1. A SELECTION OF PROBLEMS INTENDED TO MAINTAIN STUDENTS ESSENTIAL ALGEBRA SKILLS INSTRUCTOR COPY Review for College Mathematics

2. Equations and Modeling A moving company charges $46 to move a certain machine 10 miles and $58 to move the same machine 30 miles. a) Find an equation that defines this relationship if it is linear. b) What will it cost to move the machine 25 miles? c) What is the minimum charge for moving the machine? d) What is the rate for each mile the machine is moved? Solution: a) b) c) d) 1 MATHEMATICS for COLLEGE READINESS C = 0.6 m + 14 $55 $40 $0.60 per mile

3. Equations and Modeling The total cost of producing a certain item consists of paying rent for the building and paying a fixed amount per unit for material. The total cost is $250 it 10 units are produced and $330 if 30 units are produced. a) Find the equation that defines this relationship if it is linear. b) What will it cost to produce 100 units? c) How much is paid in rent? d)What is the cost of the material for each unit? Solution: a) b) c) d) 2 MATHEMATICS for COLLEGE READINESS C = 4x + 210, where x = number of units produced. $610 $210 $4 per unit

4. Equations and Modeling If x 2 + 2x = 3, demonstrate algebraically which of the following is a possible value for x. a) – 3 b) –2 c) –1 d) 0 e) 3 Solution: a) – 3 3 MATHEMATICS for COLLEGE READINESS

5. Equations and Modeling How many solutions are there to the systems of equations? Solve algebraically. Solution: Infinite: the lines coincide so the solution is 4 MATHEMATICS for COLLEGE READINESS

6. Equations and Modeling The sum of the square of a number and the product of the number and 3 equals 40. a) Write an equation to model this situation. b) What are the positive answer(s)? Solution: a) b) 5 MATHEMATICS for COLLEGE READINESS n =5

7. Equations and Modeling Two truckers start on a trip at the same time. One heads west at 60 mph while the other heads east at 55 mph. Their CB radios work for a distance up to 500 miles apart. How long will they be able to communicate with each other? Model and solve algebraically. Solution: 4 hr 20 min. 6 MATHEMATICS for COLLEGE READINESS

8. Equations and Modeling Solve the system of equations algebraically. Solution: empty set: The lines are parallel. 7 MATHEMATICS for COLLEGE READINESS

9. Equations and Modeling Solve for w: p = 2 ( l + w ) Give your answer in two forms, one answer with a single fraction and the other as a sum or difference of two terms. Solutions: 8 MATHEMATICS for COLLEGE READINESS

10. Equations and Modeling Solve for a: Solution: 9 MATHEMATICS for COLLEGE READINESS

11. Equations and Modeling Solve for x: Solution: 10 MATHEMATICS for COLLEGE READINESS

12. Equations and Modeling The Boosters Club held a spaghetti dinner as a fundraiser. They sold 300 tickets and collected $2200. If an adult’s ticket cost $8.50 and a child’s ticket cost $3.50, how many of each were sold? Solution: 230 adult tickets 70 children’s tickets 11 MATHEMATICS for COLLEGE READINESS

13. Equations and Modeling Admission to a school football game was $2 for students and $3 for nonstudents. How many of each group attended if there were 900 people and $1920 collected? Model and solve algebraically. Solution: 120 nonstudents. 780 students. 12 MATHEMATICS for COLLEGE READINESS

14. Equations and Modeling Solve algebraically: x 2 – 5 = 3x Solutions: 13 MATHEMATICS for COLLEGE READINESS

15. Bill needs to make 1000 copies before the meeting that starts in 15 minutes. He begins using a machine that makes 30 copies per minute after laminates, a newer machine which copies at 45 pages per minutes become free. Using both machines, will Bill be able to make all the copies before the meeting? Explain algebraically. 14 MATHEMATICS for COLLEGE READINESS Equations and Modeling Solution: No, it will take almost 17 min to make the copies.

