Linear Inequalities Class 11 Math Quick Recall / Short Notes (2026-27)

The Logic of Ranges: Linear Inequalities

[!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 by 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, and graphical shading in 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 in mathematics results in 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 by the same positive number.
Rule 3: If both sides are multiplied/divided by 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 in Two Variables

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

Draw the Line: Replace the inequality sign with '=' and 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 in Linear Programming.

5. Practical Applications

Inequalities are used extensively in:

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 by -2 (Reverse sign): x < -3.
Solution in 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) and (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 in higher mathematics and real-world economics. Stay within your boundaries, but keep your ranges wide!