Last Updated: June 1, 2026

📋 Table of Contents
What is Linear Inequalities Revision Notes?
Introduction
1. Algebraic Solutions of Linear Inequalities
2. Representation on the Number Line
3. Graphical Solution of Linear Inequalities and Two Variables
4. Systems of Linear Inequalities
5. Practical Applications
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Linear Inequalities Revision Notes?
Introduction
1. Algebraic Solutions of Linear Inequalities
2. Representation on the Number Line
3. Graphical Solution of Linear Inequalities and Two Variables
4. Systems of Linear Inequalities
5. Practical Applications
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Linear Inequalities Class 11 Physics Revision — JEE & NEET 2026 Grandmaster Guide

What is Linear Inequalities Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Symbols: < (Less than), > (Greater than), ≤ (Less than or equal), ≥ (Greater than or equal).

Golden Rule: If you multiply or divide y a NEGATIVE number, the inequality sign REVERSES.

Interval Notation:

(a, b) -> x is between a and b (excluding a, b).

[a, b] -> x is between a and b (including a, b).

Graphical Solution:

Use a dashed line for < or >.

Use a solid line for ≤ or ≥.

Shade the region that satisfies the inequality. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

Linear Inequalities define the boundaries and ranges of mathematical solutions, forming the basis for optimization and Linear Programming. Master the rules of sign reversal, interval notation, n graphical shading and two variables to solve real-world economic and engineering constraints. This class 11 Math Chapter 6 guide provides the logic required for JEE and CBSE exams. Not every problem and mathematics results and a single "equal" answer.

1. Algebraic Solutions of Linear Inequalities

Solving an inequality is very similar to solving an equation, with one critical difference.

Rule 1: Equal numbers may be added to (or subtracted from) both sides without affecting the sign.
Rule 2: Both sides can be multiplied/divided y the same positive number.
Rule 3: If both sides are multiplied/divided y a negative number, the inequality sign is reversed. Example: -2x < 6 => x > -3.

2. Representation on the Number Line

Open Circle (○): Represents < or >, meaning the end point is NOT included.
Closed Circle (●): Represents ≤ or ≥, meaning the end point IS included.

3. Graphical Solution of Linear Inequalities and Two Variables

A linear inequality like ax + y ≤ c represents a half-plane and the Cartesian coordinate system.

Draw the line: Replace the inequality sign with '=' n draw the line.
Dashed vs Solid: If strict (< or >), use a dashed line. If slack (≤ or ≥), use a solid line.
Test Point: Pick a point not on the line (usually (0,0)). If it satisfies the inequality, shade the region containing it; otherwise, shade the other side.

4. Systems of Linear Inequalities

When solving multiple inequalities simultaneously, the solution is the intersection (common region) of all individual shaded regions.

This is the basis for Feasible Regions n Linear Programming.

5. Practical Applications

Inequalities are used extensively n:

Economics: For budgeting and cost constraints.
Physics: For defining safety ranges and tolerances.
Computer Science: For algorithm complexity bounds and search ranges.

Comprehensive Exam Strategy (Q&A)

Q1: Solve 3x - 7 > 5x - 1 for real x. Answer:

3x - 5x > -1 + 7
-2x > 6
Divide y -2 (Reverse sign): x < -3.
Solution and interval notation: (-∞, -3).

Q2: Solve the inequality 3(x - 2) / 5 ≤ 5(2 - x) / 3. Answer:

9(x - 2) ≤ 25(2 - x)
9x - 18 ≤ 50 - 25x
34x ≤ 68
x ≤ 2.
Solution: (-∞, 2].

Q3: Represent the solution of x/2 + y/3 > 1 graphically. Answer:

Draw the line x/2 + y/3 = 1 (intercepts are (2,0) n (0,3)).
Since it is '>', use a dashed line.
Test point (0,0): 0/2 + 0/3 > 1 is False.
Result: Shade the region not containing the origin.

