Download Conceptual Questions

11/11/10 AMATYC Boston, MA
Erica Kwiatkowski-Egizio
Conceptual Questions
1. Do the commutative and associative properties work for subtraction/division?
Explain by providing examples to support your answer.
2. How is the opposite or additive inverse different from the reciprocal?
3. Why is a negative number times a positive number equal to a negative number?
4. Why is a negative number times a negative number equal to a positive number?
5. Why is division by zero undefined?
6. Name the only real number that has no reciprocal, and explain why this is so.
7. Name the only real number that is its own opposite, and explain why this is so.
8. What does it mean to distribute?
9. What does it mean to factor?
10. What does it mean to simplify?
11. Explain why 32 and (3)2 simplify to different numbers.
12. Explain why 33 and (3)3 simplify to the same number.
13. Can the absolute value be zero/negative? Explain why or why not.
14. What are some real-world applications of negative numbers?
15. When (8.2)7 is evaluated, will the value be positive or negative? How can you tell
without making any calculations?
16. What is the order of operations?
17. Explain what it means to solve an equation.
18. When solving equations, what are the three possibilities for solutions?
19. What are two real-world applications of percent?
20. Are the following two inequalities equivalent: x < 4 and 4 < x? Why or why not?
21. Explain the difference between equivalent expressions and equivalent equations.
22. Define what each of the following mean: (7, 15), [7, 15], (7, 15], and [7, 15).
23. Why do you switch the direction of an inequality when multiplying/dividing by a
negative number?
24. Why doesn’t  or   have [ ] around them for interval notation?
25. Explain what is wrong with the interval notation (6,   ).
26. Explain how solving a linear inequality is similar to solving a linear equation.
27. Explain how solving a linear inequality is different from solving a linear equation.
28. Without taking any solution steps, how do you know that the absolute value
inequality |3x  2| > 9 has a solution? What is the solution?
29. Determine whether each of the following is true or false. Explain the reason for
your choice.
a.
45 + 43 = 45 + 3
true
false
b.
45  43 = 45 + 3
true
false
c.
(32)3 = 32+3
true
false
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11/11/10 AMATYC Boston, MA
Erica Kwiatkowski-Egizio
30. What are some real-world applications of scientific notation?
31. Explain in your own words when exponents should be added and when they should
be multiplied.
32. Is it better to evaluate a polynomial before or after like terms have been collected?
Why?
33. Is the sum of two binomials ever a trinomial? Explain.
34. Under what conditions will the product of two binomials be a trinomial?
35. Under what conditions will the product of two binomials be a binomial?
36. How is the degree of a polynomial in one variable different from the degree of a
polynomial in several variables?
37. Do all graphs of linear equations have y-intercepts? Why or why not?
38. Do all graphs of linear equations have x-intercepts? Why or why not?
39. Define slope in your own words.
40. What is the slope of a horizontal line? Explain why.
41. What is the slope of a vertical line? Explain why.
42. Explain how merely looking at a line can tell you whether its slope is negative,
positive, undefined, or zero.
43. In your own words explain how to graph an equation using its slope and y-intercept.
44. Create a relation that is also a function.
45. Create a relation that is not a function.
46. Suppose y = f(x) and we are told that f(3) = 9. Which is not true and why?
a. When x = 3, y = 9.
b. A possible function is f(x) = x2.
c. A point on the graph of the function is (3, 9).
d. A possible function is f(x) = 2x + 4.
47. How is finding the domain/range from a graph different from finding the
domain/range without a graph?
48. What are the three possibilities for solutions when solving a system of equations? Be
able to sketch a graph of these possibilities as well.
49. How can you tell by just looking at the following system that it has no solution
(without graphing it out)?
y = 3x + 5
y = 3x  7
50. How can you tell by just looking at the following system that it has infinitely many
solutions (without graphing it out)?
x+y=5
2x + 2y = 10
51. Name two cases in which it would be difficult to find the solution to a system using
the graphing method.
52. Suppose the graph of the equations in a system of two equations in two variables
consists of a circle and a line. Discuss the possible number of solutions for this
system.
53. Write a system of three linear equations in three variables that has (2, 1, 5) as a
solution.
54. Name three types of application problems that use systems of linear equations in
either two or three variables to solve them.
55. When graphing the solution of a system of inequalities, are you graphing the union or
the intersection of the inequalities? Explain.
56. Give an example of a linear function, quadratic function, cubic function and a quartic
function.
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11/11/10 AMATYC Boston, MA
Erica Kwiatkowski-Egizio
57. How do you find the domain and range of a polynomial function?
58. How do you find the x and y-intercepts of a polynomial function?
59. What does the graph of a quadratic function look like?
60. How do you find the vertex of a parabola?
61. What does the graph of a cubic function look like?
62. What does it mean for a rational expression to be undefined?
63. If the LCM (Least Common Multiple) of a binomial and a trinomial is the trinomial,
what relationship exists between the two expressions?
64. In your own words, explain how to add rational expressions with different
denominators.
65. What are two differences between rational expressions and rational equations?
66. Why is it especially important to check the possible solutions to a rational equation?
67. Create your own rational equation word problem.
68. Define direct, inverse and joint variation in your own words.
69. Give a real-life example for direct, inverse and joint variation.
70. Determine if each of the following sets of ordered pairs represents direct variation,
inverse variation or neither. If the set of ordered pairs represent direct variation or
inverse variation, determine the “k” value or constant of variation.
a. (2, 4.5) (5.5, 3.625) (8, 3) (11, 2.25) (19, 0.25)
b. (0.5, 720) (2, 180) (8, 45) (15, 24) (120, 3)
c. (1, 0.75) (3, 2.25) (7, 5.25) (16, 12) (42, 31.5)
71. What is the difference between a square root and a principal square root?
72. Explain why  16 can be evaluated as a real number, but  16 cannot be evaluated
as a real number.
73. In finding n a n , why is it necessary to use absolute value in the answer when n is an
even positive integer? Why is the absolute value not necessary if n is an odd positive
integer?
3
27 3 27 3
74. Find and correct the error:

