🔹 1. Introduction
A linear inequality is similar to a linear equation but uses inequality signs instead of equality.
👉 Symbols used:
- < (less than)
- (greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
🔹 2. Definition
A linear inequality in one variable is of the form:
ax + b > 0, ax + b < 0, ax + b ≥ 0, ax + b ≤ 0
Where a ≠ 0
🔹 3. Solution of Linear Inequality (One Variable)
✔️ Rules
- Add/subtract same number → inequality remains same
- Multiply/divide by positive number → sign remains same
- Multiply/divide by negative number → sign reverses
✔️ Example
2x + 3 > 7
⇒ 2x > 4
⇒ x > 2
🔹 4. Representation on Number Line (Concept)
- Open circle → strict inequality (<, >)
- Closed circle → inclusive inequality (≤, ≥)
🔹 5. Interval Notation
✔️ Types of Intervals
- (a, b) → open interval
- [a, b] → closed interval
- (a, b] or [a, b) → mixed
✔️ Infinite Intervals
- (−∞, a), (a, ∞)
🔹 6. Linear Inequalities in Two Variables
✔️ General Form
ax + by + c > 0
✔️ Solution
- Solution is a region (not a point)
- Represents half-plane
🔹 7. Graphical Solution (Concept)
Steps:
- Replace inequality with equation
- Draw boundary line
- Test a point (like origin)
- Shade required region
🔹 8. System of Linear Inequalities
Multiple inequalities solved together.
✔️ Solution
👉 Common region satisfying all inequalities
🔹 9. Important Properties
✔️ Inequality sign changes when multiplied by negative
✔️ Solution set can be interval or region
✔️ Graph represents feasible region
🔹 10. Applications
✔️ Optimization problems
✔️ Economics (profit/loss)
✔️ Engineering constraints
✔️ Linear programming basics
🔹 11. JEE & CBSE Important Points
✔️ Practice sign changes carefully
✔️ Interval notation questions common
✔️ Graphical interpretation important
✔️ System of inequalities frequently asked
✔️ Word problems important
🔹 12. Common Mistakes
❌ Forgetting to reverse sign
❌ Wrong interval notation
❌ Incorrect boundary inclusion
❌ Graph errors
🔹 13. Practice Questions
- Solve linear inequality in one variable
- Represent solution in interval form
- Solve inequalities in two variables
- Find feasible region
- Solve system of inequalities
🔹 14. Quick Revision Sheet
- ax + b > 0
- Multiply by negative → reverse sign
- Open interval → ( )
- Closed interval → [ ]
- Solution in two variables → region
🔹 15. Conclusion
Linear Inequalities is a simple and scoring chapter in Class 11 Maths.
👉 Focus on:
- Rules of inequality
- Interval notation
- Graphical understanding
to score well in CBSE & JEE exams.