NCERT Plus One (Class 11) Mathematics • Chapter 5

Linear Inequalities

Q: Solve 24x < 100 when (i) x is a natural number, (ii) x is an integer.

Dividing by positive 24: x < 100 24 = 25 6 ≈ 4.17 . • (i) For x ∈ ℕ : {1, 2, 3, 4} . • (ii) For x ∈ ℤ : {..., −2, −1, 0, 1, 2, 3, 4} .

Q: Solve −12x > 30 when (i) x is a natural number, (ii) x is an integer.

Divide by −12 and FLIP sign: x < − 30 12 = −2.5 . • (i) For x ∈ ℕ : ∅ (No natural number is < −2.5) . • (ii) For x ∈ ℤ : {..., −5, −4, −3} .

Q: Solve 5x − 3 < 3x + 1 when (i) x is an integer, (ii) x is a real number.

5x − 3x < 1 + 3 ⇒ 2x < 4 ⇒ x < 2 . • (i) Integer solution: {..., −2, −1, 0, 1} . • (ii) Real solution: (−∞, 2) .

Q: Solve the inequality x/4 < (5x − 2)/3 − (7x − 3)/5 for real x and show the interval.

Multiply by positive LCM 60 : 15x 4 → Interval: (4, ∞) (open circle at 4, shade right).

Q: Solve −15 < (3(x − 2))/5 ≤ 0.

Multiply all 3 parts by 5/3 (positive, no flip): −25 < x − 2 ≤ 0 . Add 2 everywhere: −23 < x ≤ 2 → (−23, 2] .

Q: Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.

Let numbers be x and x + 2 (jump by 2!). Conditions: x > 10 AND x + (x + 2) < 40 ⇒ 2x < 38 ⇒ x < 19 . Odd integers in (10, 19) : x ∈ {11, 13, 15, 17} . Pairs: (11, 13), (13, 15), (15, 17), (17, 19) .

Q: Ravi obtained 70 and 75 marks in first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Let 3rd test mark be x . 70 + 75 + x 3 ≥ 60 ⇒ 145 + x ≥ 180 ⇒ x ≥ 35 . Minimum marks required = 35 .

Q: How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

Adding x litres of pure water (0% acid) gives total volume (1125 + x) . Pure acid stays 0.45 × 1125 : 0.25(1125 + x) 562.5 L . • Left: 0.25x < 0.20(1125) ⇒ x < 900 L . Add between 562.5 L and 900 L of water!

Strict vs slack rules, the negative sign-flip law, ℕ/ℤ/ℝ solution sets, number line dots, double intervals & word problems.

📄 Official NCERT Textbook PDF

Topic 5.2: Strict vs Slack Inequalities & Everyday Constraint Modelling

At a Thrissur bakery, you have ₹200 in your pocket and each pazhampori costs ₹25. You don't HAVE to spend every single rupee — you only need 25x ≤ 200! Real-world rules rarely use exact equals (=); they use limits like "less than", "at most", "more than", and "at least" — that's Linear Inequalities!

STRICT (<, >)Boundary is EXCLUDED (e.g. x < 5 forbids 5!)

SLACK (≤, ≥)Boundary is INCLUDED ("at most" = ≤, "at least" = ≥!)

LINEAR FORMax + b < 0, ≤ 0, > 0, or ≥ 0 (with a ≠ 0)

🧠 Memory Trick: Watch the tricky English words! "At least 60" means 60 or MORE (≥ 60); "At most 200" means 200 or LESS (≤ 200)!

💡 Simple Kid-Check: Imagine a theme-park ride sign saying "Height must be at least 120 cm". Can a kid who is exactly 120 cm get on? YES! That's a slack inequality (h ≥ 120). If it says "Age under 12", a 12-year-old cannot ride — that's strict (a < 12)!

Everyday Phrase	Symbol	Type	Example
Strictly less than / below	`<`	Strict	`x < 10` (10 fails!)
Strictly greater than / above	`>`	Strict	`x > 5` (5 fails!)
At most / not exceeding / max	`≤`	Slack	`25x ≤ 200` (200 passes!)
At least / minimum / not less than	`≥`	Slack	`marks ≥ 35` (35 passes!)

