Linear Inequalities: Class 11 Maths Chapter 5 Hinglish mein

Mathematics · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 18 Oct 2026 · Updated 18 Oct 2026

50 min
28 practice questions
14 MCQs

Is chapter mein aap kya seekhenge

Roz ke statements mein inequalities pehchano aur unhe <, >, ≤ ya ≥ se likho.
Kisi inequality ko numerical, literal ya double, strict ya slack, aur linear hai ya nahi, is hisaab se classify karo.
N, Z ya R par ek variable wali linear inequality ke solutions aur solution set nikalo.
Do rules se linear inequalities solve karo, aur negative number se multiply ya divide karte waqt sign ulta karo.
Solution set ko number line par dikhao aur use interval notation mein likho.
Common solutions dhoondh kar double inequalities aur do linear inequalities ke systems solve karo.
Averages, consecutive numbers, lengths, temperature ranges aur mixtures wale word problems ko linear inequalities mein model karo aur answers ka matlab samjhao.

NCERT Mathematics Class 11, Chapter 5, pp. 89–99 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

5.1 Introduction

Ek housing society ki lift mein ek plate lagi hai: maximum 8 persons ya 544 kg. Film ka certificate kehta hai ki viewer ki umar kam se kam 18 honi chahiye. Primary health centre ka vaccine fridge 2 °C aur 8 °C ke beech rehna chahiye. Inmein se koi bhi ek single value fix nahi karta. Har ek ek limit set karta hai, aur bahut saari values us limit ko maanti hain.

Is tarah ke statements chaar signs use karte hain: $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to) aur $\geq$ (greater than or equal to). Inmein se kisi ek sign se bana statement inequality (असमिका) kehlata hai. Economists budgets ke liye inequalities use karte hain, engineers safety margins aur tolerances ke liye, doctors safe dose ranges ke liye aur statisticians error bounds ke liye.

Equation aam taur par unknown ko ek hi value par pin kar deti hai; inequality aam taur par values ka poora stretch allow karti hai (Fig. 5.1). Is chapter mein tum ek variable wali linear inequality ke liye woh poora stretch nikalna aur use number line par draw karna seekhoge.

2/2Right pan bhaari: $2 x + 1 < 7$ har $x < 3$ ke liye sach hai.

Fig. 5.1Equation sirf ek value ke liye balance hoti hai; inequality values ke poore stretch ke liye sach rehti hai.

Balance ke left pan mein x likhe do boxes aur 1 likha ek weight hai; right pan mein 7 likha ek weight hai. Pehle step mein beam seedhi hai, statement 2x + 1 = 7 padhta hai, aur neeche number line par 3 par ek bhara hua dot hai. Doosre step mein beam jhuk jaati hai aur right pan neeche ho jaata hai, statement 2x + 1 < 7 padhta hai, aur number line par 3 par ek khokhla circle hai jisse ek moti ray left ki taraf jaati hai, kyunki 3 se chhota har number kaam karta hai.

5.2 Inequalities

Ek kharidari se shuru karte hain. Tumhari cricket team ko match ke liye paani ki bottles chahiye. Ek bottle ₹20 ki hai aur team ki kitty mein ₹150 hain. Agar tum $x$ bottles kharidte ho to $20 x$ rupaye kharch hote hain, aur yeh kitty ke paison se zyada nahi ho sakta:

20 x \leq 150

(5.1)

Yahan koi equals sign ek single value force nahi karta. Asal mein tum poore ₹150 kabhi kharch hi nahi kar sakte, kyunki 150, 20 ka multiple nahi hai: saat bottles ke baad ₹10 bachte hain, aur aathvi bottle ke liye ₹10 kam padte hain (Fig. 5.2).

3/4 $x = 7$ ka kharch ₹140 hai, ₹10 bachte hain.

Fig. 5.2₹20 wali poori bottles: ₹150 mein saat aa jaati hain, aath nahi.

Ek lamba khaali frame ₹150 ke budget ko dikhata hai, neeche ₹0, ₹50, ₹100 aur ₹150 marked scale hai. Pehle step mein frame khaali hai aur statement 20x ≤ 150 dikhaya gaya hai. Doosre step mein paanch bottle blocks frame ka ₹100 bhar dete hain aur label kehta hai ki yeh budget mein fit hai. Teesre step mein saat blocks ₹140 bharte hain aur ek label ₹10 bache hone ki baat kehta hai. Chauthe step mein ek aathva block, dashed bana hua, frame ke end se bahar nikalta hai, labels ke saath jo ₹10 zyada aur budget se bahar kehte hain, kyunki 20 × 8 = 160, 150 se zyada hai.

Ab do quantities ko badalne do. Notebooks ₹50 ki ek aur pens ₹10 ka ek hain, aur tumhare paas $x$ notebooks aur $y$ pens ke liye ₹200 hain. Bill $50 x + 10 y$ rupaye ka hai, isliye

50 x + 10 y \leq 200

(5.2)

Kuch baskets har rupaya use kar leti hain: teen notebooks aur paanch pens theek ₹200 ke hain. Kuch mein change bachta hai: do notebooks aur chaar pens ₹140 ke hain. Yaani ek sign $\leq$ ke andar do alag statements chhupe hain: change bachane wali baskets ke liye strict inequality $50 x + 10 y < 200$ , aur poora paisa use karne wali baskets ke liye equation $50 x + 10 y = 200$ .

