Algebra Lecture Notes

Topics Covered

• Solving Equations
• Linear Quadratic Equations
• Factoring
• Working with Exponents
• Graphing Equations

Number Systems

Types of Numbers

• Natural Numbers: Whole numbers greater than zero (e.g., 1, 2, 3, ...).
• Whole Numbers: Natural numbers including zero.
• Integers: Whole numbers that can be negative.
• Rational Numbers: Numbers that can be expressed as a fraction of integers (e.g., 2/3, 4).
• Irrational Numbers: Numbers that cannot be expressed as a fraction of integers (e.g., √5).
• Imaginary Numbers: Involves the square root of negative numbers (e.g., √-1 = i).

Basic Operations

Addition and Subtraction

• Use a number line for simplifying operations with negative numbers.
• Example: Adding 5 + 4 by moving 4 units to the right from 5 (result is 9).

Multiplication and Division

• Multiplication involves adding zeros and carrying over numbers.
• Division can be solved using long division.

Fractions

Operations with Fractions

• Addition: Find a common denominator or use cross multiplication.
• Subtraction: Same as addition but ensure to maintain order of terms.
• Multiplication: Multiply across numerators and denominators.
• Division: Use "keep, change, flip" method.

Conversion

• Improper Fraction to Mixed Number: Divide the numerator by the denominator.
• Decimal to Fraction: Place decimal over 1 and adjust.
• Percentage to Decimal: Divide percentage by 100.

Variables and Exponents

• Multiplying Variables: Add exponents.
• Dividing Variables: Subtract exponents.
• Exponentiation: Multiply exponents.
• Negative Exponents: Indicate reciprocal (e.g., x^-1 = 1/x).

Solving Algebraic Equations

Solving for x

• Simple Equations: Isolate x using basic operations.
• Quadratic Equations: Use factoring or the quadratic formula.
• Exponential Equations: Use logarithms to solve.
• With Fractions: Clear fractions using multiplication by denominators.

Functions

Evaluating Functions

• Substitute given values into the function.
• Composite functions involve evaluating the inner function first.

Logs and Exponents

Properties of Logs

• Product Rule: ln(a) + ln(b) = ln(ab)
• Quotient Rule: ln(a) - ln(b) = ln(a/b)
• Power Rule: ln(a^c) = c * ln(a)*

Logarithmic Equations

• Convert logarithmic equations to exponential form to solve.

Factoring Techniques

• GCF: Factor out the greatest common factor.
• Trinomials: Look for two numbers that multiply to the constant term and add to the middle term.
• Difference of Squares: a^2 - b^2 = (a+b)(a-b).

Graphing and Functions

• Functions can be graphed by determining key points and plotting them.
• Understand the basic structure of linear, quadratic, and polynomial functions.

This summary captures key points and methods discussed in the algebra lecture, including basic operations, types of numbers, manipulations of fractions, variables, exponents, equations, and functions.