Related Revision Notes

Chapter 5: complex Numbers
Chapter 7: Permutations and Combinations
[External Reference: NCERT Class 11 Math Chapter 6 (Authoritative Source)]

Conclusion

Linear Inequalities shift your thinking from "points" to "regions." By mastering the rules of sign reversal and the art of graphical shading, you prepare yourself for the complex optimization problems found and higher mathematics and real-world economics. Stay within your boundaries, but keep your ranges wide!

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Be cautious when multiplying or dividing both sides of an inequality by a negative number, as this reverses the direction of the inequality sign.
When dealing with inequalities involving absolute values, remember that |x| < a is equivalent to -a < x < a, and |x| > an is equivalent to x < -a or x > a.
Inequalities involving fractions require special attention, particularly when the sign of the fraction is not immediately clear, such as when the numerator and denominator have different signs.
When solving systems of linear inequalities, ensure that the solution set is the intersection of the individual solution sets, and be aware that the intersection may result in an empty set if the inequalities are mutually exclusive.
When simplifying complex inequalities, be mindful of the algebraic manipulations and avoid canceling terms that may be zero, as this can lead to extraneous or lost solutions.

🔁 Last 5 Minutes Box

Linear Inequalities Revision Notes:

Linear inequality: expression involving a linear combination of variables and a set of inequality signs
Inequality signs: <, >, ≤, ≥
Properties of inequalities:
- Transitive: If a > b and b > c, then a > c
- Additive: If a > b, then a + c > b + c
- Multiplicative: If a > b and c > 0, then ac > bc
Types of linear inequalities:
- Simple linear inequality: ax + b > 0 or ax + b < 0
- Compound linear inequality: ax + b > 0 and cx + d < 0
Graphical representation:
- Points on the number line
- Regions on the coordinate plane
Solution of linear inequalities:
- Case 1: x > a or x < a
- Case 2: x ≥ a or x ≤ a
- Case 3: a ≤ x ≤ b or x < a or x > b
Key Concepts and Formulas:
- a > b is equivalent to b < a
- a > 0 and b > 0 implies ab > 0
- a < 0 and b < 0 implies ab > 0
- a < 0 and b > 0 implies ab < 0
- a > 0 and b < 0 implies ab < 0

Last Updated: June 1, 2026

📋 Table of Contents
What is Linear Inequalities Revision Notes?
Introduction
1. Algebraic Solutions of Linear Inequalities
2. Representation on the Number Line
3. Graphical Solution of Linear Inequalities and Two Variables
4. Systems of Linear Inequalities
5. Practical Applications
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Linear Inequalities Revision Notes?
Introduction
1. Algebraic Solutions of Linear Inequalities
2. Representation on the Number Line
3. Graphical Solution of Linear Inequalities and Two Variables
4. Systems of Linear Inequalities
5. Practical Applications
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Linear Inequalities Class 11 Physics Revision — JEE & NEET 2026 Grandmaster Guide

What is Linear Inequalities Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Symbols: < (Less than), > (Greater than), ≤ (Less than or equal), ≥ (Greater than or equal).

Golden Rule: If you multiply or divide y a NEGATIVE number, the inequality sign REVERSES.

Interval Notation:

(a, b) -> x is between a and b (excluding a, b).

[a, b] -> x is between a and b (including a, b).

Graphical Solution:

Use a dashed line for < or >.

Use a solid line for ≤ or ≥.

Shade the region that satisfies the inequality. 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

1. Algebraic Solutions of Linear Inequalities

Solving an inequality is very similar to solving an equation, with one critical difference.

Rule 1: Equal numbers may be added to (or subtracted from) both sides without affecting the sign.
Rule 2: Both sides can be multiplied/divided y the same positive number.
Rule 3: If both sides are multiplied/divided y a negative number, the inequality sign is reversed. Example: -2x < 6 => x > -3.