16. 15 MATHEMATICS for COLLEGE READINESS Equations and Modeling Find the equation of a line with slope of 0 through (6, 10). Solution:y = 10

17. Function Theory MATHEMATICS for COLLEGE READINESS 16 Which of the following are functions? a.x 3 y = 12 b.x 2 + y 2 = 20 c.x 3 + y = 15 d.x (x+2) = 5 + y e. f.y(y + 2) = 4 + x g. x = 4 h. y = 5 Solution: c, d, e, h

18. Function Theory MATHEMATICS for COLLEGE READINESS 17 If f(x) = x 2 – 5, find f(b) – f( c). Solution:b 2 – c 2

19. Function Theory MATHEMATICS for COLLEGE READINESS 18 If f(x) = 2x 2 – x + 3, find f(x + h). Solution:

20. Function Theory MATHEMATICS for COLLEGE READINESS 19 If f(x) = x 2 – 4, find f(a + 5). Solution:a 2 +10a+21

21. Function Theory MATHEMATICS for COLLEGE READINESS 20 If f(x) = x 2 – 3 x + 5, find f(x + h) – f(x). Solution: 2xh + h 2 – 3h

22. Function Theory MATHEMATICS for COLLEGE READINESS 21 If f(x) = 3x 4 – 4x 3 -2x 2 +x - 5, find f(-x). Solution: 2xh + h 2 – 3h

23. Expressions with Exponents MATHEMATICS for COLLEGE READINESS 22 Simplify: Solution:

24. Simplify. Give the answer with positive exponents only: 23 MATHEMATICS for COLLEGE READINESS Expressions with Exponents Solution:

25. Simplify. Give the answer with positive exponents only: 24 MATHEMATICS for COLLEGE READINESS Expressions with Exponents Solution :

26. Multiply a. b. c. 25 MATHEMATICS for COLLEGE READINESS d. e. Solution: Expressions with Exponents

27. 26 MATHEMATICS for COLLEGE READINESS Solution: Expressions with Exponents Multiply

28. 27 MATHEMATICS for COLLEGE READINESS Solution: Expressions with Exponents Multiply

29. If the area of a circle if what is the area of a circle if the radius is 3a? 28 MATHEMATICS for COLLEGE READINESS Expressions with Exponents Solution:

30. If the volume of a sphere is what is the volume of a sphere with radius 3a? 29 MATHEMATICS for COLLEGE READINESS Expressions with Exponents Solution:

31. If the volume of a cylinder is find the volume of a cylinder with radius 3x and a height that is twice the radius. 30 MATHEMATICS for COLLEGE READINESS Expressions with Exponents Solution:

32. If, then x = a. 3 + ln t b. 31 MATHEMATICS for COLLEGE READINESS Exponential and Logarithmic Functions c. d. ln ( t – 3 ) e. ln t – 3 Solution:e. ln t – 3

33. If 7 x = 3, then x = a. b. 32 MATHEMATICS for COLLEGE READINESS c. d. e. Solution: Exponential and Logarithmic Functions

34. 33 MATHEMATICS for COLLEGE READINESS Simplifying Expressions Simplify a. b. c. d. Solution: a

35. 34 MATHEMATICS for COLLEGE READINESS Simplifying Expressions Simplify a. b. c. d. Solution: c

36. 35 MATHEMATICS for COLLEGE READINESS Simplifying Expressions Simplify Solution: 5k 2 + 250k -16

37. 36 MATHEMATICS for COLLEGE READINESS Simplifying Expressions Evaluate the following expression using x = -3.1 and y = 4.23 4x -2 + y 3 Solution: 76.103

38. Factor completely using integer coefficients. 37 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Solution: 12 m 2 n 3 (m 2 + 4)

39. Factor completely using integer coefficients. 38 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Solution: (4m – 3) (3m + 2)

40. 39 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Which of the following is a factor of a. b. c. d. not factorable Solution: c. (4x + 3)

41. 40 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Which of the following is a factor of a. b. c. d. Solution:b. (x + 2)

42. 41 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Write in simplest radical form. Solution :

43. 42 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Write in simplest radical form. Solution: 7x 2 y

44. 43 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Write in simplest radical form. Solution :

45. 44 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Write in simplest radical form. Solution :

