Introduction to Linear Inequalities for JEE Main

The chapter Linear Inequalities is a key topic in Class 11 Maths and is extensively used in algebra, coordinate geometry, and optimization problems in JEE Main.

A linear inequality is similar to a linear equation, but instead of equality, it involves inequality signs: <, ≤, >, ≥.

• Example: 2x + 3 > 7 or x – 5 ≤ 2

Solving linear inequalities and understanding their representation is crucial for both conceptual and numerical JEE Main questions.

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Linear Inequalities in One Variable

1. Definition:
A linear inequality in one variable x is of the form ax + b < 0, ax + b ≤ 0, ax + b > 0, ax + b ≥ 0, where a ≠ 0.

2. Algebraic Method of Solving:

• Isolate the variable x on one side:

  • Example: 3x – 7 ≤ 5 → 3x ≤ 12 → x ≤ 4

• Important Rule: Multiply or divide by a negative number → reverse the inequality sign

3. Solution Set Representation:

• Interval Notation: x ≤ 4 → (–∞, 4]

• Graphical Representation: Shade the solution region on the number line

4. Combined Inequalities:

• Conjunction (AND): x ≥ 2 and x ≤ 5 → 2 ≤ x ≤ 5

• Disjunction (OR): x ≤ 1 or x ≥ 4 → x ∈ (–∞, 1] ∪ [4, ∞)

Linear Inequalities in Two Variables

1. General Form: ax + by < c, ax + by ≤ c, ax + by > c, ax + by ≥ c

2. Graphical Method:

• Convert inequality to equation: ax + by = c → straight line

• Determine boundary line type:

  • Solid line for ≤ or ≥

  • Dashed line for < or >

• Test a point (usually origin) to decide which half-plane to shade

• The shaded region represents all solutions

3. System of Linear Inequalities:

• Multiple inequalities → solution is intersection of individual shaded regions

• Example:

  • x + y ≤ 4

  • x ≥ 0, y ≥ 0

  • Feasible region → triangle in the first quadrant

4. Applications:

• Feasible region problems in optimization and word problems

• Basis for linear programming problems in higher studies

Properties of Linear Inequalities

1. Transitive Property: If a > b and b > c → a > c

2. Additive Property: If a > b → a + c > b + c

3. Multiplicative Property:

   • If a > b and c > 0 → ac > bc

   • If a > b and c < 0 → ac < bc (inequality reverses)

Understanding these properties is essential for solving complex inequalities in JEE Main.

Applications of Linear Inequalities in JEE Main

• Solving word problems involving constraints

• Representing feasible solutions in linear programming

• Solving optimization problems using graphical methods

• Handling multi-variable constraints in algebraic and geometric forms

Linear inequalities are frequently tested in MCQs, integer-type questions, and reasoning-based problems in JEE Main.

Important JEE Main Preparation Tips for Linear Inequalities

• Memorize basic rules and properties of inequalities

• Practice solving inequalities in one and two variables

• Use graphical representation for better understanding

• Solve previous year JEE Main MCQs and numerical problems

• Focus on feasible region identification in systems of inequalities

Why Study Linear Inequalities from Studentbro.in?

Studentbro.in provides:

• Step-by-step explanations of linear inequalities in one and two variables

• Solved examples for algebraic and graphical methods

• Diagrams and charts for quick revision

• MCQs and PYQs aligned with JEE Main syllabus

Conclusion

The Linear Inequalities chapter is fundamental and scoring in Class 11 Maths for JEE Main. Mastery of algebraic and graphical methods, properties, and feasible regions helps students solve conceptual and numerical problems efficiently.

Studentbro.in provides structured, easy-to-understand, and exam-focused content to master Linear Inequalities effectively and boost JEE Main scores.