Definition

Inequality

Aisa statement jismein do quantities ke beech $<$ , $>$ , $\leq$ ya $\geq$ mein se koi ek sign ho, aur har quantity ek real number ya algebraic expression ho.

Example $7 > - 2$ aur $3 x - 1 \leq 8$ dono inequalities hain.

Sirf numbers ke beech comparison, jaise $π > 3$ ya $- 5 < - 2$ , ek numerical inequality hai: yeh seedha true ya false hoti hai. Jismein variable ho, woh literal inequality hai. 60 km/h limit wali road par speed $v$ ko $v \leq 60$ maanna hoga, aur yeh sach hai ya nahi, yeh $v$ ki value par depend karta hai. Double inequality kisi quantity ko do bounds ke beech fasa deti hai. Marks band $40 \leq m < 60$ ko aise padhte hain: " $m$ kam se kam 40 hai aur 60 se kam hai", isliye 40 band mein aata hai aur 60 nahi.

Signs $<$ aur $>$ strict inequality banate hain, jo boundary value ko bahar rakhti hai; signs $\leq$ aur $\geq$ slack inequality banate hain, jo use andar aane deti hai. Inequality tab linear hoti hai jab kisi bhi side par variable ki power 1 se zyada na ho aur koi variable doosre variable se multiply na ho. Table 5.1 mein kuch examples sort kiye gaye hain.

Table 5.1Inequality ke prakaar

Example	Kis type ki?
$3 x - 7 < 0$	linear, ek variable, strict
$5 - 2 t \geq 0$	linear, ek variable, slack
$2 x + 3 y > 6$	linear, do variables, strict
$4 x - y \leq 10$	linear, do variables, slack
$x^{2} \leq 9$	quadratic, ek variable, slack
$1 < 2 p \leq 9$	linear, ek variable, double

Definition

Linear inequality

Ek variable $x$ mein, expression $a x + b$ jismein $a \neq 0$ ho, ko chaar signs mein se kisi ek se 0 ke saath compare kiya jaata hai. Do variables mein, $a x + b y + c$ ko 0 se compare karte hain, jahan $a \neq 0$ aur $b \neq 0$ .

Example $5 - 2 t \geq 0$ , $t$ mein linear hai; $x^{2} \leq 9$ linear nahi hai.

$x^{2}$ wali row ek quadratic inequality hai, jo is chapter se bahar hai. Ab se hum ek variable wali linear inequalities par kaam karenge. Do variables wali inequalities Class 12 mein wapas aati hain, jahan linear programming mein unke solutions plane ke poore regions bhar dete hain.

5.3 Ek Variable wali Linear Inequalities ke Algebraic Solutions aur unka Graphical Representation

Bottles par wapas chalo, $20 x \leq 150$ . Yahan $x$ bottles ginta hai, isliye woh sirf 0, 1, 2 waghairah ho sakta hai. Itne kam candidates ke saath, sabse imaandaar pehla kadam hai unhe try karna (Table 5.2).

Table 5.2

20 x \leq 150

mein whole numbers test karna

$x$	$20 x$	Kya $20 x \leq 150$ ?
0	0	true
1	20	true
2	40	true
3	60	true
4	80	true
5	100	true
6	120	true
7	140	true
8	160	false

0 se 7 tak har value pass hoti hai, aur 8 fail hoti hai; badi values bhi fail hoti hain, kyunki $20 x$ sirf badhta hai. Test pass karne wali har value inequality ka solution (हल) kehlati hai.

Definition

Solution set

Kisi inequality ke saare solutions ka set, un values mein se liya gaya jo variable le sakta hai.

ExampleBottles ki whole numbers ke liye, $20 x \leq 150$ ka solution set hai ${0, 1, 2, \dots, 7} .$

Values try karna sirf isliye kaam karta hai kyunki yahan answers kam aur whole hain. Agar $x$ koi bhi real number ho sakta, to koi table kabhi saare solutions list nahi kar paati, isliye humein ek method chahiye. Equations ke liye tumhare paas pehle se ek hai: dono sides mein same number add ya subtract karo, ya dono sides ko same non-zero number se multiply ya divide karo, jab tak $x$ akela na reh jaaye. Inequalities lagbhag yahi moves allow karti hain, bas ek twist ke saath.

Do rules

LawRule 1 (add ya subtract karna)

Dono sides ko same amount se shift karna (har side mein ek number add karna, ya har side se ek number ghatana) inequality ki direction nahi badalta: agar $a < b$ , to har real number $c$ ke liye $a + c < b + c$ .

LawRule 2 (multiply ya divide karna)

Dono sides ko positive number se scale karna (multiply ya divide karna) inequality ko waisa hi rakhta hai; negative number se scale karna use ulta kar deta hai. Symbols mein, agar $a < b$ , to $c > 0$ ke liye $a c < b c$ , lekin $c < 0$ ke liye $a c > b c$ .

