1-1 Variables and expressions
Variable – symbols that represent unspecified numbers or values.
X, n, y, r, ………….
Algebraic Expression – a mathematic statement that contains one or more numbers or
variables with at least one operation.
3x – 7,
3ab / 5 cd,
5n
Product - The result of two or more numbers or variables multiplied together.
2x,
xy, (x)(y)
Factor – the numbers multiplied to get a product.
2xy, the factors are 2, x, and y
Power – Multiplying a number by itself a set number of times.
34
means 3 to the forth power or 3 x 3 x 3 x 3
Base – The number that is multiplied when raising it to a power.
The 3 in 34
Exponent – The number of times a number is multiplied by itself in a power.
The 4 in 34
Evaluate – Finding the solution for an algebraic expression.
Examples - 43 is 64
25 is 32
Skills
 Write algebraic expression for a verbal expression
o Eight more than a number n
o 7 less than the product of 4 and a number x
o One third the size of a
o The difference of 4 and x squared
 Write verbal expressions for algebraic expressions
o n+4
o 3n – 6
o 4m3
o c2 + 21d
Section 1-2
Order of Operations
Rule for order of operation
1.
2.
3.
4.
Grouping ( enter most first)
Exponents
Multiply / Divide ( Left to Right )
Add / Subtract
Example
3+2*3+5
15 / 3 * 5 – 42
2(5) + 3(4+3)
2[ 5 + ( 30 / 6) 2]
6 + 42
32 * 4
Evaluating an expression – replacing the variable with an indicated number and then
evaluating
2n + 5 for n = 6
3x + 2y for x = 5 and y = 4
a2 + 2 for a = 6
a2 – (b3 – 4c) if a = 7, b = 3 and c = 5
1-3 Open Sentence
Open Sentence – A mathematical statement that contains one or more variable
Equation – An open sentence that contains an equal sign.
Solution – A number that makes an equation true.
Set – A group of numbers or objects
Replacement Set – The set of possible solutions for an equation.
Solution Set – The set of numbers form the replacement set that makes the equation true.
Inequality – The comparing of numbers or variables
> Greater then
< Less than
> Greater then or equal to
< Less than or equal to
1-4 Identity and Equality Properties
Additive Identity – Any number plus zero is the number
4+0=4
n+0=n
Multiplicative Identity – Any number time 1 is the number
( 5 )( 1 ) = 5
1n=n
Multiplication Property of Zero – Any number times zero is zero.
(6)(0)=0
0n=0
Multiplicative Inverse (Reciprocals) – Two numbers whose product is equal to one are
reciprocal of each other.
Properties of Equality
Reflexive – Any quantity is equal to itself.
5=5
n=n
Symmetric – If one quantity is equal to a second quantity, then the second quantity is equal
to the first.
3 + 2 = 5 then 5 = 3 + 2
n + 2 = 7 then 7 = n + 2
Transitive – If one quantity is equal to a second quantity, and the second quantity is equal to
a third quantity, then the first quantity is equal to the third.
4 = 3 + 1, 3 + 2 = 5 – 1, then 4 = 5 – 1
a = b, b = c then a = c
Substitution – Any quantity can be substituted into an equation for its equal.
4 + 6 = 10, 2 x 3 = 6 then 4 + ( 2 x 3 ) = 10
y = x + 2, x = 6 then y = 6 + 2
Section 1-5
Distributive Property
Distributive Property – Multiplication through Addition
- Multiplication through Subtraction
For any numbers a, b, and c
a(b + c) = ab + ac
a(b - c) = ab - ac
Example
3 (2 + 5) = (3)(2) + (3)(5)
4 (9 – 7) = (4)(9) – (4)(7)
6 (n + 1) = 6n + (6)(1) or 6n + 6
4(n – 8) = 4n – (4)(8) or 4n – 32
Term – A number or variable or a product or quotient of numbers
and variables.
n, 3y, n/2, 5g2k
Like Terms – Terms that contain the same variables, with the corresponding variables having
to the same power.
3a + 4a – 6b + 8b2 + 3b2 + 3cb + 4c
Equivalent Expressions – Expressions that denote the same number.
5n + 7n and 12n
8a – 3a and 5a
Simplest Form – The Simplest form of an expression has no like terms and no parentheses.
15x + 8x
23x
10n + 3n2 + 9n2
10n + 11n2
8 (n + 6) + 2n
8n + 48 + 2n
10n + 48
Coefficient – The coefficient of a term is the numerical factor of the term.
15n the coefficient is 15
½ n2 the coefficient is ½
Section 1-6
Commutative and Associative Properties
Commutative Property of Multiplication and Addition –
The order in which you add or multiply numbers does not change the sum or product.
a+b=b+a
(a)(b) = (b)(a)
6+5+4=4+6+5
(2)(4)(5) = (5)(2)(4)
Associative Property of Multiplication and Addition
The way you group three or more numbers when adding or multiplying does not change
the sum or product.
(a + b) + c = a + (b + c)
(ab)c =
a(bc)
(2 + 4) + 6 = 2 + (4 + 6)
(3 x 5) x 4 = 3 x (5 x 4)
Example -
3c + 5(2 + c)
5½ + 8 + 2½ + 6
(.5)(2.4)(4)
4a2 + 6a + 3a2 + 2a
3(x + 2y) + 4(2x + 3y)
Section 1-7
Logical Reasoning
Conditional Statements – A statement that is written in if/then form.
Hypothesis – The “if” part of a conditional statement.
Conclusion – The “then” part of the conditional statement
If an animal is a dog, then it has four legs
If 2n + 4 = 12, then n = 4
Truth Value – Whether a conditional statement is true or false.
 The truth value is determined by the conclusion of the conditional statement.
Counter example – An example that shows a conditional statement is false.
If it is Monday, then we have school
If an animal is a dog, then it has four legs
If an animal has 4 legs, then it is a dog.
Deductive Reasoning – The process of using facts, rules, definitions, or properties to make valid
conclusions.
1-8
Graphs and Functions
Function – The relationship between the input and output of an expression. The output depends
on the input of the expression. In a function there is exactly one output for each input.
Coordinate System – System created by intersecting two number lines perpendicularly at the
zero point on both lines.
The origin (0,0) is the point of intersection of the two number lines.
y-axis – The vertical number line in the coordinate plane.
x-axis – The horizontal number line in the coordinate plane.
Ordered Pair – A set of numbers written in (x, y) form.
x coordinate - The first element in an ordered pair.
y coordinate – The second element of an ordered pair.
Independent Variable (x) – The value that you choose to place into the expression or equation.
Dependent Variable (y) – The result of the expression after the expression had been evaluated.
Relation – Set of ordered pair
Domain – The set of independent variables (x) used in an expression.
Range – The set of all dependent variables (y) that are possible solutions for an expression.