Introduction to Algebra

Vocabulary and Basic Concepts

Algebra is a branch of mathematics that uses symbols, usually letters, to represent numbers and express mathematical relationships. Understanding the foundational vocabulary is essential for success in algebra.

• Variable: A letter that represents a number (e.g., x, y).

• Constant: A number that does not change.

• Operation: Mathematical actions such as addition, subtraction, multiplication, and division.

• Evaluate: To substitute values for variables and calculate the result.

• Algebraic Expression: A combination of constants, variables, and operations (e.g., 2x + 3).

• Algebraic Equation: An algebraic expression set equal to another expression (e.g., 2x + 3 = 7).

Translation of Words to Algebraic Expressions:

• Addition: "added to," "sum of," "plus," "more than," "increased by" (e.g., "three more than x" is x + 3).

• Subtraction: "subtracted," "minus," "difference," "less than," "decreased by" (e.g., "difference of our ages").

• Multiplication: "multiplied by," "product of," "times," "twice," "of" (e.g., "double the cost" is 2c).

• Division: "divided by," "quotient of," "ratio of," "per" (e.g., "divide the bag of candy between 5 friends" is c/5).

• Equality: "equal," "is," "are" (e.g., "what number added to 73 is 201?" is x + 73 = 201).

Examples:

• The product of 9 and twice m: 9 × (2m) = 18m

• Thirteen less than one quarter of some number: (1/4)x – 13

• What number added to 73 is 201? x + 73 = 201

Common Laws Found in Algebra

Properties of Real Numbers

Algebra relies on several fundamental laws that govern the manipulation of numbers and expressions.

• Commutative Law:

  • Addition:

  • Multiplication:

• Associative Law:

  • Addition:

  • Multiplication:

• Distributive Law:

Examples:

• Commutative Law:

• Associative Law:

• Distributive Law:

• Factoring:

Fraction Notation

Prime and Composite Numbers

Understanding fractions begins with recognizing prime and composite numbers.

• Prime Number: A number greater than 1 with only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).

• Composite Number: A number with more than two factors (e.g., 4, 6, 8, 9, 10).

Examples:

• Prime Factorization of 180:

• Prime Factorization of 210:

Fraction Notation and Properties

• Numerator: The top number in a fraction.

• Denominator: The bottom number in a fraction.

• Identity Property of 1:

Examples:

• Simplify:

• Simplify:

Operations with Fractions

• Multiplication:

• Division:

• Addition/Subtraction (same denominator):

Examples:

Positive and Negative Real Numbers

Integers and Their Opposites

Integers include all positive and negative whole numbers, as well as zero. Each integer has an opposite, which is the same distance from zero on the number line but in the opposite direction.

Inequalities and Absolute Value

• Inequality Symbols:

  • <: Less than (e.g., x < 3)

  • ≤: Less than or equal to (e.g., x ≤ -4)

  • >: Greater than (e.g., x > 0)

  • ≥: Greater than or equal to (e.g., x ≥ -3)

• Absolute Value: The distance a number is from zero on the number line, always non-negative.

Examples:

Adding and Subtracting Real Numbers

Using the Number Line

To add on a number line, start at the first number and move right for positive numbers or left for negative numbers. Subtraction is interpreted as adding the opposite.

• Add:

• Subtract:

Without the Number Line

• Add:

• Subtract:

• Subtract:

• Simplify:

• Combine like terms:

• Combine like terms:

Rules for Addition and Subtraction

• Positive numbers: Add as usual; the answer is positive.

• Negative numbers: Add absolute values and make the answer negative.

• One positive and one negative: Subtract the smaller absolute value from the larger, and use the sign of the larger number.

Identity Property of 0:

Removing Parentheses:

• Like Signs: ,

• Unlike Signs: ,

Multiplying and Dividing Real Numbers

Rules and Examples

• Multiply:

• Divide:

• Multiply:

• Divide: is undefined;

Rules for Multiplication and Division:

• Multiply or divide the absolute values.

• If the signs are the same, the answer is positive.

• If the signs are different, the answer is negative.

Division Involving Zero:

• For any real number , is undefined.

• For , .

Exponential Notation and Order of Operations

Exponents

Exponential notation is a way to represent repeated multiplication of the same factor.

• For any natural number , means (n factors).

Order of Operations

To evaluate expressions correctly, follow the order of operations:

1. Simplify inside grouping symbols: ( ), [ ], { }, | |, and fraction bars.

2. Simplify all exponential expressions.

3. Perform all multiplications and divisions from left to right.

4. Perform all additions and subtractions from left to right.

Opposite of a Sum:

Examples:

• Simplify:

• Evaluate: when ,

• Simplify:

• Simplify: