An inequality is a mathematical expression that shows the relationship between two quantities using the symbols "greater than" (>), "less than" (<), "greater than or equal to" (≥), and "less than or equal to" (≤). Inequalities are used to determine the range of possible values for a variable and are widely applied in algebra, calculus, economics, and physics.

Fundamental Properties of Inequalities:

1. Reflexivity: Any number is always equal to itself, meaning an inequality of the form a ≥ a is always true.

2. Transitivity: If a > b and b > c, then a > c.

3. Antisymmetry: If a ≥ b and b ≥ a, then a = b.

4. Additive Property: If a > b, then a + c > b + c (the inequality remains unchanged if the same number is added to both sides).

5. Multiplicative Property: If a > b and c > 0, then ac > bc; however, if c < 0, the inequality sign reverses: ac < bc.

Basic Rules for Transforming Inequalities:

1. Addition and Subtraction: The same number can be added or subtracted from both sides of an inequality without changing the inequality's direction.
   Example: If x + 3 > 5, then x > 2.
2. Multiplication and Division: Both sides of an inequality can be multiplied or divided by the same positive number without changing the inequality's sign. However, when multiplying or dividing by a negative number, the inequality sign must be reversed.
   Example: If -2x ≤ 6, then x ≥ -3 (dividing both sides by -2 and reversing the sign).
3. Applying Functions: If a function is monotonically increasing, the inequality remains unchanged. If the function is monotonically decreasing, the inequality sign is reversed.
   Example: If x > y, then squaring both sides (for positive numbers) results in x² > y².
4. Variable Substitution: In some cases, introducing a new variable simplifies the expression and helps solve the inequality.
5. Absolute Value Inequalities: When dealing with absolute values, two cases must be considered: one where the expression inside the absolute value is positive and one where it is negative.
   Example: Solving |x - 4| < 3 results in two inequalities: -3 < x - 4 < 3, leading to 1 < x < 7.
6. System of Inequalities: Sometimes, inequalities need to be solved simultaneously, requiring finding the common solution set.
   Example: Solve x - 2 > 0 and x + 3 < 5 together. The solutions are x > 2 and x < 2, meaning there is no solution.

Examples:

Example 1: Linear Inequality
Solve 3x + 5 > 2:
3x > -3
x > -1.

Example 2: Quadratic Inequality
Solve x² - 4 ≤ 0. This means finding the values of x for which the square of a number does not exceed 4. Factorizing: (x - 2)(x + 2) ≤ 0. Using the graphical method or the interval method, we get the solution: -2 ≤ x ≤ 2, meaning x belongs to the interval [-2, 2].