Sets and ExpressionsNumber SetsNatural (Counting) numbers – {1,2,3,…}Whole numbers – Natural numbers and 0, that

is {0,1,2,3,….}Integers – Positive or negative whole numbersRational numbers – any number that can be

expressed as a/b, where a and b are integersIrrational numbers – any number that cannot

be expressed as a fraction of two integersReal Numbers – The set of all numbers in each

of the previous sets

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

Set Builder NotationA way of writing a set according to conditions,

the general form of which is: 

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 ØDo not use {Ø}, this means a set containing

the element ØWrite the following in set builder

notation: {…,0, 1, 2}

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:

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 ofa. Natural Numbersb. Whole Numbersc. Integersd. Rational Numberse. Irrational Numbersf. Real Numbers

Algebraic ExpressionsAn algebraic expression is any statement

containing numbers and letters connected by an operator. Variable – A letter used to represent an

unknown numberTerm – A number or product of a number and a

variableCoefficient – A numerical factor of a termConstant – A term with no variables

Example - Complete the table belowExpression Variables Terms Coefficient

sConstants

a. 3x – 5y + 3   

     

b. -4x2 + 5y - 10

  

     

c. 7ab – c   

     

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

Order of OperationsP- Parentheses or other grouping symbols

E- ExponentsM- Multiplication D- DivisionA- AdditionS- Subtraction

From Left to Right

From Left to Right

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 = -2yx

yx

2

||23

Commutative PropertiesAddition: a+b = b+aMultiplication: a∙b = b∙a 

Associative PropertiesAddition: a+(b+c) = (a+b)+cMultiplication: (a∙b)∙c = a∙(b∙c) 

Addition and Multiplication IdentitiesAdditive Identity: a + 0 = 0 + a = aMultiplicative Identity: a∙1 = 1∙a = a

Addition and Multiplication InversesAdditive Inverse: a + (-a) = 0

Multiplicative Inverse: a ∙ =1 

Distributive Property of Multiplicationa(b+c) = a∙b + b∙c 

Properties of Real Numbers:

a

1

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 _____________________

Examples:Name the identity illustrated:a. 5 + (-5) = 0 _____________________

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)

Simplifying Algebraic Expressions

 a. 3y + 8y – 7 + 2   

b. -7a + 3b – 6 + 12a – 17b – 20

c. 5(3x + 7) + 4x + 10    

d. - (3a – 7) – 2(a + 8)

Algebraic expressions can be simplified by combining like terms. Like terms are two or more terms that have:

a. The exact same variablesb. Variables are raised to the same power

Two or more like terms can be added by adding their coefficients. Examples – Simplify each algebraic expression: