⚖️ Linear Inequalities

1. Concept of Inequalities

Two real numbers or algebraic expressions related by symbols <, >, ≤, or ≥ form an inequality. Inequalities involving variables are called literal inequalities.

Strict vs Slack

Strict Inequalities: Use only < or > (e.g., ax + b < 0).

Slack Inequalities: Use ≤ or ≥ (e.g., ax + b ≥ 0).

2. Algebraic Solutions of Linear Inequalities in One Variable

Rules for solving inequalities are similar to equations, with one major exception:

⚠️ The Golden Rule of Inequalities

When you multiply or divide both sides of an inequality by a negative number, the sign of the inequality is REVERSED.

Example: -2x > 4 becomes x < -2.

The solution is represented on a number line using open circles for <, > and solid dots for ≤, ≥.

3. Graphical Solution in Two Variables

A linear inequality in two variables (ax + by < c) divides the Cartesian plane into two half-planes.

Step 1: Replace inequality sign with '=' and draw the line (dotted for strict, solid for slack).
Step 2: Take a test point (usually origin 0,0) not on the line.
Step 3: If the test point satisfies the inequality, shade the half-plane containing the point. Otherwise, shade the other half-plane.

4. System of Linear Inequalities

To solve a system of inequalities graphically, graph each inequality on the same axes. The solution is the common shaded region (intersection) of all the individual half-planes.