course Mth 158
R.1 Real Numbers Objectives: 1.) Classify Numbers
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2.) Evaluate Numerical Expressions
3.) Work with Properties of Real Numbers
Classification of Numbers
•Counting Numbers, or natural numbers, are used to count. {1, 2, 3, 4, …}
•Whole Numbers are counting numbers with 0. {0, 1, 2, 3, 4, …}
•Integers include negative numbers. {…, -3, -2, -1, 0 , 1, 2, 3, …}
•Rational Number is a fraction made up of two integers. {3/4, ½, 0/4}
•Irrational Number is decimals or square roots of numbers. {0.25, 0.5}
•Real Numbers are all of these kinds of numbers as described above.
Approximations
•Truncation – drop the digits that follow the specified final digit in the decimal.
•Rounding – round up if the next digit is 5 or more and round down if the next digit is 4 or less from the final specified digit in the decimal.
Order of Operations
•Begin with the innermost parentheses and work outward. When dividing two expressions the numerator and the denominator are treated like they were in parentheses.
•Do multiplication and division working from left to right.
•Do addition and subtraction working from left to right as well.
Properties of Real Numbers
•The reflexive property says that a number will always equal itself, b = b.
•The symmetric property says that if c = b the b = c.
•The transitive property says that if b = c and c = d the b = d
•The principle of substitution says that if b = c then you can substitute c for b in any expression containing b.
Now Work Problems:
11. A. Natural Numbers – {1}
B. Integers – {0, 1}
C. Rational Numbers – {0, 1, ½, 1/3, ¼}
D. Irrational Numbers – {None}
E. Real Numbers – {0, 1, ½, 1/3, ¼}
15. A. 18.953
B. 18.952
27. 3 + 2 = 5
39. -6 + 4 * 3
-6 + 12
6
45. 6 – [3 * 5 + 2 * (3 – 2)]
6 – [15 + 2 * 1]
6 - [15 + 2]
6 – 17
-11
53. 4 + 8 / 5 – 3
12/2
6
55. 3/5 * 10/21
30/105
2/7
59. ¾ + 2/5
15/20 + 8/20
23/20
63. 5/18 + 1/12
10/36 + 3/36
13/36
69. 5/18 / 11/27
5/18 * 27/11
135/198
15/22
71. 6(x + 4)
6x + 24
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R.2 Algebra Review
Objectives: 1.) Graph Inequalities
2.) Find Distance on the Real Number Line
3.) Evaluate Algebraic Expressions
4.) Determine the Domain of a Variable
5.) Use the Laws of Exponents
6.) Evaluate Square Roots
7.) Use a Calculator to Evaluate Exponents
8.) Use Scientific Notation
The Real Number Line
•The origin is the point in the center of the number line.
•The real number line consists of different classes of real numbers: negative real numbers, zero, and positive real numbers.
Inequalities
•1 < 2
Absolute Value
•|a| = a if a >(or equal to) 0
•|a| = -a if a < 0
Square Roots
•Negative numbers do not have square roots.
•The square root of 0 is 0.
•The square root of a positive number is positive.
•If a >(or equal to) 0 then (sqrt(a))^2 = a
Scientific Notation
•Count the number of places a decimal point must move in order to make a number less then 10. Then you write the number(x) x 10^number of decimal places.
Now Work Problems
15. -1 > -2
25. x < 2
37. d(A,E) = d(-3, 3) = |3 + 3| = |6| = 6
39. -2 + 2 * 3
-2 + 6
4
47. |3 – 2| = |1| = 1
57. A. 3^2 – 1 / 3
8/3
B. 1^2 – 1 / 1
0/1
0
C. 0^2 -1 / 0
-1/0 This is excluded because you cannot have 0 in the denominator.
D. -1^2 – 1 / -1
1 – 1 / -1
0/-1
75. 4^-2
1 / 4^2
1/16
77. 3^-6 * 3^4
3^(-6 + 4)
3^-2
1 / 3^2
1/9
83. sqrt(-4^2)
|-4|
4
87. (x^2y^-1)^2
(x^2 / y)^2
x^4 / y^2
95.2 * 2 *-1^-1
4 / -1
-4
113. 6.1^-3
0.004
119. 4.542 x 10^2
127. 61,500
151. 4000,000,000 meters
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R.3 Geometry Review
Objectives: 1. Use the Pythagorean Theorem and Its Converse
2. Know Geometry Formulas
Pythagorean Theorem
•Used for right triangles.
•c^2 = a^2 + b^2
Geometry Formulas
•Rectangle:
Area = length x width
Perimeter = 2 * length x 2 * width
•Triangle:
Area = ½ base x height
•Circle:
Area = pie(r)^2
Circumference = 2(pie)(r) = pie(diameter)
•Closed Rectangular Box
Volume = length x width x height
Surface Area = 2(length x width) + 2(length x height) + 2(width x height)
•Sphere
Volume = 4/3 pie(r)^3
Surface Area = 4 pie(r)^2
•Cylinder
Volume = pie(r)^2 x height
Surface Area = 2(pie)(r)^2 + 2(pie)(r)(height)
Now Work Problems
9. 10^2 + 24^2
100 + 576
676 take sqrt of 676
c = 26
17. 25^2 = 7^2 + 24^2
625 = 49 + 576
625 = 625
It is a right triangle and the hypotenuse is 25.
25. A = pie (5)^2
A = 25pie meters^2
C = 2(pie)(5)
C = 10pie meters
39. 10^2 – 6^2
100 – 36
64 ft^2
43. 20 ft / 5280 ft = 0.003788 mile
d^2 = (3960 + 0.003788)^2 – 3960^2
d^2 = 30
d = 5.477 miles
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R.4 Polynomials
Objectives: 1.) Recognize Monomials
2.) Recognize Polynomials
3.) Add and Subtract Polynomials
4.) Multiply Polynomials
5.) Know Formulas for Special Products
A monomial is in the form of ax^k, where a is the constant, x is the variable, and k is the integer. a is called the coefficient and k is the degree of the monomial.
A polynomial is in the form of anx^(n) + a(n – 1)x^(n – 1).
Adding and Subtracting Polynomials
•Combine like terms.
Multiplying Polynomials
•Use the FOIL method, first, outer, inner, last.
Now Work Problems
7. It is a monomial. The variable is x; coefficient is 2; and the degree is 3
17. Yes it is a polynomial; it’s degree is 2.
29. (x^3 – 2x^2 + 5x + 10) – (2x^2 – 4x + 3)
x^3 – 2x^2 + 5x + 10 – 2x^2 + 4x – 3
x^3 – 4x^2 + 9x + 7
41. x(x^2 + x – 4)
x^3 + x^2 – 4x
47. (x + 2)(x + 4)
x^2 + 4x + 2x + 8
x^2 + 6x + 8
55. (2x + 3)(x – 2)
2x^2 – 4x + 3x – 6
2x^2 – x – 6
65. (x – 7)(x + 7)
x^2 + 7x – 7x – 49
x^2 – 49
67. (2x + 3)(2x – 3)
4x^2 – 6x + 6x – 9
4x^2 – 9
69. (x + 4)(x + 4)
x^2 + 4x + 4x + 16
x^2 + 8x + 16
79. (3x + y)(3x – y)
9x^2 – 3xy + 3xy – y^2
9x^2 – y^2
85. (x – 2)(x – 2)(x – 2)
(x – 2)(x^2 – 4x + 4)
x^3 – 4x^2 + 4x – 2x^2 + 8x – 8
x^3 – 6x^2 + 12x – 8
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