1. Real Numbers and Algebraic Expressions

2. The set {1, 3, 5, 7, 9} has five elements.
The Basics About Sets
The set {1, 3, 5, 7, 9} has five elements.
A set is a collection of objects whose contents can be clearly determined.
The objects in a set are called the elements of the set.
We use braces to indicate a set and commas to separate the elements of that set.

3. Real Numbers
The real number line
The set of natural (counting) numbers
The set of Whole numbers

4. The set of even counting numbers is a subset of the set of counting numbers, since each element of the subset is also contained in the set.

5. Set Builder Notation Example:

6. See Page 5. Intersection and Unions of two sets.

8. Integers
The set of integers

9. Important Subsets of the Real Numbers
-15, -7, -4, 0, 4, 7
Integers Z
0, 4, 7, 15
Whole Numbers W
4, 7, 15
Natural Numbers N
Examples
Name

10. Expressed as an integer divided by a nonzero integer:
Decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers.
Irrational Numbers I
Expressed as an integer divided by a nonzero integer:
Rational numbers can be expressed as terminating or repeating decimals.
Rational
Numbers Q
Examples
Description
Name

11. Definition
Rational Numbers – A number that can be written in the form a / b where a and b are integers.

12. The Real Numbers Rational numbers Irrational numbers Integers
The set of real numbers is formed by combining the rational numbers and the irrational numbers.
Rational numbers
Irrational numbers
Integers
Whole numbers
Natural numbers

13. The Real Number Line Negative numbers
Units to the left of the origin are negative.
Positive numbers
Units to the right of the origin are positive.
the Origin

14. Absolute Value
Absolute value describes the distance from 0 on a real number line. If a represents a real number, the symbol |a| represents its absolute value, read “the absolute value of a.”
For example, the real number line below shows that
|-3| = 3 and |5| = 5.
The absolute value of –3 is 3 because –3 is 3 units from 0 on the number line.
|–3| = 3
The absolute value of 5 is 5 because 5 is 5 units from 0 on the number line.
|5| = 5

15. Definition of Absolute Value
The absolute value of x is given as follows:
|x| =
x if x > 0
-x if x < 0 {

16. Properties of Absolute Value
For all real number a and b,
1. |a| > 0
2. |-a| = |a|
3. a < |a| a b
|a|
|b|
4. |ab| = |a||b|
= , b not equal to 0

17. CE
Evaluate the following

18. Homework
Page – 33 odd
52 – 66 Even

19. Section P.2
Exponents

20. Definition of Positive Exponents
If n is a positive integer and b is any real number, then
Where b is the base and n is the exponent.

21. Rules of Exponents

22. CE
Evaluate

23. CE
Evaluate

24. CE
Evaluate

25. Definition
If b is a real number not equal to zero, then

26. CE
Evaluate

27. CE
Evaluate

28. Definition
If n is an integer and b is a real number not equal to zero, then

29. CE
Evaluate

30. CE
Evaluate

31. CE
Evaluate

32. CE
Evaluate

33. CE
Evaluate

34. CE
Evaluate

35. CE
Evaluate

36. Homework
Page – 62 even

37. The Wonderful World Of Radicals

38. Definition of a Radical
is a radical;
n is the index
a is called the radicand.

39. Simplifying For any real number a,
The principal square root of is the absolute value of a.
See page 32

40. CE Simplify

41. Definition of Let a be a real number and n a positive integer 2.
If a > 0, and then for some positive integer x.
See the bottom of page 37

42. Continued
If a < 0 and n is odd, then is the negative number x such that .
If a < 0 and n is even, then is not a real number

43. CE Evaluate the expression, indicate that the root is not real.

44. Evaluate the expression, indicate that the root is not real.

45. The Product Rule for Square Roots
If a and b represent nonnegative real numbers, then:
See page 32 at the bottom

46. CE Using the product Rule (or Prime Factorization) to Simplify Square Roots

47. CE Evaluate the expression, indicate that the root is not real.

48. Definition For a real number a and positive integer n, Furthermore,
See the bottom of page 39

49. CE Simplify a. b. c.

50. Definition of Let be a rational number with
if b is a real number such that is defined, then
Reference Page 41 at the top

51. CE Simplify Using The Properties of Exponents.

52. CE Simplify Using The Properties of Exponents
CE Simplify Using The Properties of Exponents. We reviewed these the other day. Check your notes! .

53. The Quotient Rule for Square Roots
If a and b represent nonnegative real numbers, and then:
See page 33 at the bottom

54. CE Using the Quotient rule to Simplify Square roots

55. Finding the nth Root of Perfect nth Powers
If n is odd
If n is even
See top of page 38. This will be discussed in more detail in later classes.

56. CE Evaluate
Note: Absolute value is not needed with odd roots, but is necessary with even roots!!!

57. Product and Quotient rules for nth Roots
For all real numbers a and b.

58. CE Simplify

59. CE Evaluate the expression, indicate that the root is not real.

60. Addition of Radicals
In order to add or subtract radicals, they must have the same index and the same radicand.

61. CE
Add or subtract as indicated

62. CE Combining Radicals That First Require Simplification
Add

63. CE
Subtract

64. CE Subtract

65. To rationalize a denominator you must multiply both the numerator and the denominator by the smallest number that produces the square root of a perfect square

66. CE Rationalize the Denominator
First ask yourself what number can I multiply 8 by to result in a perfect square?
Once you determine the number, multiply both the numerator and denominator by the square root of that number

67. CE Rationalize the Denominator

68. Rational expressions that involve the sum and difference of the same two terms are called conjugates.
Here is an example of conjugates:

69. CE a. Multiply the Conjugates b. What do you notice about the solution?

70. Rationalize with two terms
If the denominator has two terms:
Multiply the numerator and denominator by the conjugate of the radical in the denominator.

71. CE Rationalize the denominator

72. CE Rationalize the denominator

73. Due Friday Sept. 6th
HW Page 53 13,17,29,35,47,53,77
Page odd
Vidéo Project due Tuesday Sept. 10 at start of class.

74. Factoring Algebraic Expressions Containing Fractional and Negative Exponents

75. Expressions with fractional and negative exponents are not polynomials, but they can be factored using similar techniques. Find the greatest common factor with the smallest exponent in the terms.

76. Example
Factor and simplify:

77. Section P6 Rational Expressions

78. Rational Expressions

79. A rational expression is the quotient of two polynomials
A rational expression is the quotient of two polynomials. The set of real numbers for which an algebraic expression is defined is the domain of the expression. Because division by zero is undefined, we must exclude numbers from a rational expression’s domain that make the denominator zero. See examples below.

80. Example
What numbers must be excluded from the domain?

81. Simplifying Rational Expressions

83. Example
Simplify and indicate what values are excluded from the domain:

84. Example
Simplify and indicate what values are excluded from the domain:

85. Multiplying Rational Expressions

87. Example
Multiply and Simplify:

89. Dividing Rational Expressions

90. We find the quotient of two rational expressions by inverting the divisor and multiplying.

91. Example
Divide and Simplify:

93. Adding and Subtracting Rational Expressions with the Same Denominator

94. (1) Adding or subtracting the numerators,
Add or subtract rational expressions with the same denominator by
(1) Adding or subtracting the numerators,
(2) Placing this result over the common denominator, and
(3) Simplifying, if possible.

95. Example
Subtract:

96. Adding and Subtracting Rational Expressions with Different Denominators

98. Example
Add:

103. Example
Add:

104. Complex Rational Expresisons

105. Complex rational expressions, also called complex fractions, have numerators or denominators containing one or more rational expressions. Here are two examples of such expressions listed below:

106. Example
Simplify:

107. Example
Simplify:

108. Simplify:

110. Divide

112. HW Update
Page 66 93,99
Page 79 – 80
7 – 55 every other odd

113. Precalculus Section P7 Equations

114. Objectives P7 Solve linear equations in one variable.
Solve linear equations containing fractions.
Solve rational equations with variables in the denominators.
Solve a formula for a variable.
Solve equations involving absolute value.
Solve quadratic equations by factoring.
Solve quadratic equations by the square root property. .

115. Solving Linear Equations in One Variable

117. Example

118. Linear Equations with Fractions

119. Example
Solving with Fractions

121. Rational Equations

122. First list restrictions
Example

124. First list restrictions
Example

126. Solving a Formula for One of Its Variables

128. Example

129. Example

130. Equations Involving Absolute Value

131. Example

132. Example

134. Quadratic Equations and Factoring

135. Example

136. Example

138. Example

139. Quadratic Equations and the Square Root Property

141. Example

143. Example

145. Quadratic Equations and Completing the Square

147. Obtaining a Perfect Square Trinomial
Start
Add
Result
Factored Form

148. Completing the Square

150. Example

153. Example

155. Example

156. HW Show work for credit
Page 98 – 99
9,13,21,25,31,41,47,49,57,63,65,69,73.
Optional
Any other odd problems, check in the back of the book

157. Objectives P7 cont. and P9
Solve quadratic equations by completing the square.
Solve quadratic equations using the quadratic formula.
Use the discriminant to determine the number and type of solutions.
Determine the most efficient method to use when solving a quadratic equation.
Solve radical equations
Use interval notation.
Find intersections and unions of intervals.
Solve linear inequalities.
Solve absolute value inequalities

158. Derive The Quadratic Formula

161. Example

163. Example

165. Quadratic Equations and the Discriminant

167. Example

168. Example

169. Radical Equations

170. A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. We solve the equation by squaring both sides.

171. This new equation has two solutions, -4 and 4
This new equation has two solutions, -4 and 4. By contrast, only 4 is a solution of the original equation, x=4. For this reason, when raising both sides of an equation to an even power, check proposed solutions in the original equation.
Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation are called extraneous solutions or extraneous roots.

174. Example
Solve and check your answers:

175. Linear Inequalities and Absolute Value Inequalities

176. Example

177. Interval Notation

181. Unbounded Intervals

182. Example
Express the interval in set builder notation and graph:

183. Intersections and Unions of Intervals

186. Example
Find the set:

187. Example
Find the set:

188. Solving Linear Inequalities in One Variable

193. Example
Solve and graph the solution set on a number line:

195. Example
Solve the inequality:

196. Solving Inequalities with Absolute Value

198. Example
Solve and graph the solution set on a number line.

199. Example
Solve and graph the solution set on a number line.

200. Applications

201. Example
A national car rental company charges a flat rate of $320 per week for the rental of a 4 passenger sedan. The same car can be rented from a local car rental company which charges $180 plus $ .20 per mile. How many miles must be driven in a week to make the rental cost for the national company a better deal than the local company?

202. HW
Page ,81,87,101,113,117,121
Page
5,9,17,19,31,36,41,63,69,77,89

203. Test Chapter P