Sabse zyada galtiyan Rule 2 mein hoti hain, aur number line dikhati hai ki aisa kyun hona hi chahiye (Fig. 5.3). Positive number add, subtract ya usse multiply karna line ko slide ya stretch karta hai, ulta nahi karta. $- 1$ se multiply karna har point ko 0 mein reflect karta hai, isliye jo point left par tha woh right par pahunch jaata hai. Yahi palatna $\leq$ aur $\geq$ par bhi lagu hota hai: $- 2 x \geq 6$ ko $- 2$ se divide karne par $x \leq - 3$ milta hai.

4/4Rule 2, negative: har point 0 mein reflect hota hai aur order palat jaata hai: $6 > - 3$ .

Fig. 5.3Shift aur stretch karne se do numbers ka order wahi rehta hai; negative number se multiply karne par order ulta ho jaata hai.

−8 se 8 tak ki number line par ek gol point a aur ek square point b hai. Pehle step mein a 2 par aur b 5 par hai, isliye 2 < 5. Doosre step mein dono 4 units left khisak kar −2 aur 1 par aate hain, a ab bhi left mein hai, aur label kehta hai order kept. Teesre step mein dono ko 3 se multiply kiya jaata hai aur woh −6 aur 3 par pahunchte hain, phir se a left mein aur order wahi. Chauthe step mein dono ko −1 se multiply kiya jaata hai: a 6 par aur b −3 par kood jaata hai, 0 par ek dashed mirror line dikhti hai, aur label kehta hai order reversed, kyunki ab a right mein hai.

Ab se har step us rule ka naam batayega jo woh use karta hai. Rule 1 hi tumhe kisi term ka sign badal kar use doosri side le jaane deta hai, bilkul waise hi jaise equations mein karte ho.

Example 5.1NCERT Example 1 · p. 92

Natural numbers ya integers

$12 x \leq 75$ ka solution set nikalo (i) natural numbers mein, (ii) integers mein.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 5.2NCERT Example 2 · p. 92

Integers ya real numbers

$7 x + 2 \leq 4 x + 14$ ka har solution nikalo (i) integers mein, (ii) real numbers mein.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Real numbers par solutions ek poora interval (अंतराल) bhar dete hain, jise Chapter 1 ke notation mein likhte hain. Round bracket end point ko bahar rakhta hai aur square bracket use andar rakhta hai; $\infty$ ke saath hamesha round bracket lagta hai kyunki woh number nahi hai. Jab tak question kuch aur na kahe, ab se $x$ ek real number hai.

Example 5.3NCERT Example 3 · p. 93

Negative coefficient

Solve karo: $3 x - 5 > 7 x + 3.$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 5.4NCERT Example 4 · p. 93

Fractions hatana

Solve karo: $\frac{2 - x}{3} \geq \frac{x + 4}{2} .$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

4 mein se 1

- 3 x \leq 12

solve karo.

x \leq - 4

x \leq 4

x \geq 4

x \geq - 4

Answer dekho

Answer (D) $x \geq - 4$

- 3

se divide karne par sign ulta hota hai, isliye

x \geq - 4

. Sign ko waisa hi rakhna (option A) sabse common galti hai.

Number line par solutions dikhana

Solution set ka sketch aksar khud inequality se zyada batata hai. Number line par boundary number ko ek circle se mark karo. Phir jis hisse mein solutions hain uspar ek moti line kheencho, aur arrowhead lagao taaki pata chale ki woh hamesha aage chalti rehti hai. Circle batata hai ki boundary set mein hai ya nahi (Fig. 5.4):

$<$ aur $>$ ke liye hollow (open) circle, kyunki boundary solution nahi hai;
$\leq$ aur $\geq$ ke liye filled (solid) circle, kyunki boundary solution hai;
$x < a$ aur $x \leq a$ ke liye left ki taraf shading, aur $x > a$ aur $x \geq a$ ke liye right ki taraf.

2/42 par filled circle, left shaded: $(- \infty, 2] .$

Fig. 5.4Hollow circle: boundary bahar rehti hai. Filled circle: woh shaamil hoti hai.NCERT Figs 5.1, 5.2 · p. 93

−5 se 5 tak ki number line. Pehle step mein 2 par ek hollow circle se moti ray left jaati hai, label x < 2, minus infinity se 2 tak round bracket wala interval, aur note ki 2 bahar hai. Doosre step mein wahi ray 2 par filled circle ke saath hai, label x ≤ 2 square bracket ke saath, aur note kehta hai ki 2 shaamil hai. Teesre step mein −3 par hollow circle se ray right jaati hai, label x > −3. Chauthe step mein woh circle filled hai, label x ≥ −3, −3 par square bracket ke saath.

Example 5.5NCERT Example 5 · p. 93

Solve karo, phir draw karo

$6 x + 5 > 2 x - 7$ solve karo aur solutions ko number line par dikhao.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 5.6NCERT Example 6 · p. 93

Pehle fractions, phir graph

$\frac{3 x + 1}{2} \leq \frac{x + 9}{4} + 2$ solve karo aur solutions ko number line par dikhao.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Neeche wala lab $a x + b < c x + d$ (ya $\leq$ , $>$ , $\geq$ ke saath) form ki koi bhi inequality ek-ek rule karke solve karta hai. Dekho ki jab tum jis coefficient se divide karte ho woh negative ho jaata hai tab sign ke saath kya hota hai, phir integers ya natural numbers par switch karo aur dekho ki shaded ray alag-alag dots mein toot jaati hai.

