Class 11 Mathematics Chapter 5 Linear Inequalities — Formulas & Key Points

Core Inequality Symbols and Definitions

• < means 'less than' (strict inequality, open interval)
• > means 'greater than' (strict inequality, open interval)
• ≤ means 'less than or equal to' (non-strict, closed on that end)
• ≥ means 'greater than or equal to' (non-strict, closed on that end)
• Negation: the opposite of < is ≥, and the opposite of ≤ is >

Rules for Solving Linear Inequalities (One Variable)

Interval Notation and Set-Builder Form

• Use ( or ) when the endpoint is NOT included (< or >)
• Use [ or ] when the endpoint IS included (≤ or ≥)
• ∞ and −∞ always appear with round brackets
• Union symbol ∪ combines disjoint intervals (e.g. x < 2 or x > 5)

Graphical Solution of Two-Variable Linear Inequalities

• Boundary line: ax + by = c (replace inequality symbol with =)
• Dashed line for < or > (points on the line are NOT solutions)
• Solid line for ≤ or ≥ (points on the line ARE solutions)
• Test point method: substitute (0,0) or any point not on the line
• Shade the half-plane where the inequality is satisfied

System of Linear Inequalities — Feasible Region

• Graph each inequality one by one on the same coordinate plane
• Feasible region = common shaded area satisfying all inequalities
• Bounded region: closed polygon; unbounded region: open area extending to infinity
• Corner points (vertices) are found by solving pairs of boundary-line equations simultaneously
• Always label the feasible region clearly in your answer

Key Terminology and Definitions

• Linear inequality in one variable: ax + b < c (or >, ≤, ≥), a ≠ 0
• Linear inequality in two variables: ax + by < c, a and b not both zero
• Solution set: all values or ordered pairs that make the inequality true
• Half-plane: region on one side of a line in the Cartesian plane
• Boundary line: ax + by = c (equality form of the inequality)
• Strict inequality: < or > (boundary not included, dashed line)
• Non-strict inequality: ≤ or ≥ (boundary included, solid line)

Common Sign, Notation and Graphing Mistakes

• Forgetting to flip the sign when multiplying/dividing by a negative number
• Drawing a solid line for < or > (should be dashed)
• Writing [a, ∞] instead of [a, ∞) — infinity takes round bracket only
• Shading inequalities on separate graphs instead of finding the common region
• Choosing the test point on the boundary line (it must be off the line)
• Omitting the colon in set-builder notation: write {x: x < 3}, not {x < 3}

Memory Tricks and Mnemonics

• 'Negative flips the sign' — multiplying or dividing by negative reverses inequality
• 'Dashed for Dangerous' — strict inequalities (< or >) use dashed boundary lines
• 'Test the origin' — substitute (0,0) to decide which half-plane to shade
• 'Round for open, Square for closed' — interval notation brackets
• For systems, think 'overlap wins' — only the common shaded region is the solution

Three Solved Mini-Examples Applying the Formulas

Last-Minute Revision Box — One-Glance Summary

• Symbols: < (less), > (greater), ≤ (less or equal), ≥ (greater or equal)
• Sign reversal rule: multiply/divide by negative ⇒ flip the inequality sign
• Interval notation: ( ) open, [ ] closed; ∞ always gets ( )
• Graphical method: (1) boundary line, (2) dashed or solid, (3) test point, (4) shade half-plane
• System: graph all inequalities on one plane, feasible region = overlap
• Common mistakes: forgetting to flip sign, wrong line style, wrong bracket for infinity