When do I use a dashed line vs a solid line in graphical problems?

Use a SOLID line for ≤ or ≥ (the boundary is included). Use a DASHED line for (the boundary is excluded). Think: dashed = dotted = not included.

Chapter 5 of 14

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CBSE Class 11 Mathematics★ 4-6 marks · CBSE Class 11 Mathematics Chapter 5

Linear Inequalities

Not every relationship is an EQUATION. Sometimes — often — it's an INEQUALITY. This chapter covers solving and graphing linear inequalities in one and two variables, and the crucial rule: when you multiply or divide by a NEGATIVE, the sign FLIPS. CBSE Class 11 Mathematics Chapter 5.

⏱ 18 min readEasy difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Solve linear inequalities in one variable and represent solutions using interval notation and number lines
2Apply the sign-flip rule correctly when multiplying or dividing an inequality by a negative number
3Solve compound (simultaneous) inequalities by finding the intersection of two solution sets
4Graph linear inequalities in two variables by identifying the correct half-plane using a test point
5Solve systems of linear inequalities graphically by finding the intersection of all half-planes

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Why this chapter matters

Linear inequalities introduce the concept of solution regions rather than solution points — the foundation for linear programming in Class 12. The sign-flip rule when multiplying by a negative is one of the most frequently tested algebraic rules in board exams.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Relations and Functions

Algebraic manipulation and coordinate plane representation used in graphical solutions

Linear Inequalities

"Life is not an equation. It's an inequality. Constraints, not equalities, define what's possible."

1. Chapter Overview

So far in math, you've mostly solved EQUATIONS (equal to...). But many real-world relationships are INEQUALITIES: less than, greater than, at most, at least. This chapter covers: solving linear inequalities in ONE variable, representing solutions on the NUMBER LINE, solving inequalities in TWO variables, and representing the solution as a REGION on the coordinate plane (linear programming preview).

2. Inequalities — Basic Rules

Inequality Symbols

a < b: a is LESS THAN b
a > b: a is GREATER THAN b
a ≤ b: a is less than OR EQUAL TO b
a ≥ b: a is greater than OR EQUAL TO b

The GOLDEN RULE — Sign Flip

When you MULTIPLY or DIVIDE both sides of an inequality by a NEGATIVE NUMBER, the inequality sign REVERSES.

Example: -2x < 6 → Divide both sides by -2 → x > -3 (sign flipped from < to >)

Other Rules (No Sign Flip)

Adding/subtracting the same number to both sides: sign DOES NOT change
Multiplying/dividing both sides by a POSITIVE number: sign DOES NOT change

3. Solving Linear Inequalities in One Variable

Method

Solve like an equation: isolate the variable on one side
BE CAREFUL: if dividing/multiplying by a negative → FLIP the sign

Representing the Solution

Algebraic: x > 3
Interval notation: (3, ∞)
Number line: OPEN circle at 3 (not included), arrow pointing RIGHT

Types of Intervals

Inequality	Interval	Number Line
x > a	(a, ∞)	Open at a
x ≥ a	[a, ∞)	Closed at a
x < b	(-∞, b)	Open at b
x ≤ b	(-∞, b]	Closed at b
a < x < b	(a, b)	Open at both ends
a ≤ x ≤ b	[a, b]	Closed at both ends

4. Compound (Simultaneous) Inequalities

A < B < C → solve as TWO separate inequalities: A < B AND B < C
The solution is the INTERSECTION

Example: -3 ≤ 2x — 1 < 5 → Solve: -3 ≤ 2x — 1 → -2 ≤ 2x → x ≥ -1 → Solve: 2x — 1 < 5 → 2x < 6 → x < 3 → Solution: -1 ≤ x < 3 (intersection)

5. Linear Inequalities in Two Variables

The Inequality

ax + by + c > 0 (or <, ≥, ≤)

Graphical Solution

Replace the inequality sign with = and GRAPH the line
- If strict (<, >): DASHED line (line itself NOT included)
- If non-strict (≤, ≥): SOLID line (line itself IS included)
Choose a TEST POINT (usually the origin (0,0) — if it's not on the line)
Plug into the inequality:
- If TRUE → shade the side CONTAINING the test point
- If FALSE → shade the OPPOSITE side

System of Linear Inequalities

Each inequality defines a HALF-PLANE. The solution is the INTERSECTION of all half-planes.
This is the basis of LINEAR PROGRAMMING (Class 12)

6. Exam Focus

Solving one-variable linear inequalities — especially the sign flip rule
Representing solution on number line
Interval notation
Solving compound inequalities
Graphical solution of two-variable inequalities (dashed vs solid line, test point)

7. Common Mistakes

Forgetting to flip the sign when multiplying/dividing by a negative — THE #1 error. -2x < 6 → x > -3 (NOT x < -3!)
Using a solid line for strict inequality — If < or > (no equal sign), use DASHED line. The line itself is NOT part of the solution.

8. Conclusion

Inequalities describe CONSTRAINTS — and constraints define the real world:

SIGN FLIP RULE: Multiply/divide by negative → flip inequality sign
NUMBER LINE: Open for strict inequality, closed for non-strict
TWO VARIABLES: The solution is a REGION (half-plane). Graph the line (dashed or solid), test a point, shade the correct side.