InteractiveInequality lab

Chaar sliders integers a, b, c aur d ko −9 se 9 tak set karte hain, buttons sign <, ≤, > ya ≥ chunte hain, aur teen buttons chunte hain ki x real number hai, integer hai ya natural number. Lab inequality likhta hai aur solving steps list karta hai, Rule 1 ya Rule 2 ka naam leta hai aur us step ko highlight karta hai jahan negative number se divide karne par sign ulta hota hai; jab x-terms cancel ho jaate hain to batata hai ki har value kaam karti hai ya koi nahi. −10 se 10 tak ki number line hollow ya filled circle ke saath solution ray shade karti hai (ya integer aur natural-number solutions ke liye dots lagati hai) aur solution set batati hai. Ek test slider kisi bhi x par marker rakhta hai aur dono sides aur statement true hai ya nahi, yeh batata hai.

Fig. 5.5Banao

a x + b ? c x + d,

steps follow karo, aur number line par

x

ki values test karo.

Word problems

At least, at most, no more than aur exceeds jaise phrases chhupe hue inequality signs hain. At least (kam se kam) matlab $\geq$ , at most (zyada se zyada) aur no more than (isse zyada nahi) matlab $\leq$ , aur exceeds (se zyada) matlab $>$ . Unknown ko naam do, condition likho aur solve karo. Phir check karo ki answer kahani mein sense banata hai: log aur bottles whole numbers mein aate hain, aur marks maximum se zyada nahi ho sakte.

Example 5.7NCERT Example 7 · p. 94

Average banaye rakhna

Ek school tournament mein Ishita ne apni pehli teen innings mein 42, 35 aur 61 runs banaye. Har innings ka average kam se kam 50 runs rakhne ke liye use chauthi innings mein kitne runs banane honge?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 5.8NCERT Example 8 · p. 94

5 ke consecutive multiples

5 ke do consecutive multiples mein se har ek 20 se bada hai, aur dono ka jod 75 se kam hai. Kaun se pairs possible hain?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

3 mein se 1

Ek number line

- 1

par filled circle aur right ki taraf shading dikhati hai. Kaunsi inequality draw ki gayi hai?

x > - 1

x \leq - 1

x \geq - 1

x < - 1

Answer dekho

Answer (C) $x \geq - 1$

Filled circle:

- 1

shaamil hai. Right shading: badi values. Dono milakar,

x \geq - 1

. Option A ke liye hollow circle chahiye hota.

