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Algebra 101

Absolute Value:

The absolute value of a number is always the positive value of the number. e.g. ABS value of -5 and 5 is 5. 
ABS value is indicated by two thin lines |5|


Coefficient is a number in front of a variable. e.g. 5 is the coefficient in the expression 5x.


An expression is a mathematical statement that does not contain an = sign. An expression can contain numbers, variables and/or operators, such as +, - , x , . /. e.g. x +y and 5x -2y are expressions.


Mathematical statement containing an equal sign (=) . The = indicates that both sides of the equation are equal. An equation whose highest exponent is the equation is one, such as x +y=10, is known as a linear equation. An equation whose highest exponent in the equation is 2, such as x^2 + 4x = 16, is known as a quadratic equation.


A formula is a statement expressing a general mathematical truth and can be used to solve or reorganize mathematical problems. i.e. a2 - b2 always equals (a+b) (a-b), so a2 - b2 = (a+b) (a-b) is a formula you can use to work with problems.


An exponent indicates the number of times a number is multiplied by itself. it. 23 equals 2x2x2. The number to which the exponent is attached , such as 2 in 23 ,is called the base.

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A fraction such as 1/2 or 3/4 is a division problem written with a fraction bar (-) instead of a division sign. e.g. You can write 3 divided by 4 as 3/4. A fraction has two parts-the top number in a fraction is called the numerator and the bottom is called the denominator.


Factors are numbers or terms that are multiplied together to arrive at a specific number. e.g. 3 and 4 are factors of 12. When you factor an expression in algebra, you break the expression into pieces, called factors, that you can multiply together to give you the original expression.

Prime Numbers:

A Prime number is a positive number that you can only evenly divide by itself and the number 1. The number 1 is not considered a prime number since it can only be evenly divided by one number. the number 2 is the smallest prime number. Prime # examples. 2,3,5,7,11,13,17,19,23,29

Composite Numbers:

A composite number is a number that you can evenly divide by itself, the number 1 an one or more other numbers.
The numbers you can evenly divide into another number are called factors. for example, the factors for the number 12 are 1,2,3,4 and 6. composite number examples: 4,6,8,10,12,14,15,16,18
An inequality is a mathematical statement in which one side is less than, greater than, or possibly equal to the other side. Inequalities use four different symbols (< Less than, ≤ Less than or equal to, > Greater than , ≥ Greater than or equal to..


A ridical is a symbol √ that tells you to find the root of a number. In algebra, you will commonly fin the square root of numbers.
To find the square root of a number, you need to determine which number multiplied by itself equals teh number under the radical sign. √ 25 equals 5


A matrix is a collection of numbers, called elements, which are arranged in horizonatal rows and vertical columns. The collection of numbers is surrounded by brackets or parentheses. Matrices is the term used to indicate mor than one matrix


A whole number or whole number with a negative sign eg. 1 -1


Terms are numbers, variables, which are separated by addition(+), subtraction (-), multiplication (x), or (.) or division (/) signs. For example, the expression xy contains one term, whereas the expression x + Y contains two terms.

When you are asked to solve a problem, you need to find the answer or answers to the problem. eg. when you solve the equation 2x -6 =0, you determine that x equals 3

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System of equations

A system of equations is a groupe of two or more related equations. You are often asked to find the value of each variable that solves all the equations in the group.


A variable is a letter, sucah x or y, which represents an unknown number. For example, if x represents emilys age, then x + 5 represents the age of emiliys sister who is 5 years older.


A polynomial is an expression that consists of one or more terms, which can be a combination of numbers and/or variables, that are added together or subtracted from one another. For example. 2x^2 + 3x is a polynomial. A polynomial with only one term, such as 5x is called a monomial. A polynomial with two terms, such as 5x +7, is called a binomial. A polynomial with three terms, such as 2x^2 + 5x + 7, is called a trinomial.