2. Representation on the Number Line

Open Circle (○): Represents < or >, meaning the end point is NOT included.
Closed Circle (●): Represents ≤ or ≥, meaning the end point IS included.

3. Graphical Solution of Linear Inequalities and Two Variables

A linear inequality like ax + y ≤ c represents a half-plane and the Cartesian coordinate system.

Draw the line: Replace the inequality sign with '=' n draw the line.
Dashed vs Solid: If strict (< or >), use a dashed line. If slack (≤ or ≥), use a solid line.
Test Point: Pick a point not on the line (usually (0,0)). If it satisfies the inequality, shade the region containing it; otherwise, shade the other side.

4. Systems of Linear Inequalities

When solving multiple inequalities simultaneously, the solution is the intersection (common region) of all individual shaded regions.

This is the basis for Feasible Regions n Linear Programming.

5. Practical Applications

Inequalities are used extensively n:

Economics: For budgeting and cost constraints.
Physics: For defining safety ranges and tolerances.
Computer Science: For algorithm complexity bounds and search ranges.

Comprehensive Exam Strategy (Q&A)

Q1: Solve 3x - 7 > 5x - 1 for real x. Answer:

3x - 5x > -1 + 7
-2x > 6
Divide y -2 (Reverse sign): x < -3.
Solution and interval notation: (-∞, -3).

Q2: Solve the inequality 3(x - 2) / 5 ≤ 5(2 - x) / 3. Answer:

9(x - 2) ≤ 25(2 - x)
9x - 18 ≤ 50 - 25x
34x ≤ 68
x ≤ 2.
Solution: (-∞, 2].

Q3: Represent the solution of x/2 + y/3 > 1 graphically. Answer:

Draw the line x/2 + y/3 = 1 (intercepts are (2,0) n (0,3)).
Since it is '>', use a dashed line.
Test point (0,0): 0/2 + 0/3 > 1 is False.
Result: Shade the region not containing the origin.

Related Revision Notes

Chapter 5: complex Numbers
Chapter 7: Permutations and Combinations
[External Reference: NCERT Class 11 Math Chapter 6 (Authoritative Source)]

Conclusion

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Be cautious when multiplying or dividing both sides of an inequality by a negative number, as this reverses the direction of the inequality sign.
When dealing with inequalities involving absolute values, remember that |x| < a is equivalent to -a < x < a, and |x| > an is equivalent to x < -a or x > a.
Inequalities involving fractions require special attention, particularly when the sign of the fraction is not immediately clear, such as when the numerator and denominator have different signs.
When solving systems of linear inequalities, ensure that the solution set is the intersection of the individual solution sets, and be aware that the intersection may result in an empty set if the inequalities are mutually exclusive.
When simplifying complex inequalities, be mindful of the algebraic manipulations and avoid canceling terms that may be zero, as this can lead to extraneous or lost solutions.

🔁 Last 5 Minutes Box

Linear Inequalities Revision Notes:

Linear inequality: expression involving a linear combination of variables and a set of inequality signs
Inequality signs: <, >, ≤, ≥
Properties of inequalities:
- Transitive: If a > b and b > c, then a > c
- Additive: If a > b, then a + c > b + c
- Multiplicative: If a > b and c > 0, then ac > bc
Types of linear inequalities:
- Simple linear inequality: ax + b > 0 or ax + b < 0
- Compound linear inequality: ax + b > 0 and cx + d < 0
Graphical representation:
- Points on the number line
- Regions on the coordinate plane
Solution of linear inequalities:
- Case 1: x > a or x < a
- Case 2: x ≥ a or x ≤ a
- Case 3: a ≤ x ≤ b or x < a or x > b
Key Concepts and Formulas:
- a > b is equivalent to b < a
- a > 0 and b > 0 implies ab > 0
- a < 0 and b < 0 implies ab > 0
- a < 0 and b > 0 implies ab < 0
- a > 0 and b < 0 implies ab < 0