Algebra Study Guide

Math 20 Exam 1 Study Guide

1 Vocabulary

There will be a short vocabulary section. Please know the definition of each of the following phrases/words.

Natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), irrational numbers (H), real numbers (R), solve, evaluate, variable, independent variable, dependent variable, linear equation, standard form, point-slope form, slope-intercept form, algebraic expression, equation, formula, perimeter, circumference, area, volume, relation, function, domain, and range.

2 Problems

Translate each phrase into an algebraic expression. You do not need to solve.

1. Eleven less than a number.

2. Three more than double a number.

3. A number decreased by negative one.

4. Fifty-four percent of a number is five.

5. Ten fewer than triple a number is five percent of fifty.

6. One-half the sum of a number and four.

7. The product of twenty-six and a number is seventy-eight.

8. The quotient of triple a number and seven is fifteen.

9. One hundred increased by fifteen percent of a number.

10. Eleven times the total of a number, eight, and two.

Determine whether each of the following numbers is rational.

11. 123.8080080008 . . .

12. 7

15

13. π

14.

√ 25

121

15. 0.9̄ = 0.999 . . .

16. 5.535353 . . .

1

{ − 2, −

7

5 , π, 1,

1

3 , √ 7, 5, 12.343434 . . . , 6.72

} Which of the elements in the above set belong in the

17. set of natural numbers?

18. set of whole numbers?

19. set of integers?

20. set of rational numbers?

21. set of irrational numbers?

22. set of real numbers?

Use order of operations to simplify the following.

23. 15 + 3[−4 − 5(6 − 4)2]

24. 3 − 6(5 − 3)3

23 − 4

Evaluate each expression for the given values.

25. √

(x2 − x1)2 + (y2 − y1)2 for x1 = −2, x2 = 10, y1 = 3, and y2 = 8.

26. −b +

√ b2 − 4ac 2a

for a = 3, b = 6, and c = −9.

27. x2

a2 +

y2

b2 for x = −3, y = −4, a = 5, and b = −5.

28. −s2 + 1 + 16r2

32 − 2 for s = −10 and r = 1

4 .

Solve each of the following. If there is no solution, write ∅. If there are infinitely many solutions, write R. (Note: if you find, for example, x = 2, you would write that as your answer. Only use ∅ if you reach a contradiction, like −1 = 7, and only use R if you reach something that is true for all values of x like 2 = 2.)

29. 3(x − 2) + 4 = 3x − 2

30. t − 2 5

+ 5t = 7

5 −

t − 2 2

31. 2(x − 2) = 2 3 (3x + 8) − 2

2

Solve the following formulas for the specified variable.

32. d = rt for t

33. A = 1 2 h(b1 + b2) for b1.

34. S = n(a1 + an)

2 for n.

The next few problems are from section 1.7 and 1.8 problems. On the exam, like on quiz 2 and quiz 21 2 ,

you will have options as to which type of problem you prefer. However, I cannot specify how problems like those below will be partitioned within the options for each problem on the exam.

35. A collection of coins consists of only dimes and quarters. The total value of the coins is $6.75, and there are twice as many dimes as quarters. How many quarters and dimes are in the collection?

36. An office supply store generated $5238 in calculator sales. Twenty-four more graphing calculators were sold than scientific calculators. How many of each type of calculator did the bookstore sell if scientific calculators cost $18 each and graphing calculators cost $87 each.

37. A board that is 23 feet long is cut into two pieces. One piece is 2 feet shorter than four times the length of the other. Find the length of each piece.

38. You are considering buying a printer, and you have narrowed your selection down to two options. One printer costs $99, and its ink cartridges are only $17 each. The other printer costs $80, and its ink cartridges are $30 each. How many ink cartridges would you have to buy for each printer to be equally as costly to you?

39. A couple is planning their wedding, and they have narrowed their section to two locations. One location costs $2,500 plus $32 per guest, and the other location is $1,550 plus $51 per guest. How many guests would have to be invited for both venues to be equally costly?

40. How many ounces of water should be mixed with nine fluid ounce of a 20% alcohol solution to obtain the mix that is 18% alcohol?

41. A train heads south from Huntington Beach at a rate of 12mph. Meanwhile, a train in San Diego heads north at a rate of 30mph. Suppose the tracks of the two trains run parallel to one another. If Huntington Beach is 98 miles north of San Diego, how long until the trains pass by each other? (This is a variation of a classic problem. However, usually the trains collide, which I think is a little too violent.)

42. Two marathon runners leave the starting gate, one running 13mph and the other 11.5mph. If they maintain the pace, how long will it take for them to be one-quarter of a mile apart?

Plot and label each point on the xy-coordinate system.

43. P(3, 0)

44. Q

( − 7

2 , 3

) 45. R(−5, −2)

46. S(−1, 4)

3

Find the midpoint of TU where

47. T

( − 3

5 , 7

) and U(1, 2).

48. T (−4, 7) and U ( 1,

2

3

)

Find the coordinates of V . If

49. the midpoint of V W is M(3, 4) and W(−3, 5).

50. the midpoint of V W is M(1, −3) and W ( − 5

9 , 7

) .

Graph each of the following. Make sure to use at least three points.

51. 2x + 3y = 12

52. y − 3 = −(x + 4)

53. y = x

3 − 4

54. y = x

55. y = −3

56. x = 2

Find the x- and y-intercepts of

57. The equation in problem 50.

58. The equation in problem 51.

59. The equation in problem 52.

60. The equation in problem 53.

What is the slope of

61. a line that passes through (7, 2) and (3, 4)?

62. the line x = 7?

63. the line y = −3?

64. the line with equation 5x − 2y = 4?

4

Find the equation of a line that passes through (−1, 3) that is (a) parallel and (b) perpendicular to the segment

65. AB such that A(−3, 2) and B(1, 2).

66. PQ such that P(1, 4) and Q(−1, −1).

.....

−4

.

−3

.

−2

.

−1

.

1

.

2

.

3

.

4

.

−4

.

−3

.

−2

.

−1

.

1

.

2

.

3

.

4

.

x

.

y

Find the equation of

67. the black line.

68. the blue line.

69. the red line.

70. the green line.

5

Determine whether each of the following is a function.

71. {(3, 4), (3, −4), (4, 3), (4, −3)}

72. {(−1, 1), (−3, 1), (−5, 1), (−7, 1), (−9, 1)}

73. x 1 2 3 4 5

y 7 15 23 16 8

74.

.....

x

.

y

Compute f(2) and f(−1) for

75. f(x) = x2 − 2

76. f(x) = 1 2 x − 3

77. f(x) = 2

x + 2

Graph each of the following.

78. f(x) = (x + 3)2 − 3

79. g(x) = |x − 2| + 1

80. h(x) = −(x + 1)2

6