1. Review for College Mathematics
A selection of problems intended to maintain students essential algebra skills
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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.
Find an equation that defines this relationship if it is linear.
What will it cost to move the machine 25 miles?
What is the minimum charge for moving the machine?
What is the rate for each mile the machine is moved?
Solution: a) b) c) d)
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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)
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4. Equations and Modeling
If x2 + 2x = 3, demonstrate algebraically which of the following is a possible value for x.
– 3 –2 –1 3
Solution:
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5. Equations and Modeling
How many solutions are there to the systems of equations? Solve algebraically. Solution:
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6. Equations and Modeling
The sum of the square of a number and the product of the number and 3 equals 40.
Write an equation to model this situation.
What are the positive answer(s)?
Solution: a) b)
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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:
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8. Equations and Modeling
Solve the system of equations algebraically. Solution:
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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:
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10. Equations and Modeling
Solve for a: Solution:
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11. Equations and Modeling
Solve for x: Solution:
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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? Model and solve algebraically. Solution:
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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:
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14. Equations and Modeling
Solve algebraically: x2 – 5 = 3x Solutions:
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15. Equations and Modeling
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.
Solution:
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16. Equations and Modeling
Find the equation of a line with slope of 0 through (6, 10).
Solution:
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17. Function Theory Which of the following are functions? x3 y = 12
x (x+2) = 5 + y e.
y(y + 2) = 4 + x
g. x = 4
h. y = 5
Solution:
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18. Function Theory If f(x) = x2 – 5, find f(b) – f( c) . Solution:
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19. Function Theory If f(x) = 2x2 – x + 3, find f(x + h) . Solution:
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20. Function Theory If f(x) = x2 – 4, find f(a + 5) . Solution:
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21. Function Theory If f(x) = x2 – 3 x + 5, find f(x + h) – f(x) .
Solution:
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22. Function Theory If f(x) = 3x4 – 4x3 -2x2 +x - 5, find f(-x). Solution:
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23. Expressions with Exponents
Simplify:
Solution:
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24. Expressions with Exponents
Simplify. Give the answer with positive exponents only:
Solution:
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25. Expressions with Exponents
Simplify. Give the answer with positive exponents only:
Solution:
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26. Expressions with Exponents
Multiply a. b. c. d. e.
Solution:
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27. Expressions with Exponents
Multiply
Solution:
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28. Expressions with Exponents
Multiply
Solution:
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29. Expressions with Exponents
If the area of a circle if what is the area of a circle if the radius is 3a?
Solution:
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30. Expressions with Exponents
If the volume of a sphere is what is the volume of a sphere with radius 3a?
Solution:
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31. Expressions with Exponents
If the volume of a cylinder is find the volume of a cylinder with radius 3x and a height that is twice the radius.
Solution:
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32. Exponential and Logarithmic Functions
If , then x =
a ln t b. c.
d. ln ( t – 3 )
e. ln t – 3
Solution:
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33. Exponential and Logarithmic Functions
If 7x = 3, then x = a. b. c. d. e.
Solution:
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34. Simplifying Expressions
Simplify a. b. c. d.
Solution:
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35. Simplifying Expressions
Simplify a. b. c. d.
Solution:
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36. Simplifying Expressions
Simplify
Solution:
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37. Simplifying Expressions
Evaluate the following expression using x = -3.1 and y = 4.23
4x-2 + y3
Solution:
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38. Algebraic Expressions
Factor completely using integer coefficients.
Solution:
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39. Algebraic Expressions
Factor completely using integer coefficients.
Solution:
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40. Algebraic Expressions
Which of the following is a factor of a. b. c.
d. not factorable
Solution:
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41. Algebraic Expressions
Which of the following is a factor of a. b. c. d.
Solution:
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42. Algebraic Expressions
Write in simplest radical form.
Solution:
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43. Algebraic Expressions
Write in simplest radical form.
Solution:
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44. Algebraic Expressions
Write in simplest radical form.
Solution:
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45. Algebraic Expressions
Write in simplest radical form.
Solution:
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46. Algebraic Expressions
Perform the indicated operations and reduce answers to lowest terms. a. b. c. d.
Solution:
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47. Quadratic Equations Solve algebraically. Solution:
Solution:
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48. Quadratic Equations Solve algebraically. Solution:
Solution:
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49. Quadratic Equations Solve algebraically. Solution:
Solution:
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50. Quadratic Equations Solve algebraically. Solution:
Solution:
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51. Quadratic Equations Solve algebraically. Solution:
Solution:
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52. Rational Expressions State the restricted values: Solution:
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53. Rational Expressions
Over the set of real numbers, state where the function is undefined:
Solution:
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54. Rational Expressions Find the product: Solution:
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55. Rational Expressions
Calculate and write the answer in scientific notation. a. b. c. d.
Solution:
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56. Rational Expressions
Find the area of a rectangle if the length is and the width is
Solution:
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57. Rational Expressions Add and simplify Solution:
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58. Rational Expressions Simplify Solution:
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59. Rational Expressions Simplify Solution:
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60. Rational Expressions Simplify: Solution:
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61. Rational Expressions Simplify: a. b. c. d. Solution:
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62. Rational Expressions Simplify: a. b. c. d. Solution:
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63. Rational Expressions Simplify: a. c. b. d. Solution:
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64. Rational Expressions What values for x are not allowed? a. c. b. d.
Solution:
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65. 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:
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66. Miscellaneous
Find the length of the hypotenuse of the right triangle whose legs are 5 and
Solution:
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67. Miscellaneous Find the distance between (-3,5) and (-7,8). Solution:
Solution:
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68. Miscellaneous (-4,6) (-1,-4) (1,6) (-4,-1) Solution:
Find the coordinates of the point midway between (-5,2) and (3,-10).
(-4,6)
(-1,-4)
(1,6)
(-4,-1)
Solution:
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69. Miscellaneous 24 18 4 8 Find y in the diagram below. Solution: 9 6 12
Solution: 12 6 9 y
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70. Miscellaneous Find y in the diagram below. Solution: 12 8 y 15
Solution: y 8 12 15
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71. Variation
Find an equation of variation where y varies inversely as and when
Solution:
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72. Variation
Y varies directly as the cube root of x. Y is 30 when x is 1. Find Y when x is 27. 10 90 15 30
Solution:
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73. Variation
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?
18 hours
10 hours
8.5 hours
8 hours
Solution:
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74. Content Guide
Equations and Modeling Slides 1 – 15 Function Theory Slides 16 – 21 Expressions with Exponents Slides 22 – 30 Exponential and Logarithmic Functions Slides 31 – 32 Simplifying Expressions Slides 33 – 36 Algebraic Expressions Slides 37 – 45 Quadratic Equations Slides 45 – 50 Rational Expressions Slides 51 – 63 Miscellaneous Slides 64 – 69 Variation Slides 70 – 72
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