Practice · 14 questions

Practice 5.1: answers ke saath

1
Short answerEasyNCERT Exercise 5.1, Q1
$14 x \leq 90$ ke saare solutions nikalo
1. natural numbers mein;
  Solution dikhao
  Answer ${1, 2, \dots, 6}$
  $x \leq \frac{90}{14} = \frac{45}{7},$ lagbhag 6.43. 6.43 tak ke natural numbers 1 se 6 hain.
2. integers mein.
  Solution dikhao
  Answer ${\dots, 4, 5, 6}$
  Bound wahi hai, $x \leq \frac{45}{7} .$ 6 tak ka har integer qualify karta hai, neeche koi end nahi.
2
Short answerEasyNCERT Exercise 5.1, Q2
$- 15 x \geq 48$ ke solutions list karo
1. $𝐍;$ mein
  Solution dikhao
  Answerkoi nahi: $\emptyset$
  $- 15$ se divide karo aur sign ulta karo: $x \leq - \frac{48}{15} = - 3.2.$ Koi bhi natural number negative nahi hota, isliye solution set khaali hai.
2. $𝐙 .$ mein
  Solution dikhao
  Answer ${\dots, - 5, - 4}$
  $- 3.2$ se zyada na hone wale integers hain $- 4, - 5, - 6, \dots$
3
Short answerEasyNCERT Exercise 5.1, Q3
$6 x - 5 < 13$ solve karo
1. integers par;
  Solution dikhao
  Answer ${\dots, 0, 1, 2}$
  $6 x < 18$ , isliye $x < 3$ . 3 se neeche ke integers hain $2, 1, 0, - 1, \dots;$ khud 3 bahar hai.
2. real numbers par.
  Solution dikhao
  Answer $(- \infty, 3)$
  3 se chhota har real number, yaani interval $(- \infty, 3) .$
4
Short answerEasyNCERT Exercise 5.1, Q4
Kaun si values $2 x + 11 \geq 4$ ko satisfy karti hain?
1. Integer values ke liye.
  Solution dikhao
  Answer ${- 3, - 2, - 1, \dots}$
  $2 x \geq - 7$ , isliye $x \geq - 3.5$ . Kam se kam $- 3.5$ wala sabse chhota integer $- 3$ hai.
2. Real values ke liye.
  Solution dikhao
  Answer $[- \frac{7}{2}, \infty)$
  $- 3.5$ se upar ka har real number, $- 3.5$ ko milakar.
5
Short answerEasyNCERT Exercise 5.1, Q5–Q8
Real $x$ ke liye solve karo.
1. $7 x + 2 < 9 x - 6$
  Solution dikhao
  Answer $(4, \infty)$
  $- 2 x < - 8$ . $- 2$ se divide karo aur sign ulta karo: $x > 4$ .
2. $5 x + 4 > 8 x - 11$
  Solution dikhao
  Answer $(- \infty, 5)$
  $- 3 x > - 15$ , isliye $x < 5$ .
3. $4 (x + 2) \geq 3 (x - 1)$
  Solution dikhao
  Answer $[- 11, \infty)$
  $4 x + 8 \geq 3 x - 3,$ isliye $x \geq - 11$ .
4. $5 (1 - x) < 2 (4 - x)$
  Solution dikhao
  Answer $(- 1, \infty)$
  $5 - 5 x < 8 - 2 x,$ isliye $- 3 x < 3$ aur $x > - 1$ .
6
Short answerMediumNCERT Exercise 5.1, Q9–Q12
Real $x$ ke liye solve karo.
1. $x - \frac{x}{3} + \frac{x}{4} < 22$
  Solution dikhao
  Answer $(- \infty, 24)$
  12 se multiply karo: $12 x - 4 x + 3 x < 264,$ isliye $11 x < 264$ aur $x < 24$ .
2. $\frac{x}{5} < \frac{x}{2} + 3$
  Solution dikhao
  Answer $(- 10, \infty)$
  10 se multiply karo: $2 x < 5 x + 30,$ isliye $- 3 x < 30$ aur $x > - 10$ .
3. $\frac{2 (x + 1)}{3} \geq \frac{3 (x - 4)}{4}$
  Solution dikhao
  Answer $(- \infty, 44]$
  12 se multiply karo: $8 (x + 1) \geq 9 (x - 4),$ isliye $8 x + 8 \geq 9 x - 36.$ Phir $- x \geq - 44$ aur $x \leq 44$ .
4. $\frac{1}{3} (\frac{x}{2} + 5) < \frac{1}{4} (x - 2)$
  Solution dikhao
  Answer $(26, \infty)$
  12 se multiply karo: $4 (\frac{x}{2} + 5) < 3 (x - 2),$ isliye $2 x + 20 < 3 x - 6$ aur $x > 26$ .
7
Short answerMediumNCERT Exercise 5.1, Q13–Q16
Real $x$ ke liye solve karo.
1. $3 (x - 2) + 10 \leq 5 (x + 1)$
  Solution dikhao
  Answer $[- \frac{1}{2}, \infty)$
  $3 x + 4 \leq 5 x + 5,$ isliye $- 2 x \leq 1$ aur $x \geq - \frac{1}{2} .$
2. $25 - (2 x - 3) < 7 x - 4 (x + 2)$
  Solution dikhao
  Answer $(\frac{36}{5}, \infty)$
  Left side $28 - 2 x$ hai aur right side $3 x - 8$ hai. Isliye $- 5 x < - 36$ aur $x > \frac{36}{5} .$
3. $\frac{x}{3} > \frac{3 x + 1}{4} - \frac{x - 2}{6}$
  Solution dikhao
  Answer $(- \infty, - \frac{7}{3})$
  12 se multiply karo: $4 x > 3 (3 x + 1) - 2 (x - 2),$ aur right side $7 x + 7$ hai. Isliye $- 3 x > 7$ aur $x < - \frac{7}{3} .$
4. $\frac{3 x}{5} \leq \frac{x + 1}{2} - \frac{1 - x}{4}$
  Solution dikhao
  Answer $[- \frac{5}{3}, \infty)$
  20 se multiply karo: $12 x \leq 10 (x + 1) - 5 (1 - x),$ yaani $12 x \leq 15 x + 5.$ Isliye $- 3 x \leq 5$ aur $x \geq - \frac{5}{3} .$
8
DiagramMediumNCERT Exercise 5.1, Q17–Q20
Har inequality solve karo aur number line par uske graph ko describe karo.
1. $4 x + 1 < 3 x + 5$
  Solution dikhao
  Answer $x < 4$
  $x < 4$ : 4 par hollow circle, left ki taraf shaded.
2. $6 x - 2 > 4 x - 8$
  Solution dikhao
  Answer $x > - 3$
  $2 x > - 6$ , isliye $x > - 3$ : $- 3$ par hollow circle, right ki taraf shaded.
3. $4 (2 - x) \leq 3 (x + 5)$
  Solution dikhao
  Answer $x \geq - 1$
  $8 - 4 x \leq 3 x + 15,$ isliye $- 7 x \leq 7$ aur $x \geq - 1$ : $- 1$ par filled circle, right ki taraf shaded.
4. $\frac{x}{3} \geq \frac{2 x + 1}{2} - \frac{x + 3}{4}$
  Solution dikhao
  Answer $x \leq \frac{3}{5}$
  12 se multiply karo: $4 x \geq 6 (2 x + 1) - 3 (x + 3),$ aur right side $9 x - 3$ hai. Isliye $- 5 x \geq - 3$ aur $x \leq \frac{3}{5}$ : $\frac{3}{5}$ par filled circle, left ki taraf shaded.
NumericalEasyNCERT Exercise 5.1, Q21
Kavya ke monthly tests har ek 50 marks ke hote hain. Pehle do mein usne 34 aur 41 score kiye. Kam se kam 38 ka average paane ke liye teesre test mein use kam se kam kitne marks chahiye?
Solution dikhao
Answer39 marks
$\frac{34 + 41 + x}{3} \geq 38$ se milta hai $75 + x \geq 114,$ isliye $x \geq 39$ . Test 50 marks ka hai, isliye 39 se 50 tak koi bhi score chalega.
NumericalEasyNCERT Exercise 5.1, Q22
Ek cycling club chhe rides mein har ride ka average kam se kam 25 km hone par badge deta hai. Arjun ki pehli paanch rides 22, 27, 24, 21 aur 26 km thi. Badge paane ke liye uski chhathi ride kitni lambi honi chahiye?
km
Solution dikhao
AnswerKam se kam 30 km
Paanch rides ka total 120 km hai. Phir $\frac{120 + x}{6} \geq 25$ se milta hai $120 + x \geq 150,$ isliye $x \geq 30$ .
Short answerMediumNCERT Exercise 5.1, Q23
Do consecutive odd natural numbers dono 18 se chhote hain aur unka jod 20 se zyada hai. Aise har pair ki list banao.
Solution dikhao
Answer(11, 13), (13, 15), (15, 17)
Dono 18 se chhote hain, matlab $x + 2 < 18$ , isliye $x < 16$ . Jod wali condition $2 x + 2 > 20$ se $x > 9$ milta hai. $9 < x < 16$ wale odd numbers 11, 13 aur 15 hain.
Short answerMediumNCERT Exercise 5.1, Q24
4 ke consecutive multiples ke aise pairs chahiye jinmein dono numbers 10 se zyada hon aur total 50 se neeche rahe. Sab dhoondho.
Solution dikhao
Answer(12, 16), (16, 20), (20, 24)
$x$ aur $x + 4$ ke saath: $x > 10$ aur $2 x + 4 < 50$ , isliye $10 < x < 23$ . Is range mein 4 ke multiples 12, 16 aur 20 hain.
NumericalMediumNCERT Exercise 5.1, Q25
Ek triangle mein ek side shortest side se 6 cm lambi hai, aur longest side shortest ke dugne se 1 cm zyada hai. Perimeter kam se kam 55 cm hai. Shortest side ki sabse kam possible length nikalo.
cm
Solution dikhao
Answer12 cm
$x + (x + 6) + (2 x + 1) \geq 55$ se milta hai $4 x + 7 \geq 55,$ isliye $x \geq 12$ . Triangle sach mein ban sakta hai: do chhoti sides ka jod $2 x + 6$ hai, jo hamesha $2 x + 1$ se bada hai. $x = 12$ ke liye sides 12, 18 aur 25 cm hain.
Short answerHardNCERT Exercise 5.1, Q26
Ek tailor 150 cm ke ribbon se teen tukde kaatta hai (kuch ribbon bach bhi sakta hai). Doosra tukda sabse chhote se 10 cm lamba hai, aur teesra sabse chhote ka teen guna hai. Teesra tukda doosre se kam se kam 20 cm lamba hona chahiye. Sabse chhote tukde ki kaun si lengths possible hain?
Solution dikhao
Answer15 cm se 28 cm tak
Maan lo sabse chhota tukda $x$ cm hai. Fit hona: $x + (x + 10) + 3 x \leq 150,$ isliye $5 x \leq 140$ aur $x \leq 28$ . Fark: $3 x \geq (x + 10) + 20,$ isliye $2 x \geq 30$ aur $x \geq 15$ . Isliye $15 \leq x \leq 28.$