Rational Numbers

Rational numbers include integers and fractions. A rationsl number is also a number that you can write with decimal values that either end or have a pattern that repeats forever. eg. 5, -5, 1/2, 13, 9.25, .24242424...

Irrational numbers

Irrational numbers do not include integers or fractions. An irrational number has decima values that continue forever without a pattern.
the most well know irrational number is pi

Real numbers

Real numbers include natural numbers, whole numbers and irratinal numbers. Real numbers can include fractions as well as numbers with or without decimal places.

Natural numbers

natural numbers are the numbers 1,2,3,4,5, and so on. Natural numbers are also called the counting numbers

Whole numbers

Whole numbers are the numbers 0,1,2,3,4,5 and so on. Whole numbers are the same as natural numbers with the addition of the number 0
=== Tips ===
When multiplying or dividing a problem with a lot of numbers, simply count the number of negative numbers. an even number of negative numbers means the answer will be positive. while an odd number of negative numbers will result in a negative answer.
If the two numbers or variables you are multiplying or dividing have the same sign (+or -), the result will always be positive.
If the two numbers or variables you are multiplying or dividing have different sighns (+ and -), then the result will always be negative
eg: (+3)x(-2) and (-3)x (+2) both equal -6

Order of Operations

Parenteses () or other grouping symbols. such as brackets [] or braces {}
Exponents, such as 2^4
Multiplication and division
Addition and subtraction
Remember to work with numbers in parentheses first, then caluclate exponents, then multiply and divide and then add and subtract.
When working on the parentheses level and one set of parentheses appeares within another, start with the innermost set and work your way to the outside of the problem.
When working on the multiplication and division level, multiply or divede, whichever comes first, from left to right.
When working on the additon and subtraction level, add subtract, whichever comes first, from left to right.

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The Commutative Property

The commutative property states that when calculation addition or multiplaction problems, you will arrive at the same answer no matter what order the terms are arranged in.
- when you add or multiply two or more numbers together, the order in which you add or multiply the numbers will not change the result. This is know as the commutative property of addition and multiplication.
The commutative property does not apply to subtraction and division problems, which must be carried out in the order that they are given.

Associative property

The associative property states that when calculation addition or multiplication problems, you will arrive at the same answer no matter what order the operations are performed in.
eg. both (2x5)x6 and 2x(5x6) equals 60
eg. x x(2xy) = (x x2) x y

The Distributive Proberty

The distributive property is actually a very simply concept to learn and apply. It will allow you to simplify something like 3(6x + 4), where you have a number being multiplied by a set of parenthesis. Let's start with a simple problem:
6(4 + 2)
Based on the order of operations, you know that anything inside parenthesis should be done first. Adding 4 + 2 is simple enough, resulting in this:
When you see a number next to parenthesis like this, it means multiplication, so what we really have here is this (remember that * means multiplication):
6 * 6 = 36
That was easy enough, but what about a more difficult problem? Let's suppose that the 4 was really 4x, meaning 4 times the variable x. The distributive property allows you to simplify an expression like this, where you cannot just do the parenthesis and multiply.
6(4x + 2)
What this expression seems to say is that we want 6 times the sum of 4x + 2. It can also be expressed in a different way using the distributive property:
(6 * 4x) + (6 * 2)
We can do this because with 6(4x + 2), the 6 is distributed to the 4x and the 2. That expression can now be simplified to 24x + 12, which is easier to use that the original 6(4x + 2). Now try simplifying this expression:
-2(4y - 8)
This is no more difficult so simplify than the last one. Just distribute the -2 to the 4y and the -8:
(-2 * 4y) + (-2 * -8)
-8y + 16
16 - 8y

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How can i make problems with several positive (+) and negative (-) signs less complicated?