46. 45 MATHEMATICS for COLLEGE READINESS Algebraic Expressions Perform the indicated operations and reduce answers to lowest terms. a. b. c. d. Solution : d

47. Solve algebraically. 46 MATHEMATICS for COLLEGE READINESS Quadratic Equations Solution:

48. Solve algebraically. 47 MATHEMATICS for COLLEGE READINESS Quadratic Equations Solution:

49. Solve algebraically. 48 MATHEMATICS for COLLEGE READINESS Quadratic Equations Solution:

50. Solve algebraically. 49 MATHEMATICS for COLLEGE READINESS Quadratic Equations Solution:

51. Solve algebraically. 50 MATHEMATICS for COLLEGE READINESS Quadratic Equations Solution:

52. Rational Expressions MATHEMATICS for COLLEGE READINESS 51 State the restricted values: Solution: x ≠ –3, 2

53. Over the set of real numbers, state where the function is undefined: 52 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution: The expression is undefined over the set of real numbers where -2<x<2 or x = -3 or 3.

54. Find the product: 53 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution :

55. Calculate and write the answer in scientific notation. a. b. c. d. 54 MATHEMATICS for COLLEGE READINESS Rational Expressions aSolution:

56. Find the area of a rectangle if the length is and the width is 55 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution:

57. Add and simplify 56 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution:

58. Simplify 57 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution:

59. Simplify 58 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution:

60. Simplify: 59 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution:

61. Simplify: a. b. c. d. 60 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution: b

62. Simplify: a. b. c. d. 61 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution : a

63. Simplify: a.c. b. d. 62 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution: d

64. What values for x are not allowed? a.c. b.d. 63 MATHEMATICS for COLLEGE READINESS Rational Expressions Solution:b

65. 64 MATHEMATICS for COLLEGE READINESS Miscellaneous Which of the following sets of numbers could NOT be the measures of the sides of a right triangle? a.c. b.d. Solution: a

66. 65 MATHEMATICS for COLLEGE READINESS Miscellaneous Find the length of the hypotenuse of the right triangle whose legs are 5 and Solution:6

67. 66 MATHEMATICS for COLLEGE READINESS Miscellaneous Find the distance between (-3,5) and (-7,8). Solution: 5

68. 67 Miscellaneous MATHEMATICS for COLLEGE READINESS Find the coordinates of the point midway between (-5,2) and (3,-10). a. (-4,6) b. (-1,-4) c. (1,6) d. (-4,-1) Solution: b. (-1, -4)

69. 68 MATHEMATICS for COLLEGE READINESS Miscellaneous MATHEMATICS for COLLEGE READINESS Find y in the diagram below. a. 24 b. 18 c. 4 d. 8 Solution: 12 6 9 y a

70. 69 MATHEMATICS for COLLEGE READINESS Miscellaneous MATHEMATICS for COLLEGE READINESS Find y in the diagram below. Solution: y 8 12 15 7.5

71. c. d. 70 MATHEMATICS for COLLEGE READINESS Variation Find an equation of variation where y varies inversely as and when a. b. Solution:d

72. 71 MATHEMATICS for COLLEGE READINESS Variation Y varies directly as the cube root of x. Y is 30 when x is 1. Find Y when x is 27. a. 10 b. 90 c. 15 d. 30 Solution: b. 90

73. The number of hours (H) it takes to do the job in inversely proportional to the number of people (P) working on it. It takes 4 people 12 hours to do the job. How long would it take 6 people to do the job? a. 18 hours b. 10 hours c. 8.5 hours d. 8 hours 72 MATHEMATICS for COLLEGE READINESS Variation Solution: d

74. Equations and Modeling Slides 1 – 15 Function TheorySlides 16 – 21 Expressions with ExponentsSlides 22 – 30 Exponential and Logarithmic FunctionsSlides 31 – 32 Simplifying ExpressionsSlides 33 – 36 Algebraic ExpressionsSlides 37 – 45 Quadratic EquationsSlides 45 – 50 Rational ExpressionsSlides 51 – 63 MiscellaneousSlides 64 – 69 VariationSlides 70 – 72 MATHEMATICS for COLLEGE READINESS Content Guide