Algebra Lecture Notes

Overview

• Covers topics in algebra including:
  • Solving equations
  • Linear and quadratic equations
  • Factoring
  • Working with exponents
  • Graphing equations
  • Understanding numbers (natural, whole, integers, rational, irrational, imaginary)

Numbers

• Natural Numbers: Whole numbers > 0 (e.g., 1, 2, 3,...)
• Whole Numbers: Natural numbers including 0
• Integers: Whole numbers including negatives
• Rational Numbers: Can be expressed as a fraction of integers (e.g., 2/3, 4)
• Irrational Numbers: Cannot be expressed as a fraction of integers (e.g., √5)
• Imaginary Numbers: Involves 'i', where i = √-1

Basic Operations

• Addition/Subtraction:
  • Use number line for negative numbers
• Addition Example: 5 + 4 = 9 using number line
• Subtraction Example: 2 - 5 = -3 using number line
• Multiplication/Division:
  • Multiply across for fractions
  • Use long division for division

Fractions

• Adding Fractions: Find common denominators
• Subtracting Fractions: Consider order when subtracting
• Multiplying Fractions: Multiply numerators and denominators
• Dividing Fractions: Keep-Change-Flip method

Converting Numbers

• Fractions to Decimals: Use long division
• Decimals to Fractions: Multiply by powers of 10
• Percentages to Decimals: Divide by 100

Expressions and Exponents

• Manipulating Exponents:
  • Add when multiplying same base
  • Subtract when dividing
  • Multiply when raising a power to a power
• Negative Exponents: Invert the base to make the exponent positive

Solving Equations

• Basic Equations: Isolate the variable
• Quadratic Equations: Factor or use the quadratic formula
• Logarithmic Equations: Convert to exponential form

Functions

• Evaluating Functions: Substitute values
• Composite Functions: Work from the inside function out

Graphs and Inequalities

• Graphing Equations: Focus on linear and quadratic
• Inequalities: Techniques for solving

Complex Topics

• Exponential Equations: Common bases and logs
• Logs and Properties:
  • Learn to expand and condense log expressions
  • Use properties to simplify

Practice Problems

• Solving Equations: Examples using different techniques
• Functions: Evaluate composite functions
• Logs: Solve equations involving logs
• Factoring: Various types of factoring problems