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**Be prepared to take notes when the bell rings.**

P.1 Real Numbers Be prepared to take notes when the bell rings.

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Real Numbers Set of numbers formed by joining the set of rational numbers and the set of irrational numbers Subsets: (all members of the subset are also included in the set) {1, 2, 3, 4, …} natural numbers {0, 1, 2, 3, …} whole numbers {…-3, -2, -1, 0, 1, 2, 3, …} integers

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**Rational and Irrational Numbers**

A real number that can be written as the ratio 𝑝 𝑞 of two integers, where q ≠0 Example: 1 3 =0.3333 Repeats 1 8 = 0.125 Terminates = =1. 126 A real number that cannot be written as the ratio of two integers *infinite non-repeating decimals Example: 2 = … 𝜋= …

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**Non-Integer Fractions**

Real Numbers Irrational Numbers Rational Numbers Non-Integer Fractions Integers Negative Integers Whole Numbers Natural Numbers Zero

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**Real Number Line Negative Positive Origin Coordinate:**

Positive Negative Coordinate: Every point on the real number line corresponds to exactly one real number called its coordinate

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**Ordering Real Numbers Inequalities <𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 >𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛**

<𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 >𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 ≤𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 ≥𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑜𝑟 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 a b Example: a < b − 14 3 _______− 26 >

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**Describe the subset of real numbers represented by each inequality.**

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**Interval: subsets of real numbers used to describe inequalities**

Notation 𝑎,𝑏 𝑎, 𝑏 Interval Type Closed Open Inequality 𝑎≤𝑥≤𝑏 𝑎<𝑥<𝑏 𝑎≤𝑥<𝑏 𝑎<𝑥≤𝑏 The endpoints of a closed interval ARE included in the interval. The endpoints of an open interval are NOT included in the interval. *Unbounded intervals using infinity can be seen on page 4

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**Properties of Absolute Value**

𝑎 ≥0 −𝑎 = 𝑎 𝑎𝑏 = 𝑎 𝑏 𝑎 𝑏 = 𝑎 𝑏 , 𝑏≠0

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**Absolute value is used to define the distance (magnitude) between two points on the real number line**

Let a and b be real numbers. The distance between a and b is: 𝑑 𝑎,𝑏 = 𝑏−𝑎 = 𝑎−𝑏 The distance between -3 and 4 is: −3−4 = −7 =7

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**Algebraic Expressions**

Variables: letter that represents an unknown quantity Constant: Real number term in an algebraic expression Algebraic Expression: Combination of variables and real numbers (constants) combined using the operations of addition, subtraction, multiplication and division Examples of algebraic expressions: 5𝑥, 2𝑥−3, 4 𝑥 2 +2 , 7𝑥+𝑦 Terms: Parts of an algebraic expression separated by addition i.e. 𝑥 2 −5𝑥+8= 𝑥 2 + −5𝑥 +8 𝑥 2 , −5𝑥:𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑡𝑒𝑟𝑚𝑠 8:𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚 Coefficient: Numerical factor of a variable term Evaluate: Substitute numerical values for each variable to solve an algebraic expression

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**Examples of Evaluation**

Expression −3𝑥+5 3 𝑥 2 +2𝑥−1 Value of Variable 𝑥=3 𝑥=−1 Substitute −3(3)+5 3 (−1) 2 +2(−1)−1 Value of Expression −9+5=−4 3−2−1=0 Used Substitution Principle: If a=b, then a can be replaced by b in any expression involving a.

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**Basic Rules of Algebra 4 Arithmetic operations: Addition, +**

Subtraction, - Division, / ÷ Multiplication, × ∙ Addition and Multiplication are the primary operations. Subtraction is the inverse of Addition and Division is the inverse of Multiplication.

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**Basic Rules of Algebra 𝑎−𝑏=𝑎+ −𝑏**

Subtraction: add the opposite of b Division: multiply by the reciprocal of b; if b≠0, then 𝑎−𝑏=𝑎+ −𝑏 −𝑏 is called the additive inverse (opposite of a real number) 𝑎 𝑏 =𝑎 1 𝑏 = 𝑎 𝑏 1 𝑏 is called the multiplicative inverse (reciprocal of a real number) 𝑎 𝑏 : 𝑎 𝑖𝑠 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟, 𝑏 𝑖𝑠 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

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**Let a, b and c be real numbers, variables or algebraic expressions.**

Commutative Property of Addition 𝑎+𝑏=𝑏+𝑎 Commutative Property of Multiplication 𝑎𝑏=𝑏𝑎 Associative Property of Addition 𝑎+𝑏 +𝑐=𝑎+ 𝑏+𝑐 Associative Property of Multiplication 𝑎𝑏 𝑐=𝑎 𝑏𝑐 Distributive Property 𝑎 𝑏+𝑐 =𝑎𝑏+𝑎𝑐 Additive Identity Property 𝑎+0=𝑎 Multiplicative Identity Property 𝑎∗1=𝑎 Additive Inverse Property 𝑎+ −𝑎 =0 Multiplicative Inverse Property 𝑎 1 𝑎 =1

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**Let a, b and c be real numbers, variables or algebraic expressions.**

Properties of Negation and Equality −1 𝑎=−𝑎 − −𝑎 =𝑎 −𝑎 𝑏=− 𝑎𝑏 =𝑎 −𝑏 −𝑎 −𝑏 =𝑎𝑏 −(𝑎+𝑏)=(−𝑎)+(−𝑏) 𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎+𝑐=𝑏+𝑐 𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐=𝑏𝑐 𝐼𝑓 𝑎+𝑐=𝑏+𝑐, 𝑡ℎ𝑒𝑛 𝑎=𝑏 𝐼𝑓 𝑎𝑐=𝑏𝑐, 𝑡ℎ𝑒𝑛 𝑎=𝑏 𝑖𝑓 𝑐≠0

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**Let a, b and c be real numbers, variables or algebraic expressions.**

Properties of Zero 𝑎+0=𝑎, 𝑎−0=𝑎 𝑎∗0=0 0 𝑎 =0, 𝑖𝑓 𝑎≠0 𝑎 0 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝐼𝑓 𝑎𝑏=0, 𝑡ℎ𝑒𝑛 𝑎=0 𝑜𝑟 𝑏=𝑜

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Homework Problems Page 9 #’s 1-25 odd, 29, odd, 43-47, 51-55, 59, , ,

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