Definition
A linear inequality is a relationship between two linear algebraic expressions involving .
- How to read: “Less than, greater than, less than or equal to, or greater than or equal to.”
- Meaning: Mathematical operators that define a boundary or range instead of an exact equivalency.
Unlike an equation, the solution is typically a range of values rather than a single point.
Why It Matters
The real world is rarely a single “point”—it is a range. Inequalities provide the language for expressing these boundaries (safety zones, budget limits, tolerances), allowing us to define the “feasible region” of a practical problem.
Core Concepts
- Properties of Inequalities:
- Addition/Subtraction: Adding or subtracting the same value preserves the inequality direction.
- Multiplication/Division by Positive: Preserves the inequality direction.
- Multiplication/Division by Negative: The inequality sign must be reversed.
- Combined Inequalities: Expressions like can be solved by applying identical operations to all three parts simultaneously, maintaining the compound relationship.
- How to read: “a is less than f of x, which is less than b.”
- Meaning: f(x) is strictly bounded between the values of a and b.