Linear Inequalities — Mathematics Class 11 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Linear Inequalities (Class 11) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Linear Inequalities (CBSE & HBSE)
Compact chapter combining algebra with coordinate geometry. CBSE asks 4–5 marks: one number-line solution and one graphical system. HBSE keeps it at 3–4 marks.
Linear Inequalities in One Variable
Inequality Symbols
| Symbol | Meaning |
|---|---|
| < | strictly less than |
| > | strictly greater than |
| ≤ | less than or equal |
| ≥ | greater than or equal |
Rules for Manipulation
For real a, b, c:
- Add/subtract the same number to both sides — inequality preserved.
- Multiply/divide both sides by a positive number — preserved.
- Multiply/divide both sides by a negative number — flip the inequality.
- Taking reciprocals of positive numbers reverses direction.
- Squaring is allowed only when both sides are non-negative.
Solving a Linear Inequality (One Variable)
Goal: isolate the variable.
Example: 3x − 5 < 7. 3x < 12 ⇒ x < 4 ⇒ solution = (−∞, 4).
Representation
| Form | Looks Like |
|---|---|
| Number line | open circle for strict, closed for ≤ / ≥ |
| Interval notation | (a, b), [a, b], (a, ∞), [a, ∞) |
| Set-builder | {x ∈ ℝ : x < 4} |
Solution Set Conventions
- Single inequality → an interval (one piece).
- Compound (and / or) → an intersection or union of intervals.
- An empty solution: write ∅ or "no solution".
Compound Inequalities
a < x and x < b ⇒ a < x < b ⇒ open interval (a, b). x < a or x > b ⇒ (−∞, a) ∪ (b, ∞).
CBSE Tip: Always show every step where you multiply/divide by a negative — that's where students lose marks for flipping incorrectly.
Linear Inequalities in Two Variables — Graphical
Form
ax + by + c ≤ 0 (or any of <, >, ≥). Represents a half-plane in the xy-plane.
Steps to Plot
- Replace inequality with equality and draw the line ax + by + c = 0.
- Use a solid line for ≤ / ≥ (boundary included) and a dotted line for < / > (boundary excluded).
- Pick a test point not on the line (commonly (0, 0) if possible).
- Substitute into the inequality. If true ⇒ shade the side containing the test point; if false ⇒ shade the opposite side.
Common Inequalities of Coordinate Axes
- x ≥ 0 → right half-plane (and y-axis)
- y ≥ 0 → upper half-plane
- x ≤ 0, y ≤ 0 → left / lower half-plane
Boundary Lines as Locus
For 3x + 4y ≤ 12: draw 3x + 4y = 12 (intercept form: x/4 + y/3 = 1), shade including the origin if 3(0) + 4(0) = 0 ≤ 12 (true) → shade the origin side.
Quick Cross-Check
- Strict inequality (<, >) → boundary line not part of solution → use dotted line.
- Non-strict (≤, ≥) → boundary included → solid line.
Solution Type
The solution of a single linear inequality in two variables is a half-plane — infinitely many points.
HBSE Tip: Label both axes and write the inequality on the shaded region — examiners deduct for missing labels.
Systems of Linear Inequalities & Word Problems
System of Inequalities (Two Variables)
Graph each inequality separately; the common shaded region (intersection) is the solution.
Strategy
- Plot each boundary line.
- Shade the half-plane for each inequality.
- The overlap is the feasible region.
- Mark vertices (intersections of boundary lines) — useful for linear programming preview.
Bounded vs Unbounded Feasible Region
If the feasible region can be enclosed in a rectangle, it is bounded; otherwise unbounded.
Common Constraint Pairs
- x ≥ 0, y ≥ 0 → restrict to first quadrant (typical for cost / quantity problems).
- ax + by ≤ k (with a, b, k > 0) → caps the feasible region.
Modelling Word Problems
- Define variables (x = number of …, y = number of …).
- Translate each constraint into a linear inequality (cost ≤ budget, total time ≤ available hours, x, y ≥ 0).
- Plot and find the feasible region.
- Identify the vertices (corner points).
Sample Word Problem
A company produces x items of type A and y of type B. Each A needs 2 hours, each B 3 hours; total ≤ 60 hours. Profit constraints: x ≥ 0, y ≥ 0.
Inequalities:
- 2x + 3y ≤ 60
- x ≥ 0, y ≥ 0
Feasible region: triangle with vertices (0, 0), (30, 0), (0, 20).
Use in Class 12 Linear Programming
This chapter is the warm-up for Linear Programming (Class 12). The feasible region you sketch here becomes the constraint set for an objective function.
CBSE Tip: For word problems, always include the non-negativity constraints x ≥ 0 and y ≥ 0 — even if not stated, they apply to physical quantities.
Frequently asked questions
Are these Linear Inequalities notes free?
Yes — the Linear Inequalities notes for Mathematics (Class 11) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Linear Inequalities notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Linear Inequalities chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Linear Inequalities.