Algebra: Variables, Expressions, Equations, and Inequalities

Algebra

Variables and Expressions

• 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

Equations and Inequalities

• 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

Solving Equations and Inequalities

• 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

Geometry

Points, Lines, and Planes

• 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

Angles and Measurements

• 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

Algebra

Variables and Expressions

• 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.

Equations and Inequalities

• 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).

Solving Equations and Inequalities

• 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.

Geometry

Points, Lines, and Planes

• 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.

Angles and Measurements

• 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).