Miscellaneous Examples

Aakhri examples rules ko milakar use karte hain. $p < a x + b \leq q$ jaisi double inequality ek hi line mein do inequalities hai, aur tum teeno parts par ek saath koi rule laga sakte ho, bas shart yeh hai ki use har part par lagao. Kai inequalities jo ek saath sach honi chahiye, ek system of inequalities banati hain; iska solution set alag-alag solution sets ka common hissa (intersection) hota hai.

Example 5.9NCERT Example 9 · p. 96

Teen parts par kaam

Solve karo: $- 7 < 4 x + 5 \leq 13.$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 5.10NCERT Example 10 · p. 96

Palatne wali double inequality

Solve karo: $- 4 \leq \frac{1 - 3 x}{2} < 5.$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Fig. 5.6 mein end points ko follow karo. $- 9$ wala closed end 3 wala closed end ban jaata hai, aur 9 wala open end $- 3$ wala open end ban jaata hai. Negative number se divide karne par dono ends jagah badal lete hain, aur har end apna type rakhta hai.

4/4 $x$ , $(- 3, 3]$ mein hai: closed end 3 par aur open end $- 3$ par chala jaata hai.

Fig. 5.6Negative number se divide karne par interval ke ends jagah badal lete hain; har end apna type rakhta hai.

−10 se 10 tak ki number line. Pehle step mein −4 (filled end) se 5 (hollow end) tak ka segment beech wale part (1 − 3x)/2 ki values dikhata hai. Doosre step mein, 2 se multiply karne ke baad, segment −8 (filled) se 10 (hollow) tak hai. Teesre step mein, 1 subtract karne ke baad, yeh −9 (filled) se 9 (hollow) tak hai. Chauthe step mein, −3 se divide karne ke baad, x ka segment −3 (hollow) se 3 (filled) tak hai; purana segment dashed dikhaya gaya hai aur do cross karte arrows −9 ko 3 par aur 9 ko −3 par le jaate hain, jo dikhata hai ki ends side badalte hain aur apna type rakhte hain.

