Linear Inequalities
Uttar Pradesh Board · Class 11 · Mathematics
Quick revision notes for Linear Inequalities — Uttar Pradesh Board Class 11 Mathematics. Key concepts, formulas, and definitions for last-minute revision.
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Introduction to Inequalities
- Inequalities arise naturally in real-life situations involving comparisons and constraints
- Examples: height restrictions, budget limitations, capacity constraints
- Inequalities use symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to)
Types of Inequalities
- Strict inequalities: Use < or > symbols, do not include the boundary value
- Slack (Non-strict) inequalities: Use ≤ or ≥ symbols, include the boundary value
- Linear inequalities in one variable: ax + b < 0, ax + b > 0, etc. (where a ≠ 0)
Rules for Solving Linear Inequalities
- Rule 1: Equal numbers may be added to or subtracted from both sides without changing the inequality sign
- Rule 2: Both sides can be multiplied or divided by the same positive number without changing the inequality sign
- CRITICAL Rule 2 Exception: When multiplying or dividing by a negative number, the inequality sign must be reversed
Algebraic Solutions of Linear Inequalities in One Variable
- Solution of an inequality: Any value of the variable that makes the inequality a true statement
- Solution set: The set of all solutions of an inequality
- Unlike equations which usually have one solution, inequalities typically have infinitely many solutions
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