Schooling · Class 11 · Mathematics · Linear Inequalities · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Linear Inequalities: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- An inequality uses one of the…
- A linear inequality in one variable…
- Adding or subtracting the same…
- Multiplying or dividing both sides…
- The solution set of a linear…
- A linear inequality in two…
- A system of linear inequalities in…
- Formulas and facts
- If a < b, then a + c < b + c
- If a < b and c > 0, then ac < bc
- If a < b and c < 0, then ac > bc
- ax + b > 0 ⇒ x > −b/a (when a > 0)
- ax + b > 0 ⇒ x < −b/a (when a < 0)
- Interval (p, ∞)
- Interval [p, ∞)
- Worked examples
- Solving a linear inequality in one…
- Inequality with a negative coefficient
- Graphing a linear inequality in two…
- Watch out for
- Forgetting to reverse the inequality…
- Using a solid line for strict…
- Testing a point that lies on the…
- Writing the solution set…
- Confusing interval notation
What this chapter is about
In earlier classes, you learnt to solve equations where two expressions are equal. This chapter extends that idea to situations where one quantity is greater than, less than, or at most equal to another. Such statements are called inequalities, and when the variable appears only to the first power they are linear inequalities.
Key ideas
- An inequality uses one of the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to) instead of the equals sign.
- A linear inequality in one variable x has the general form ax + b < 0, ax + b > 0, ax + b ≤ 0, or ax + b ≥ 0, where a ≠ 0.
- Adding or subtracting the same number on both sides of an inequality does not change the direction of the inequality sign.
- Multiplying or dividing both sides by a positive number keeps the inequality sign unchanged; multiplying or dividing by a negative number reverses the inequality sign.
- The solution set of a linear inequality in one variable is usually an infinite interval on the number line, shown with an open circle for strict inequalities and a closed circle when the endpoint is included.
- A linear inequality in two variables, such as 2x + 3y ≤ 12, has a solution set that is a half-plane; the boundary line is included when ≤ or ≥ is used, and excluded when < or > is used.
- A system of linear inequalities in two variables has a solution region that is the intersection of the half-planes corresponding to each inequality.
Formulas and facts to remember
- Rule / Form: If a < b, then a + c < b + c · Meaning: Adding the same number to both sides preserves the inequality.
- Rule / Form: If a < b and c > 0, then ac < bc · Meaning: Multiplying by a positive number preserves direction.
- Rule / Form: If a < b and c < 0, then ac > bc · Meaning: Multiplying by a negative number reverses direction.
- Rule / Form: ax + b > 0 ⇒ x > −b/a (when a > 0) · Meaning: Isolate x; direction stays the same for positive coefficient.
- Rule / Form: ax + b > 0 ⇒ x < −b/a (when a < 0) · Meaning: Direction reverses when coefficient of x is negative.
- Rule / Form: Interval (p, ∞) · Meaning: All real numbers greater than p (p not included).
- Rule / Form: Interval [p, ∞) · Meaning: All real numbers greater than or equal to p (p included).
- Rule / Form: Half-plane for ax + by ≤ c · Meaning: Region on one side of the line ax + by = c, including the line.
Common mistakes
- Forgetting to reverse the inequality when multiplying or dividing by a negative number → always check the sign of the multiplier; if negative, flip the inequality symbol.
- Using a solid line for strict inequalities (<, >) when graphing → use a dashed line for < or > and a solid line only for ≤ or ≥.
- Testing a point that lies on the boundary line to decide which half-plane to shade → pick any point clearly off the line, such as the origin if it is not on the line.
- Writing the solution set incorrectly, for example writing x > 8 when the algebra gives x ≤ 8 → re-check each step, especially when adding or subtracting terms.
- Confusing interval notation: writing (8, ∞] instead of (8, ∞) → infinity is never included, so always use a round bracket next to ∞ or −∞.
Quick revision
- Adding or subtracting the same quantity keeps the inequality direction; multiplying or dividing by a negative reverses it.
- Solution of a linear inequality in one variable is an interval on the number line; use open circle for < or >, closed for ≤ or ≥.
- Solution of a linear inequality in two variables is a half-plane; shade the side where a test point satisfies the inequality.
- Dashed boundary for strict inequality, solid boundary when the inequality includes equality.
- Intersection of half-planes gives the solution region for a system of inequalities.