Example 5.11NCERT Example 11 · p. 96

Do inequalities ka system

System solve karo: $4 x - 3 \geq x - 6$ aur $9 - 2 x > 1,$ aur solutions ko number line par dikhao.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

3/3Dono $[- 1, 4)$ par sach hain.

Fig. 5.7System ka solution alag-alag solution sets ka overlap hota hai.NCERT Fig. 5.3 · p. 97

−3 se 6 tak ki teen number lines ek ke upar ek rakhi hain. Pehle step mein upar wali line x ≥ −1 ko −1 par filled circle aur right jaati ray se dikhati hai. Doosre step mein beech wali line x < 4 ko 4 par hollow circle aur left jaati dashed ray se jodti hai. Teesre step mein −1 aur 4 se dashed guides neeche wali line tak aate hain, jo sirf overlap dikhati hai: −1 par filled circle se 4 par hollow circle tak ka segment, label −1 ≤ x < 4.

Example 5.12NCERT Example 12 · p. 97

Vaccine fridge

Kai vaccines ko 2 °C se 8 °C ke beech store karna padta hai. Degrees Fahrenheit mein yeh range kya hai? Use karo: $F = \frac{9}{5} C + 32.$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

3/3Wahi range hai $35.6 \leq F \leq 46.4.$

Fig. 5.8Wahi safe range, do scales par naapi gayi.

Do parallel scales: upar 0 se 10 tak degrees Celsius aur neeche 32 se 50 tak degrees Fahrenheit, aise line up kiye gaye ki same temperature same horizontal position par ho. Pehle step mein Celsius scale par ek band 2 se 8 tak mark karta hai. Doosre step mein dashed arrows 2 aur 8 se seedhe neeche Fahrenheit scale tak aate hain, aur scales ke beech formula F = 9C/5 + 32 likha hai. Teesre step mein Fahrenheit scale par matching band shaded hai aur uspar 35.6 aur 46.4 likha hai.

Example 5.13NCERT Example 13 · p. 97

Do fertiliser solutions milana

Ek kisan ke tank mein 10 % nutrient wala 500 litre liquid fertiliser hai. Woh ise 40 % nutrient wale ek strong fertiliser se bharegi, aur chahti hai ki mix ka nutrient content strictly 16 % aur 20 % ke beech ho. Strong fertiliser ki kaun si maatra chalegi?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Fig. 5.9 dikhata hai ki answer ek window kyun hai. Jaise-jaise zyada strong solution daalte hain, mix ki strength 10 % se 40 % ki taraf badhti hai, 125 litre par 16 % aur 250 litre par 20 % paar karti hai.

3/3Curve band ke andar hai jab $125 < x < 250.$

Fig. 5.9Mix tabhi 16% aur 20% ke beech hota hai jab 125 aur 250 litre ke beech milaya jaaye.

Ek graph jismein across 0 se 400 tak 40% solution ke milaye gaye litres hain aur upar 10% se 24% tak mix ki strength. Pehle step mein curve kuch na milane par 10% se shuru hota hai aur dheere-dheere aur dheere badhta hai, 400 litre par lagbhag 23% tak pahunchta hai. Doosre step mein 16% aur 20% lines ke beech ek shaded band dikhta hai. Teesre step mein jahan curve band mein ghusta aur nikalta hai wahan se dashed lines horizontal axis tak aati hain, aur 125 aur 250 litre ke beech ka hissa hollow end circles se marked hai, kyunki dono conditions strict hain.

Quick check

3 mein se 1

Solve karo:

1 \leq 3 - 2 x < 7.

(- 2, 1]

[- 2, 1)

(- 1, 2]

[1, 3)

Answer dekho

Answer (A) $(- 2, 1]$

- 2 \leq - 2 x < 4;

- 2

se divide karne par milta hai

1 \geq x > - 2,

isliye

(- 2, 1]

. Option B bhool jaata hai ki signs ulte hone par ends jagah badalte hain.

Recap

Summary aur key points

Inequality do numbers ya expressions ko $<$ , $>$ , $\leq$ ya $\geq$ se compare karti hai; at least (kam se kam) matlab $\geq$ aur at most (zyada se zyada) matlab $\leq$ (Section 5.1).
Inequalities numerical ya literal, single ya double, strict ( $<$ , $>$ ) ya slack ( $\leq$ , $\geq$ ), aur linear ya non-linear hoti hain (Section 5.2).
Solution inequality ko true banata hai; solution set un sabko ikattha karta hai aur is par depend karta hai ki $x$ , $𝐍$ , $𝐙$ ya $𝐑$ mein se kis par chalta hai (Section 5.3).
Rule 1: dono sides mein same number add ya subtract karne se sign wahi rehta hai (Law).
Rule 2: positive number se multiply ya divide karne par sign wahi rehta hai; negative number se karne par sign ulta ho jaata hai (Law).
Kabhi aise expression se multiply ya divide mat karo jiska sign pata nahi, ya zero se (Socho zara).
Number line par $<$ aur $>$ ke liye hollow circle, $\leq$ aur $\geq$ ke liye filled circle use karo, aur jis side solutions hain use shade karo (Number line par solutions dikhana).
$𝐑$ par answers intervals hote hain, jaise $(- \infty, a),$ $(- \infty, a],$ $(a, \infty)$ aur $[a, \infty)$ (Example 5.2).
Double inequality teeno parts par kaam karke solve hoti hai; negative divisor dono signs ulte kar deta hai aur ends ki jagah badal deta hai (Example 5.10).
System ka solution set alag-alag solution sets ka overlap hota hai, aur woh khaali bhi ho sakta hai (Example 5.11).
Word problems mein check karo ki answer kahani mein fit baithta hai: whole numbers, maximum marks, physical limits (Word problems).

