Definitions (Linear Equations and Inequalities)

Chapter 5 - Linear Equations and Inequalities

1. Linear Equation

An equation of the form \( ax + b = 0 \), where \( a \) and \( b \) are constants, \( a \neq 0 \), and \( x \) is a variable, is called a linear equation in one variable.

Note: In a linear equation, the highest power of the variable is always 1.

2. Inequality

A mathematical statement that expresses a relationship between two expressions that are not equal.

Inequalities are expressed using the following symbols:

• > Greater than
• < Less than
• ≥ Greater than or equal to
• ≤ Less than or equal to

3. Linear Inequality in One Variable

A linear inequality in one variable \( x \) is of the form:

We may replace the symbol \( < \) by \( > \), \( \leq \) or \( \geq \).

4. Problem Constraints

In real-world problem solving, each inequality associated with a particular problem is called a problem constraint.

The collection of these linear inequalities for a given problem is referred to as problem constraints.

5. Decision Variables

The variables used in these systems of inequalities must satisfy non-negative constraints, meaning they can only take zero or positive values. These variables are crucial for decision-making and are therefore called decision variables.

6. Feasible Region

The area confined to the first quadrant that satisfies all given constraints is known as the feasible region.

7. Feasible Solution

Every point within the feasible region represents a valid feasible solution to the system of linear inequalities.

8. Corner Point (Vertex)

A point of a solution region where two of its boundary lines intersect is called a corner point or vertex of the solution region.

9. Objective Function

A function which is to be maximized or minimized is called an objective function.

10. Optimal Solution

The feasible solution which maximizes or minimizes the objective function is called the optimal solution.