Chapter 5 — Linear Inequalities

Chapter 5 of the Class 11 Maths NCERT textbook, "Linear Inequalities", covers statements relating real numbers or algebraic expressions using <, >, ≤, or ≥, teaching how to solve one-variable and two-variable linear inequalities algebraically and graphically.

• Inequalities describe ranges, not points — The chapter contrasts inequalities with equations: instead of a single exact answer, a solution is a whole range of values, which is why they model constraints in science, economics and statistics so naturally.
• The rules for solving them — Solving rests on two manipulation rules — adding or subtracting keeps the sign, but multiplying or dividing by a negative reverses it — and this sign-flip is the key subtlety that distinguishes inequality algebra from equation algebra.
• Representing and applying solutions — Solutions are expressed as sets, intervals or number-line graphs, and the chapter applies them to real problems like budgets, mixtures and temperature ranges, including double and simultaneous inequalities.

1. 01An inequality consists of two real numbers or algebraic expressions related by <, >, ≤, or ≥; unlike equations, inequalities represent ranges of values, not exact solutions.
2. 02The solution of an inequality is any value of the variable that makes the statement true; solutions are often represented as intervals (e.g., x ∈ (−∞, 2)) or graphically on a number line.
3. 03Rule 1: Equal numbers may be added to or subtracted from both sides of an inequality without changing the inequality sign; Rule 2: Multiplying or dividing both sides by a positive number preserves the sign, but by a negative number reverses it (e.g., 3 > 2 becomes −3 < −2).
4. 04Linear inequalities in one variable have the form ax + b < 0 (or >, ≤, ≥); in two variables, ax + by < c (or >, ≤, ≥); higher-degree inequalities like ax² + bx + c ≤ 0 are quadratic, not linear.
5. 05Double inequalities (e.g., −8 ≤ 5x − 3 < 7) are solved by treating them as two simultaneous inequalities; graphically, the solution is the region where both conditions overlap on the number line.
6. 06Practical applications include solving real-world problems with constraints: finding minimum test scores for target averages, determining ranges for temperature conversions (Celsius to Fahrenheit), calculating mixture ratios for acid solutions, and identifying pairs of numbers satisfying multiple conditions.