Algebra Basics

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Key Concepts:
- Variables: Letters representing numbers (e.g., x, y).
- Expressions: Combinations of variables and constants (e.g., 3x + 2).
- Equations: Mathematical statements that two expressions are equal (e.g., 2x + 3 = 7).
- Inequalities: Statements indicating one value is larger or smaller than another (e.g., x < 5).
Operations:
- Addition/Subtraction: Combining or removing values.
- Multiplication/Division: Scaling values or distributing them.
Factoring: Breaking down expressions into simpler components (e.g., x^2 - 9 = (x + 3)(x - 3)).
Linear Equations: Equations that graph as straight lines (e.g., y = mx + b).
Quadratic Equations: Second-degree polynomial equations (e.g., ax^2 + bx + c = 0), solved using factoring, completing the square, or the quadratic formula.

Ratio:
- Definition: A comparison of two quantities, showing the relative sizes.
- Notation: Can be expressed as a fraction (a/b), with a colon (a:b), or in words ("a to b").
- Types:
  - Part-to-Part: Compares different parts (e.g., 2:3).
  - Part-to-Whole: Compares a part to the total (e.g., 2 out of 5, or 2:5).
Rate:
- Definition: A specific type of ratio that compares two different units (e.g., speed = distance/time).
- Common Examples:
  - Speed: Miles per hour (mph), kilometers per hour (km/h).
  - Density: Mass per unit volume (e.g., grams per cubic centimeter).
  - Unit Price: Cost per item or cost per unit.
Conversions: Important in calculations involving rates; ensure units are consistent.
Proportions: An equation stating that two ratios are equal (e.g., a/b = c/d), often used to solve for unknowns in ratio problems.

Algebra is focused on symbols and rules for manipulating these symbols, essential for solving mathematical problems.
Variables represent unknown quantities and are denoted by letters such as x and y.
Expressions combine variables and constants, exemplified by the equation 3x + 2.
Equations assert the equality of two expressions; for instance, 2x + 3 = 7 indicates that both sides are equivalent.
Inequalities express a relationship where one quantity is greater or less than another, such as x < 5.
Basic operations in algebra include addition and subtraction for combining values, and multiplication and division for scaling and distributing values.
Factoring involves breaking down expressions into simpler components; for example, x^2 - 9 can be factored into (x + 3)(x - 3).
Linear equations are first-degree equations that graph as straight lines, generally written in the form y = mx + b.
Quadratic equations are second-degree polynomial equations, structured as ax^2 + bx + c = 0, solvable through factoring, completing the square, or the quadratic formula.

A ratio compares two quantities, demonstrating their relative sizes and can be shown as a fraction (a/b), using a colon (a:b), or in words (a to b).
There are two main types of ratios:
- Part-to-Part ratios, which compare different parts of a whole (example: 2:3).
- Part-to-Whole ratios, which compare a specific part to the entire amount (example: 2 out of 5, or 2:5).
A rate is a specific kind of ratio that compares different units, such as speed (distance/time).
Common examples of rates include:
- Speed expressed in miles per hour (mph) or kilometers per hour (km/h).
- Density represented as mass per unit volume, like grams per cubic centimeter.
- Unit price, demonstrating cost per item or cost per unit.
Unit consistency is crucial in calculations involving rates to ensure accurate conversions.
Proportions illustrate that two ratios are equivalent (example: a/b = c/d), frequently utilized to find unknown values in ratio problems.