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**Sets and Expressions Number Sets**

Natural (Counting) numbers – {1,2,3,…} Whole numbers – Natural numbers and 0, that is {0,1,2,3,….} Integers – Positive or negative whole numbers Rational numbers – any number that can be expressed as a/b, where a and b are integers Irrational numbers – any number that cannot be expressed as a fraction of two integers Real Numbers – The set of all numbers in each of the previous sets

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The symbols is read “is an element of”, it is used to denote an element in a set is read “ is not an element of”, it is used to denote an element that is not part of the set

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Set Builder Notation A way of writing a set according to conditions, the general form of which is: 𝑥 𝑥 ∈𝑠𝑒𝑡, 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 Translated as “x such that x is an element of a set, given some condition. Empty Set: also called a null set, a set that contains no elements. The empty set is represented by { } or Ø Do not use {Ø}, this means a set containing the element Ø Write the following in set builder notation: {…,0, 1, 2}

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Roster Notation: A manner of writing a set in which all elements are listed, such as: {4, 5,6, …} Write the following in roster notation: 𝑥 𝑥 ∈𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑥≤5 𝑥 𝑥 ∈𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠, 𝑥 ≥−4

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**Given the set {25, 7\3, -15, -3\4, √5, -3.7, 8.8,-99}**

List the numbers in the set that belong to the set of Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers

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**Algebraic Expressions**

An algebraic expression is any statement containing numbers and letters connected by an operator. Variable – A letter used to represent an unknown number Term – A number or product of a number and a variable Coefficient – A numerical factor of a term Constant – A term with no variables Example - Complete the table below Expression Variables Terms Coefficients Constants a. 3x – 5y + 3 b. -4x2 + 5y - 10 c. 7ab – c

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**Evaluating Algebraic Expressions**

To evaluate an algebraic expression, substitute known values for variables into an expression and use the order of operations to simplify.

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**Order of Operations P- Parentheses or other grouping symbols**

E- Exponents M- Multiplication D- Division A- Addition S- Subtraction From Left to Right From Left to Right

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**Examples:Evaluate the following expressions given the values for the variables:**

Evaluate 2a + 3b when a = 2 and b = 7 Evaluate x2 + y2 – xy when x = -3 and y = 4 Evaluate when x = 9 and y = -2

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**Properties of Real Numbers:**

Commutative Properties Addition: a+b = b+a Multiplication: a∙b = b∙a Associative Properties Addition: a+(b+c) = (a+b)+c Multiplication: (a∙b)∙c = a∙(b∙c) Addition and Multiplication Identities Additive Identity: a + 0 = 0 + a = a Multiplicative Identity: a∙1 = 1∙a = a Addition and Multiplication Inverses Additive Inverse: a + (-a) = 0 Multiplicative Inverse: a ∙ =1 Distributive Property of Multiplication a(b+c) = a∙b + b∙c

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**Examples: Give an example of each of the following properties:**

a. Commutative __________________________ b. Associative __________________________ c. Distributive __________________________ d. Additive Identity ________________________ e. Multiplicative Identity_____________________ f. Additive inverse _____________________ g. Multiplicative Inverse _____________________

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**Examples: Name the identity illustrated:**

b. -4 (6x) = (-4∙6 )x _____________________ c. y + (3+2) = (y+3) + 2 _____________________ d. 5∙ =1 _____________________ Use the distributive property to rewrite: a. 3(2x + 4) b. -2(3x + y – z) c. 5(2 + 3a – 4b)

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**Simplifying Algebraic Expressions**

Algebraic expressions can be simplified by combining like terms. Like terms are two or more terms that have: a. The exact same variables b. Variables are raised to the same power Two or more like terms can be added by adding their coefficients. Examples – Simplify each algebraic expression: a. 3y + 8y – 7 + 2 b. -7a + 3b – a – 17b – 20 c. 5(3x + 7) + 4x + 10 d. - (3a – 7) – 2(a + 8)

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