 3.
9
9
75. True or false? a  b  a  b . Explain. Be sure to provide an example with
numbers if it is false to show this.
76. True or false?  x  2  x  2  x 2  4 . Explain. Be sure to provide an
example with numbers if it is false to show this.
77. In each case, determine by which number both the numerator and denominator should
be multiplied to rationalize the denominator of the radical expression.
1
1
a. 3
b.
4
7
8
78. Is it possible to rationalize both the numerator and denominator at the same time?
Explain.
79. Explain why rationalizing the denominator does not change the value of the original
Expression.
80. Why is it especially important to check ALL solutions to radical equations?
81. How can you tell immediately that the equation 2 y  3  4 has no real solution?
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11/11/10 AMATYC Boston, MA
Erica Kwiatkowski-Egizio
82. What is the domain of the radical expression?
83.What is wrong with the following solution?
2x  5  4  x  8

2x  5  4  x

2
 82
2 x  5  4  x   64
x  9  64
x  55
84. What three conclusions does the discriminant give you about the nature and type of
solutions to a quadratic equation of the form ax 2  bx  c  0 ?
85. True or False? The maximum number of solutions that a quadratic equation can have
is two. Explain.
86. True or False? The maximum number of solutions that an equation in quadratic form
can have is two. Explain.
87. What three conclusions does the discriminant give you in regards to the number of
x-intercepts a quadratic function will have?
88. Explain how you might go about constructing a quadratic equation whose solutions
are 5 and 7.
89. True or False? Every quadratic equation can be solved using the factoring method.
Explain.
90. True or False? Every quadratic equation can be solved using the quadratic formula.
Explain.
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