Top translation trap in word problems: Reading "at least" and writing >, or reading "at most" and writing <! Both "at least" (≥) and "at most" (≤) are SLACK inequalities that INCLUDE the boundary number!

Topic 5.3: Golden Rules of Inequality (Negative Sign Flip!)

Look at two numbers: 2 < 5 (clearly 2 is smaller than 5!). Now multiply BOTH sides by −1: you get −2 and −5! Wait — which is larger on a thermometer: −2°C or −5°C? −2 > −5! Multiplying or dividing by a NEGATIVE number spins the whole number line backward — so the inequality sign MUST FLIP!

RULE 1: ADD / SUBAdd or subtract ANY number on both sides → sign stays SAME!

RULE 2: POSITIVE ×/÷Multiply or divide by a POSITIVE number → sign stays SAME!

RULE 3: NEGATIVE ×/÷Multiply or divide by a NEGATIVE number → FLIP SIGN (< ↔ >)!

🧠 Memory Trick: Negative = Mirror Flip! The instant you multiply or divide both sides by a minus number, flip < to >, and ≤ to ≥!

💡 Simple Kid-Check: Why does multiplying by −1 flip the sign? Think of debt! Having ₹2 is less than having ₹5 (2 < 5). Flipping signs turns money into debt: owing ₹2 (−2) is BETTER than owing ₹5 (−5), so −2 > −5!

1
Safe Moves (No Flip): Transposing +5 to the other side as −5, or dividing by +4 never flips the sign.
2
Danger Move (MUST Flip!): Dividing −4x ≤ 20 by −4 flips ≤ into ≥ → x ≥ −5!
3
Or Dodge the Flip: Move −4x to the other side so its coefficient stays positive: −20 ≤ 4x ⇒ −5 ≤ x (same answer!).

The #1 chapter-wide DHSE blunder: Solving −3x < 12 and writing x < −4! Wait — test x = 0: −3(0) = 0 < 12 (TRUE!), yet 0 < −4 is FALSE! Dividing by −3 MUST flip < to >, giving x > −4! Also NEVER cross-multiply by an unknown variable x whose sign might be negative!

Topic 5.3: Algebraic Solving in One Variable (ℕ, ℤ & ℝ Solution Sets)

Suppose solving 30x < 200 tells you the number of notebooks you can buy is x < 6.67. Can you ask a shopkeeper for −3 notebooks or 4.25 notebooks? NO — when x is a Natural Number (ℕ), the solution set is ONLY {1, 2, 3, 4, 5, 6}! Always check which number family x belongs to!

WHEN x ∈ ℕPositive counting numbers ONLY: {1, 2, 3, ...} (roster list!)

WHEN x ∈ ℤIncludes 0 & negative integers: {..., −2, −1, 0, 1, ...}

WHEN x ∈ ℝContinuous strip of all decimals/fractions → write as an Interval!

🧠 Memory Trick: Step 1: Clear fractions with positive LCM. Step 2: Isolate x (flip if dividing by negative!). Step 3: Filter answer through ℕ, ℤ, or ℝ!

💡 Simple Kid-Check: Two inequalities with the exact same algebra like −12x > 30 ⇒ x < −2.5 give completely different answers depending on the question's universe!
• If x ∈ ℕ: No solution (∅) (no natural number is less than −2.5!)
• If x ∈ ℤ: {..., −5, −4, −3} (notice −2 is NOT less than −2.5!)

🧮 Step-by-Step Fractional Inequality: Solve 3(x − 2)5 ≤ 5(2 − x)3 for real x:
1) Multiply both sides by positive LCM 15 (NO flip!): 9(x − 2) ≤ 25(2 − x).
2) Expand: 9x − 18 ≤ 50 − 25x.
3) Group x: 34x ≤ 68 ⇒ x ≤ 2 → Interval: (−∞, 2]!

Watch TWO sneaky Kerala exam traps: (1) For x < −2.5 with x ∈ ℤ, students write {..., −4, −3, −2} — WRONG! Check a number line: −2 is GREATER than −2.5, so the largest integer below −2.5 is −3! (2) Leaving answer as interval (−∞, 6.67) when question specifies x ∈ ℕ!

