Class 11 Maths - ISC

Linear Inequalities

The Linear Inequalities chapter in Class 11 ISC Mathematics extends your understanding of algebraic equations to situations involving inequalities. You will learn to solve linear inequalities in one and two variables algebraically and represent their solutions graphically on a number plane. Mastering this chapter is crucial as it forms the foundational technique for Linear Programming, a high-weightage topic in Class 12. In ISC board examinations, questions from this chapter frequently appear in both short and long-answer sections, testing your precision in handling inequality signs and boundary lines.

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Key Concepts

Algebraic Solution of Linear Inequalities in One Variable

Finding all real numbers that satisfy a given inequality, utilizing rules similar to linear equations with the vital exception that multiplying or dividing by a negative number reverses the inequality sign.

Graphical Representation in Two Variables

Representing a linear inequality in two variables on a Cartesian plane by first drawing a boundary line (solid or dashed) and then shading the appropriate half-plane determined by a test point.

Solution Region of a System of Inequalities

Finding the common intersection region (feasible region) for a system of two or more linear inequalities, which represents the set of all ordered pairs satisfying all conditions simultaneously.

Strict vs. Slack Inequalities

Strict inequalities use greater than or less than (<, >) and are represented by dashed lines on graphs, while slack inequalities use greater than/equal to or less than/equal to (≤, ≥) and use solid lines.

Important Formulas

If a > b, then a + c > b + c and a - c > b - c

If a > b and c > 0, then ac > bc and a/c > b/c

If a > b and c < 0, then ac < bc and a/c < b/c

For ax + b < 0 (a > 0), the solution is x < -b/a

Board Exam Info

In the ISC Class 11 Mathematics examination, Linear Inequalities typically carries around 4 to 6 marks. Questions usually include solving a system of linear inequalities graphically and finding the region that satisfies all conditions, or solving algebraic inequalities involving modulus functions.

Frequently Asked Questions

When do I reverse the inequality sign?

You must always flip the inequality sign (e.g., from < to >) whenever you multiply or divide both sides of the inequality by a negative number.

How do I know whether to use a solid line or a dashed line when graphing?

Use a solid line for inequalities containing 'less than or equal to' (≤) or 'greater than or equal to' (≥) to show that the boundary line is included in the solution. Use a dashed line for strict inequalities (< or >) to show the boundary is excluded.

What is a test point and how do I use it?

A test point is any coordinate (usually the origin, 0,0) not lying on the boundary line. You substitute its x and y values into the inequality; if the statement is true, shade the region containing that point; if false, shade the opposite side.

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More Maths Chapters - ISC Class 11

Sets Relations And Functions Trigonometric Functions Complex Numbers And Quadratic Equations Permutations And Combinations Binomial Theorem Sequences And Series Straight Lines Conic Sections Introduction To Three Dimensional Geometry Limits And Derivatives Statistics Probability