'Equations tell you where you CAN be. Inequalities tell you where you MUST be.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Sign Flip Rule

If a < b and c < 0, then ac > bc (inequality reverses when multiplied/divided by a negative)

THE most important rule in this chapter — the #1 source of errors

Interval Notation

x > a ↔ (a,∞); x ≥ a ↔ [a,∞); x < b ↔ (−∞,b); a < x < b ↔ (a,b); a ≤ x ≤ b ↔ [a,b]

Round bracket = open (endpoint excluded); square bracket = closed (endpoint included)

Distance from a Line (half-plane test)

Substitute test point (0,0) into ax+by+c>0; if true, shade the side containing (0,0)

Use a different test point if (0,0) lies on the boundary line

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Not flipping the inequality sign when dividing by a negative number

✓ Example: −2x < 6. Divide by −2 → x > −3 (sign FLIPS from < to >). Forgetting this gives the completely wrong solution set.

WATCH OUT

✗ Drawing a solid line for a strict inequality (< or >) in graphical problems

✓ Strict inequalities (< or >) use a DASHED boundary line because the line itself is not included in the solution. Use a solid line only for ≤ or ≥.

WATCH OUT

✗ Testing the point (0,0) when the line passes through the origin

✓ If (0,0) lies on the boundary line, you must choose a different test point such as (1,0) or (0,1).

WATCH OUT

✗ Forgetting to take the intersection in compound inequalities

✓ For a<x<b (AND), the solution is the intersection — both conditions must hold simultaneously. If it says OR, take the union instead.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 5.1

Exercise 5.1

Solving linear inequalities in one variable, number line representation, interval notation

Questions

Ex 5.2

Exercise 5.2

Graphical solution of linear inequalities in two variables

Questions

Ex 5.3

Exercise 5.3

Systems of linear inequalities — graphical solution and finding the feasible region

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· One Variable

Solve: −3x + 9 > 0 and represent the solution on a number line.

▶ Show solution

−3x > −9. Divide by −3 (flip sign): x < 3. Solution: x < 3, i.e., (−∞, 3). Number line: open circle at 3, arrow pointing left.

Q2MEDIUM· Compound Inequality

Solve: −5 ≤ 3x − 2 < 10 and write the solution in interval notation.

▶ Show solution

Split into two: (i) −5 ≤ 3x−2 → −3 ≤ 3x → −1 ≤ x. (ii) 3x−2 < 10 → 3x < 12 → x < 4. Intersection: −1 ≤ x < 4. Interval notation: [−1, 4).

Q3HARD· Two Variables

Solve graphically: 2x + 3y ≤ 12, x ≥ 0, y ≥ 0.

▶ Show solution

Step 1: Draw the line 2x+3y=12. x-intercept: (6,0); y-intercept: (0,4). Step 2: Line is solid (≤ includes equality). Step 3: Test (0,0): 2(0)+3(0)=0 ≤ 12 ✓, so shade the side containing the origin. Step 4: Also x≥0 (right of y-axis) and y≥0 (above x-axis). Step 5: The feasible region is the triangle with vertices (0,0), (6,0), and (0,4) — shaded.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Adding/subtracting the same value to both sides: inequality sign UNCHANGED
•Multiplying/dividing by a POSITIVE number: inequality sign UNCHANGED
•Multiplying/dividing by a NEGATIVE number: inequality sign REVERSES — the golden rule
•Open circle on number line = strict inequality (< or >); closed circle = non-strict (≤ or ≥)
•Interval: (a,b) = open both ends; [a,b] = closed both ends; [a,b) = left closed right open
•Graphical: strict inequality → dashed boundary line; non-strict → solid boundary line
•Test point method: substitute (0,0) into the inequality; shade the same side if true, opposite side if false
•Feasible region = intersection of all half-planes in a system of inequalities

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 4-6 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	1-2	Solving one-variable inequalities, number line and interval notation
Long Answer	4	1	Graphical solution of system of two-variable inequalities

Prep strategy

Practice the sign-flip rule with at least 10 examples until it becomes instinctive — it appears in every exam
For graphical problems, always write the line equation first, find intercepts, then test (0,0) — a systematic three-step approach eliminates errors
In compound inequalities, always solve both halves separately first, then take the intersection

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Linear Programming (Class 12 preview)

Factory production planning, diet optimisation, and transportation problems are solved by maximising or minimising a linear objective function subject to linear inequality constraints.

Budget Constraints in Economics

A consumer's budget constraint is an inequality (spending ≤ income). The feasible region represents all affordable consumption bundles — directly applying the graphical method.

Medical Dosage Ranges

Drug dosages are prescribed as inequalities: 'at least X mg but no more than Y mg per day' — a compound inequality ensuring both efficacy (≥ lower bound) and safety (≤ upper bound).

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Always write the inequality sign at every step — omitting it and just writing numbers is a common cause of sign-flip errors going unnoticed
For number line questions, circle the endpoint clearly (open or closed) before drawing the arrow
For two-variable graphical problems, clearly label the boundary line, the shaded region, and the test point used
In word problems, define your variable clearly, write the inequality, solve, then interpret the answer in context

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

AM-GM inequality: for positive reals a,b — (a+b)/2 ≥ √(ab) — the foundational inequality of olympiad mathematics
Quadratic inequalities: solving ax²+bx+c>0 using roots and sign charts — a natural extension of this chapter

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainMedium

NDA MathematicsHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Consider 2 < 4. Multiply both sides by −1: −2 and −4. But −2 > −4 on the number line. The relative order reverses when you reflect across zero, so the inequality sign must flip.

Use a SOLID line for ≤ or ≥ (the boundary is included). Use a DASHED line for < or > (the boundary is excluded). Think: dashed = dotted = not included.

Choose another test point that is clearly not on the line, such as (1,0) or (0,1). Then test whether it satisfies the inequality and shade accordingly.

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