Topic 5.3: Open Circles vs Solid Dots on the Real Number Line

Drawing a real solution set on a number line is like placing a security guard at the gate! When the inequality is slack (≤ or ≥), the boundary point joins the party with a solid filled dot (●). When it is strict (< or >), the boundary stays locked out with a hollow open circle (○)!

HOLLOW CIRCLE (○)Used for strict < or > → endpoint is EXCLUDED

SOLID DOT (●)Used for slack ≤ or ≥ → endpoint is INCLUDED

SHADED RAYDark line + arrow points RIGHT for >, ≥; LEFT for <, ≤!

🧠 Memory Trick: Extra ink in the symbol = extra ink in the dot! ≤ and ≥ have an extra bar underneath, so colour the dot SOLID (●)! No bar (<, >) = hollow circle (○)!

💡 Simple Kid-Check: Always isolate x on the LEFT side before choosing your arrow direction! If you have x ≥ 3, the arrow > literally points RIGHT! (Whereas 3 ≤ x points left visually even though x goes right!)

1
Put x on the left: e.g. 5 ≥ x becomes x ≤ 5.
2
Mark boundary 5 with a Solid Dot (●) for ≤, ≥ or Open Circle (○) for <, >.
3
Shade thickly along the number line in the arrow's direction (left toward −∞ or right toward +∞).

Two classic sketching errors in DHSE exams: (1) Shading the WRONG side when x sits on the right (reading 3 < x and shading left of 3 instead of right!). Always flip it so x stands on the LEFT: x > 3! (2) Drawing a solid dot for strict < or forgetting to make the dot hollow ○!

Topic 5.3: Double Inequalities & Interval Notation [a, b] vs (a, b)

Think of Goldilocks choosing soup: not too cold (T > 40°C) AND not too hot (T ≤ 65°C) — the safe zone is 40 < T ≤ 65, written sleekly as (40, 65]! Solving two inequalities together means finding the shared overlap (intersection) where BOTH conditions hold true at the exact same time!

DOUBLE CHAIN−8 ≤ 5x − 3 < 7 → do the SAME operation on all 3 parts!

BRACKET RULE[ ] = Solid dot (≤, ≥) | ( ) = Open dot (<, >)

INFINITY RULE−∞ and +∞ ALWAYS use round ( ) — NEVER [ ]!

🧠 Memory Trick: For a system of TWO separate inequalities, graph both on the same number line and take their OVERLAP (S₁ ∩ S₂)! If they don't overlap, answer is ∅ (No Solution)!

💡 Simple Kid-Check: Infinity isn't a parking spot you can touch — it keeps going forever! So x ≥ 3 is ALWAYS written [3, ∞) (closed on 3, round open on ∞!).

🧩 Double Inequality Step-by-Step: Solve −3 ≤ 4 − 7x2 ≤ 18:
1) Subtract 4 from all 3 parts: −7 ≤ −7x2 ≤ 14.
2) Multiply all 3 parts by −2/7 (NEGATIVE → flip BOTH signs!):
(−7)(−2/7) ≥ x ≥ (14)(−2/7) → 2 ≥ x ≥ −4, i.e., −4 ≤ x ≤ 2 → [−4, 2]!

Watch THREE common exam traps: (1) Writing [2, ∞] with a square bracket on ∞ — ALWAYS wrong, infinity takes )! (2) Flipping only ONE sign when multiplying a double inequality a < −x ≤ b by −1 — BOTH signs flip to −b ≤ x < −a! (3) Writing [5, 2] with larger number first — intervals ALWAYS run (smaller, larger)!

Topic 5.3: Real-Life Word Problems: Marks Average, Temperature & Acid Mixture

How many marks does Ravi need in his 3rd exam to average at least 60? What Celsius range matches 68°F to 77°F? How much 30% acid should a lab pour into 600 L of 12% acid? Every single one of these real-life Kerala DHSE 3–4 mark problems turns straight into a 1-variable inequality!

MARKS AVERAGEm₁ + m₂ + x3 ≥ Target

TEMPERATUREF = 95C + 32 | C = 59(F − 32)

ACID MIXTUREPure acid = 12%(600) + 30%(x) out of TOTAL (600 + x)!

🧠 Memory Trick: For consecutive ODD or EVEN integers, NEVER use x and x + 1! Odd/Even numbers jump by 2: always pick x and x + 2!

