Algebra: Variables, Expressions, Equations, and Inequalities

Study Notes

A variable is a symbol that represents a value that can change
An expression is a group of variables, constants, and mathematical operations combined to form a mathematical statement
Examples:
- 2x + 5 is an expression where x is a variable and 2 and 5 are constants
- 3y - 2 is an expression where y is a variable and 3 and -2 are constants

An equation is a statement that says two expressions are equal
Examples:
- 2x + 3 = 5
- x - 2 = 3
An inequality is a statement that says one expression is greater than, less than, or equal to another expression
Examples:
- 2x + 3 > 5
- x - 2 ≤ 3

Linear equations can be solved using addition, subtraction, multiplication, or division
Examples:
- 2x + 3 = 5 => 2x = 5 - 3 => 2x = 2 => x = 2/2 => x = 1
Inequalities can be solved using similar methods, but with caution when multiplying or dividing by negative numbers
Examples:
- 2x + 3 > 5 => 2x > 5 - 3 => 2x > 2 => x > 2/2 => x > 1

A point is a location in space, represented by a set of coordinates (x, y, z)
A line is a set of points that extend infinitely in two directions
A plane is a flat surface that extends infinitely in all directions
Examples:
- A point on a coordinate plane: (3, 4)
- A line on a coordinate plane: y = 2x + 1
- A plane in 3D space: z = 2x + 3y - 1

An angle is formed by two rays sharing a common endpoint (vertex)
Angles can be classified as acute (< 90°), right (90°), obtuse (> 90°), or straight (180°)
Measurements include:
- Perimeter: the distance around a shape
- Area: the amount of space inside a shape
- Volume: the amount of space inside a 3D object
Examples:
- The perimeter of a triangle with sides 3, 4, and 5 is 3 + 4 + 5 = 12
- The area of a rectangle with length 4 and width 3 is 4 × 3 = 12
- The volume of a cube with side length 2 is 2 × 2 × 2 = 8

A variable represents a value that can change and is often denoted by a symbol (e.g., x, y).
An expression combines variables, constants, and mathematical operations to form a mathematical statement.
Examples of expressions include 2x + 5 and 3y - 2, where x and y are variables and 2, 5, 3, and -2 are constants.

An equation is a statement equating two expressions (e.g., 2x + 3 = 5, x - 2 = 3).
An inequality is a statement comparing two expressions using greater than, less than, or equal to (e.g., 2x + 3 > 5, x - 2 ≤ 3).

Linear equations can be solved using addition, subtraction, multiplication, or division.
Solving examples include 2x + 3 = 5, which becomes x = 1 after isolating the variable.
Inequalities require caution when multiplying or dividing by negative numbers to maintain the correct inequality sign.

A point is a location in space represented by a set of coordinates (x, y, z).
A line is a set of points extending infinitely in two directions.
A plane is a flat surface extending infinitely in all directions.
Examples include points like (3, 4) on a coordinate plane, lines like y = 2x + 1, and planes like z = 2x + 3y - 1 in 3D space.

An angle is formed by two rays sharing a common endpoint (vertex).
Angles can be classified as acute (< 90°), right (90°), obtuse (> 90°), or straight (180°).
Measurements include perimeter, area, and volume.
Examples of measurements include the perimeter of a triangle (3 + 4 + 5 = 12), the area of a rectangle (4 × 3 = 12), and the volume of a cube (2 × 2 × 2 = 8).