kinds of marigold seeds for her display. The first type cost her $1 per packet and the second type cost $1.26 per packet. How many packets of the first type did she purchase if she bought 50 more of the $1.26 packets than the $1 packets and spent a total of $402?

150

200

250

100

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* 68. Harold used a 3% iodine solution and a 20% iodine solution to make a 95-ounce solution that was 19% iodine. How many ounces of the 3% iodine solution did he use?

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* 69. A chemist mixed a solution that was 34% acid with another solution that was 18% acid to produce a 30-ounce solution that was 28% acid. How much of the 34% acid solution did he use?

11.25

18.75

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* 70. Bob is 2 years from being twice as old as Ellen. The sum of twice Bob’s age and three times Ellen’s age is 66. How old is Ellen?

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* 71. Sam’s age is 1 less than twice Shari’s age. The sum of their ages is 104. How old is Shari?

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* 72. At the school bookstore, two binders and three pens cost $12.50. Three binders and five pens cost $19.50. What is the total cost of 1 binder and 1 pen?

$4.50

$4.00

$1.50

$5.50

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* 73. Two angles are complementary. The larger angle is 15° more than twice the smaller. Find the measure of the smaller angle.

25°

65°

90°

82.5°

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* 74. The cost of a student ticket is $1 more than half of an adult ticket. Six adults and four student tickets cost $28. What is the cost of one adult ticket?

$2.50

$3.00

$5.50

$4.00

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* 75. Three shirts and five ties cost $23. Five shirts and one tie cost $20. What is the price of one shirt?

$3.50

$2.50

$6.00

$3.00

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* 76. Noel rode 3x miles on his bike and Jamie rode 5x miles on hers. In terms of x, what is the total number of miles they rode?

15x miles

15x^2 miles

8x miles

8x^2 miles

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* 77. If the areas of two sections of a garden are 6a + 2 and 5a, what is the difference between the areas of the two sections in terms of a?

a − 2

3a + 2

a + 2

11a − 2

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* 78. Laura has a rectangular garden whose width is x^3 and whose length is x^4. In terms of x, what is the area of her garden?

2x^7

x^7

x^12

2x^12

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* 79. Jonestown High School has a soccer field whose dimensions can be expressed as 7y^2 and 3xy. What is the area of this field in terms of x and y?

10xy^2

10xy^3

21xy^3

21xy^2

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* 80. The area of a parallelogram is x8. If the base is x4, what is the height in terms of x?

x^4

x^2

x^12

x^32

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* 81. The quotient of 3d^3 and 9d^5 is

3d^2.

3d^8.

1 / 3d^8

1 / 3d^2

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* 82. The product of 6x^2 and 4xy^2 is divided by 3x^3y. What is the simplified

expression?

4y / x

8y / x

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* 83. If the side of a square can be expressed as a2b3, what is the area of the square in simplified form?

a^4b^5

a^4b^6

a^2b^6

a^2b^5

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* 84. If 3x^2 is multiplied by the quantity 2x^3y raised to the fourth power, what would this expression simplify to?

48x^14y^4

1,296x^16y^4

6x^9y^4

6x^14y^4

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* 85. Sara’s bedroom is in the shape of a rectangle. The dimensions are 2x and 4x + 5. What is the area of Sara’s bedroom?

18x

18x^2

8x^2 + 5x

8x^2 + 10x

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* 86. Express the product of −9p^3r and the quantity 2p − 3r in simplified form.

−11p^4r + 12p^3r^2

−18p^4r + 27p^3r^2

−18p^4r − 3r

−18p^3r + 27p^3r^2

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* 87. A number, x, increased by 3 is multiplied by the same number, x, increased by 4. What is the product of the two numbers in terms of x?

x^2 + 7

x^2 + 12

x^2 + 7x + 12

x^2 + x + 7

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* 88. The length of Kara’s rectangular patio can be expressed as 2x − 1 and the width can be expressed as x + 6. In terms of x, what is the area of her patio?

2x^2 + 13x − 6

2x^2 − 6

2x^2 − 5x − 6

2x^2 + 11x −6

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* 89. A car travels at a rate of (4x^2 − 2). What is the distance this car will travel in (3x − 8) hours?

12x^3 − 32x^2 − 6x + 16

12x^2 − 32x^2 −6x + 16

12x^3 + 32x^2 − 6x − 16

12x^3 − 32x^2 − 5x + 16

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* 90. The area of the base of a prism can be expressed as x2 + 4x + 1 and the height of the prism can be expressed as x − 3. What is the volume of this prism in terms of x?

x^3 + x^2 − 13x − 3

x^3 + 7x^2 − 13x − 3

x^3 − x^2 − 11x − 3

x^3 + x^2 − 11x − 3

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* 91. The dimensions of a rectangular prism can be expressed as x + 1, x − 2, and x + 4. In terms of x, what is the volume of the prism?

x^3 + 3x^2 + 6x − 8

x^3 + 3x^2 − 6x − 8

x^3 + 5x^2 − 2x + 8

x^3 − 5x^2 − 2x − 8

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* 92. The area of Mr. Smith’s rectangular classroom is x2 − 25. Which of the following binomials could represent the length and the width of the room?

(x + 5)(x + 5)

(x − 5)(x − 5)

(x + 5)(x − 5)

x(x − 25)

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* 93. The area of a parallelogram can be expressed as the binomial 2x2 − 10x. Which of the following could be the length of the base and the height of the parallelogram?

2x(x^2 − 5x)

2x (x − 5)

(2x − 1)(x − 10)

(2x − 5)(x + 2)

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* 94. A farmer’s rectangular field has an area that can be expressed as the trinomial x^2 + 2x + 1. In terms of x, what are the dimensions of the field?

(x + 1)(x + 2)

(x − 1)(x − 2)

(x − 1)(x + 2)

(x + 1)(x + 1)

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* 95. Harold is tiling a rectangular kitchen floor with an area that is expressed as x^2 + 6x + 5. What could the dimensions of the floor be in terms of x?

(x + 1)(x + 5)

(x − 1)(x − 5)

(x − 2)(x + 3)

(x + 2)(x + 3)

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* 96. The area of a rectangle is represented by the trinomial: x^2 + x − 12. Which of the following binomials could represent the length and width?

(x + 4)(x − 3)

(x − 4)(x − 3)

(x − 4)(x + 3)

(x − 6)(x + 2)

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* 97. Katie’s school has a rectangular courtyard whose area can be expressed as 3x^2 − 7x + 2. Which of the following could be the dimensions of the courtyard in terms of x?

(3x − 1)(x + 2)

(3x − 1)(x − 2)

(3x − 2)(x − 1)

(3x + 2)(x + 1)

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* 98. The distance from the sun to the earth is approximately 9.3 × 10^7 miles. What is this distance expressed in standard notation?

930,000,000

93,700,000

0.00000093

93,000,000

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* 99. The distance from the earth to the moon is approximately 240,000 miles. What is this distance expressed in scientific notation?

24 × 10^4

240 × 10^3

2.4 × 10^5

2.4 × 10^−5

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* 100. It takes light 5.3 × 10^−6 seconds to travel one mile. What is this time in standard notation?

0.00000053

0.000053

5.300000

0.0000053