if a problem contains positive or negative sighns withch are side by side, you can simplyfy the problem a little bit. When you find tow positive signs next to one another, you can replace the signs with a single (+) sign.
3 + (+5) = 3+5
6-(-3) = 6 +3
if you find two opposite signs (+ or -) side by side you can replace the signs with a single minus (-) sign
7 + (-3) = 7-3
9-(+2) = 9-2
3(2 + x) = (3x2) + 3 * X)
= 6 + 3x
-4(x - y) = (-4 x X) - (-4 * y)
= -4x - (-4y)
= -4x + 4y
a(a+b+c) = (a x a) + (a*b) + (a * c)
= a^2 + ab + ac
Variables with Exponents
3(x^2 + y^2) = (3 * X^2) + ( 3 * y^2)
= 3x^2 + 3y^2
x^2(7-x^2) = (x^2 * 7) - (x^2 * 7) - (x^2 * x^2)
= 7x^2 - x^2+2
= 7x^2 - x^4
A variable is a letter, such as x or y, which represents and unknown number. When you multyply a variable by itself you can use exponents to simplify the problem. eg. a* a = a^2

Algebra Properties

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



Commutative Property of Addition a + b = b + a 3x + x2 = x2 + 3x
Commutative Property of Multiplication ab = ba (3 - x)x2 = x2(3 - x)
Associative Property of Addition (a + b) + c = a + (b + c) (x + 3) + x2 = x + (3 + x2)
Associative Property of Multiplication (ab)c = a(bc) (3x • 2)(5) = (3x)(2 • 5)
Distributive Properties a(b + c) = ab + ac

(a + b)c = ac + bc

3x(5 + 2x) = (3x• 5) + (3x • 2x)

(y + 5)4 = (y • 4) + (5 • 4)

Additive Identity Property a + 0 = a 7x2 + 0 = 7x2
Multiplicative Identity Property a • 1 = a 8y • 1 = 8y
Additive Inverse Property a + (-a) = 0 5x2 + (-5x2) = 0
Multiplicative Inverse Property
a • 


 = 1
(x2 + 3) • 


 = 1
(x2 + 3)

Inverse Property

The inverse property outlines the two types of opposites in algebra - the aditive inverse and the multiplicative inverse.
The additive inverse property simply states that every number has an inverse number with the opposite sign. eg. the additive inverse of 12 is -12. When you add two opposites together, the numbers will effectively cancel each other out and bring you to 0.
The multiplicative inverse property states that every number, except of zero, has an opposite reciprocal number. When a number is bultiplied by its reciprocal, the answer will always be 1. A reciprocal is found by simply flipping a fraction over. For example, to find the reciprocal of 2, rewrite it as teh fraction 2/1 -keeping in mind that all positive and negative whole numbers have an invisible denominator, or bottom part of the fraction, of 1 -and flip the fraction over so that you have 1/2. When you multiply 2 x 1/2, you arrive at 1.

Additive inverse property

10 + (-10)=0
-5 + 5=0
a+(-a) =0

Multiplicative inverse

When you multiply a number by its reciprocal, called the multoplicative inverse, the result will be 1.
To find the reciprocal of a fraction, switch the top and bottom numbers in the fraction. eg. the reciprocal of 1/2 is 2/1
1/x * x/1 = 1
1/2 * 2/1 = 1
3/4 * 4/3 = 1
3/x * x/3 =1

Evaluate an expression

To evaluate an expression, you replace the variables in the expression with their numberical values, otherwise known as pluggin numbers into an expression. For the best results, you should surround the values you plug into an expression with parenteses to that positive and negative numbers don't get mixed up with the mathematical operations of the problem. For example. pluggin the value of x = -4 into the equation y + 4x would be open to errors it was written as y + 4 -4. using parentheses and writing the expression as y + 4(-4) removes any ambiguity.

Compound Intrest

Compound Intrest
P= Principal
I= Intrest Rate
N= Time in years
Results = P * (1 + I) ^ N

What if interest is paid more frequently? 
Here are a few examples of the formula: 

Annually = P × (1 + r) = (annual compounding) 

Quarterly = P (1 + r/4)4 = (quarterly compounding)

Monthly = P (1 + r/12)12 = (monthly compounding)

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