💡 Simple Kid-Check: In acid/sugar mixture problems, pouring x litres of 30% solution increases BOTH the pure acid AND the total tank volume to (600 + x)!

🧪 Classic 4-Mark Acid Mixture Setup: Add x L of 30% acid to 600 L of 12% acid so resulting mixture lies strictly between 15% and 18%:
15%(600 + x) < 12%(600) + 30%(x) < 18%(600 + x).
Multiply by 100:
• Left part: 9000 + 15x < 7200 + 30x ⇒ 1800 < 15x ⇒ x > 120.
• Right part: 7200 + 30x < 10800 + 18x ⇒ 12x < 3600 ⇒ x < 300.
Answer: Add strictly between 120 and 300 litres!

Watch TWO classic 4-mark word-problem traps: (1) In mixture problems, writing 15%(600) instead of 15%(600 + x) (forgetting that adding x litres expands total tank volume!). (2) In consecutive odd/even integers, writing x and x + 1 — consecutive odd (e.g. 5, 7) or even (e.g. 4, 6) integers ALWAYS differ by 2, so use x and x + 2!

Solved Previous Year Questions (Kerala DHSE & NCERT)

Q1 (Kerala HSE March • 2 Marks): Solve 24x < 100 when (i) x is a natural number, (ii) x is an integer.

Dividing by positive 24: x < 10024 = 256 ≈ 4.17.
• (i) For x ∈ ℕ: {1, 2, 3, 4}.
• (ii) For x ∈ ℤ: {..., −2, −1, 0, 1, 2, 3, 4}.

Q2 (Kerala HSE Say • 2 Marks): Solve −12x > 30 when (i) x is a natural number, (ii) x is an integer.

Divide by −12 and FLIP sign: x < −3012 = −2.5.
• (i) For x ∈ ℕ: ∅ (No natural number is < −2.5).
• (ii) For x ∈ ℤ: {..., −5, −4, −3}.

Q3 (Kerala HSE March • 2 Marks): Solve 5x − 3 < 3x + 1 when (i) x is an integer, (ii) x is a real number.

5x − 3x < 1 + 3 ⇒ 2x < 4 ⇒ x < 2.
• (i) Integer solution: {..., −2, −1, 0, 1}.
• (ii) Real solution: (−∞, 2).

Q4 (Kerala HSE March • 3 Marks): Solve the inequality x/4 < (5x − 2)/3 − (7x − 3)/5 for real x and show the interval.

Multiply by positive LCM 60:
15x < 20(5x − 2) − 12(7x − 3)
15x < 100x − 40 − 84x + 36 = 16x − 4
−x < −4 ⇒ x > 4 → Interval: (4, ∞) (open circle at 4, shade right).

Q5 (Kerala HSE Model • 3 Marks): Solve −15 < (3(x − 2))/5 ≤ 0.

Multiply all 3 parts by 5/3 (positive, no flip):
−25 < x − 2 ≤ 0.
Add 2 everywhere: −23 < x ≤ 2 → (−23, 2].

Q6 (Kerala HSE March • 3 Marks): Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their sum is less than 40.

Let numbers be x and x + 2 (jump by 2!).
Conditions: x > 10 AND x + (x + 2) < 40 ⇒ 2x < 38 ⇒ x < 19.
Odd integers in (10, 19): x ∈ {11, 13, 15, 17}.
Pairs: (11, 13), (13, 15), (15, 17), (17, 19).

Q7 (Kerala HSE Say • 3 Marks): Ravi obtained 70 and 75 marks in first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Let 3rd test mark be x.
70 + 75 + x3 ≥ 60 ⇒ 145 + x ≥ 180 ⇒ x ≥ 35.
Minimum marks required = 35.

Q8 (NCERT / DHSE Model • 4 Marks): How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that the resulting mixture will contain more than 25% but less than 30% acid content?

Adding x litres of pure water (0% acid) gives total volume (1125 + x).
Pure acid stays 0.45 × 1125:
0.25(1125 + x) < 0.45(1125) < 0.30(1125 + x).
• Right: 0.45(1125) < 0.30(1125) + 0.30x ⇒ 0.15(1125) < 0.30x ⇒ x > 562.5 L.
• Left: 0.25x < 0.20(1125) ⇒ x < 900 L.
Add between 562.5 L and 900 L of water!