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise 5.1, Q1	Practice 5.1 · Q1
NCERT Exercise 5.1, Q2	Practice 5.1 · Q2
NCERT Exercise 5.1, Q3	Practice 5.1 · Q3
NCERT Exercise 5.1, Q4	Practice 5.1 · Q4
NCERT Exercise 5.1, Q5–Q8	Practice 5.1 · Q5
NCERT Exercise 5.1, Q9–Q12	Practice 5.1 · Q6
NCERT Exercise 5.1, Q13–Q16	Practice 5.1 · Q7
NCERT Exercise 5.1, Q17–Q20	Practice 5.1 · Q8
NCERT Exercise 5.1, Q21	Practice 5.1 · Q9
NCERT Exercise 5.1, Q22	Practice 5.1 · Q10
NCERT Exercise 5.1, Q23	Practice 5.1 · Q11
NCERT Exercise 5.1, Q24	Practice 5.1 · Q12
NCERT Exercise 5.1, Q25	Practice 5.1 · Q13
NCERT Exercise 5.1, Q26	Practice 5.1 · Q14
NCERT Miscellaneous Exercise, Q1–Q3	Miscellaneous exercises · Q1
NCERT Miscellaneous Exercise, Q4–Q6	Miscellaneous exercises · Q2
NCERT Miscellaneous Exercise, Q7–Q10	Miscellaneous exercises · Q3
NCERT Miscellaneous Exercise, Q11	Miscellaneous exercises · Q4
NCERT Miscellaneous Exercise, Q12	Miscellaneous exercises · Q5
NCERT Miscellaneous Exercise, Q13	Miscellaneous exercises · Q6
NCERT Miscellaneous Exercise, Q14	Miscellaneous exercises · Q7

Quick check: MCQs

5.2 Inequalities4 questions
Do rules4 questions
Word problems3 questions
Miscellaneous Examples3 questions

Glossary · 11

Key terms (English + Hindi)

double inequality: Aisa statement jo kisi quantity ko do bounds ke beech rakhta hai, jaise $40 \leq m < 60$ .Chapter mein pehli baar yahan aaya →

inequalityअसमिकाasamika: Aisa statement jo do numbers ya expressions ko $<$ , $>$ , $\leq$ ya $\geq$ se compare karta hai.Chapter mein pehli baar yahan aaya →
intervalअंतरालantaral: Real line ka bina toote ek hissa, jaise $(- 3, 2]$ ya $[4, \infty)$ .Chapter mein pehli baar yahan aaya →

linear inequalityरैखिक असमिकाrekhik asamika: Aisi inequality jiski dono sides variables mein zyada se zyada degree 1 ke polynomials hon, jaise $3 x - 7 < 0$ ya $2 x + 3 y \leq 6$ .Chapter mein pehli baar yahan aaya →
literal inequality: Variable wali inequality, jaise $v \leq 60$ ; yeh sach hai ya nahi, variable ki value par depend karta hai.Chapter mein pehli baar yahan aaya →

numerical inequality: Sirf numbers ke beech ki inequality, jaise $- 5 < - 2$ ; yeh seedha true ya false hoti hai.Chapter mein pehli baar yahan aaya →

slack inequality: $\leq$ ya $\geq$ wali inequality, isliye boundary value allowed hai.Chapter mein pehli baar yahan aaya →
solutionहलhal: Variable ki woh value jo inequality ko true statement bana de.Chapter mein pehli baar yahan aaya →
solution setहल समुच्चयhal samuchchay: Variable ki allowed values mein se inequality ke saare solutions ka set.
strict inequality: $<$ ya $>$ wali inequality, isliye boundary value allowed nahi hai.Chapter mein pehli baar yahan aaya →
system of inequalities: Do ya zyada inequalities jo ek hi samay par sach honi chahiye; iske solutions woh values hain jo un sabmein common hain.Chapter mein pehli baar yahan aaya →

Linear Inequalities: Class 11 Maths Chapter 5 Hinglish mein

Is chapter mein aap kya seekhenge

5.1 Introduction

5.2 Inequalities

5.3 Ek Variable wali Linear Inequalities ke Algebraic Solutions aur unka Graphical Representation

Do rules

Number line par solutions dikhana

Word problems

Practice 5.1: answers ke saath

Miscellaneous Examples

Summary aur key points

Practice questions answers ke saath (NCERT “Miscellaneous exercises”)

Apna NCERT exercise question dhoondo

Quick check: MCQs

Key terms (English + Hindi)

Tum chapter ke end tak pahunch gaye