Basic Concepts in Algebra
1. Variables and Expressions
● A variable represents an unknown value (e.g., x,y,zx,y,z).
● An algebraic expression is a combination of numbers, variables, and operations
(e.g., 3x+53x+5)
2. Order of Operations (PEMDAS)
● Parentheses
● Exponents
● MD Multiplication and Division (left to right)
● AS Addition and Subtraction (left to right)
3. Types of Numbers
● Natural Numbers: 1,2,3,…1,2,3,…
● Whole Numbers: 0,1,2,3,…0,1,2,3,…
● Integers: …,−2,−1,0,1,2,……,−2,−1,0,1,2,…
● Rational Numbers: Numbers that can be written as abbawhere b≠0b=0.
● Irrational Numbers: Numbers that cannot be expressed as fractions (e.g., π,2,2^(½))
4. Properties of Numbers
● Commutative Property: a+b=b+a, ab=ba
● Associative Property: (a+b)+c=a+(b+c), (ab)c=a(bc)
● Distributive Property: a(b+c)=ab+ac
● Identity Property: a+0=a, a×1=a
● Inverse Property: a+(−a)=0a+(−a)=0, a×1/a=1/a×a/1=1
5. Solving Linear Equations
A linear equation is of the form: ax+b=cax + b = cax+b=c.
Steps to solve:
1. Simplify both sides.
2. Isolate the variable using inverse operations.
3. Solve for the variable.
Example:
Solve 3x+5=20
Step 1: Subtract 5 → 3x=15
Step 2: Divide by 3 → x=5
6. Solving Inequalities
Similar to solving equations, but flipping the inequality when multiplying/dividing by a
negative number.
1. Variables and Expressions
● A variable represents an unknown value (e.g., x,y,zx,y,z).
● An algebraic expression is a combination of numbers, variables, and operations
(e.g., 3x+53x+5)
2. Order of Operations (PEMDAS)
● Parentheses
● Exponents
● MD Multiplication and Division (left to right)
● AS Addition and Subtraction (left to right)
3. Types of Numbers
● Natural Numbers: 1,2,3,…1,2,3,…
● Whole Numbers: 0,1,2,3,…0,1,2,3,…
● Integers: …,−2,−1,0,1,2,……,−2,−1,0,1,2,…
● Rational Numbers: Numbers that can be written as abbawhere b≠0b=0.
● Irrational Numbers: Numbers that cannot be expressed as fractions (e.g., π,2,2^(½))
4. Properties of Numbers
● Commutative Property: a+b=b+a, ab=ba
● Associative Property: (a+b)+c=a+(b+c), (ab)c=a(bc)
● Distributive Property: a(b+c)=ab+ac
● Identity Property: a+0=a, a×1=a
● Inverse Property: a+(−a)=0a+(−a)=0, a×1/a=1/a×a/1=1
5. Solving Linear Equations
A linear equation is of the form: ax+b=cax + b = cax+b=c.
Steps to solve:
1. Simplify both sides.
2. Isolate the variable using inverse operations.
3. Solve for the variable.
Example:
Solve 3x+5=20
Step 1: Subtract 5 → 3x=15
Step 2: Divide by 3 → x=5
6. Solving Inequalities
Similar to solving equations, but flipping the inequality when multiplying/dividing by a
negative number.
