Kya Class 11 mein Linear Inequalities samajh nahi aati? Chinta mat karo! Aaj main tumhe bilkul pehli baar se samjhaungi — itne simple tarike se ki sab crystal clear ho jaye.
Sabse Pehle — Inequality Kya Hoti Hai
Class 10 mein tumne equations padhi thi — jaise:
•2x + 3 = 7 — yahan equal hai
Par real life mein hamesha equal nahi hota.
Jaise:
•Tumhari age 18 se zyada honi chahiye driving ke liye
•Exam mein 33 se zyada marks chahiye pass hone ke liye
•Temperature 0° se kam hona chahiye barf ke liye
Yeh sab Inequalities hain.
Simple matlab — do cheeze jab equal nahi hoti — inequality hoti hai.
Inequality Ke Signs — Pehle Samjho
Char tarah ke signs hote hain:
1. < (Less than — Chhota)
a < b — a, b se chhota hai
Jaise — 3 < 5 — 3, 5 se chhota hai ✅
2. > (Greater than — Bada)
a > b — a, b se bada hai
Jaise — 7 > 4 — 7, 4 se bada hai ✅
3. ≤ (Less than or equal to)
a ≤ b — a, b se chhota ya equal hai
Jaise — x ≤ 5 — x 5 ya 5 se chhota ho sakta hai
4. ≥ (Greater than or equal to)
a ≥ b — a, b se bada ya equal hai
Jaise — marks ≥ 33 — pass hone ke liye.
Linear Inequality Kya Hoti Hai
Jisme variable ki power 1 ho — woh Linear Inequality hai.
Jaise:
•2x + 3 > 7 — Linear Inequality ✅
•x – 5 ≤ 10 — Linear Inequality ✅
•3x + 2y < 12 — Linear Inequality — two variables ✅
•x² > 4 — Linear nahi — x ki power 2 hai ❌
One Variable Mein Linear Inequality — Solve Karna
Bilkul equation ki tarah solve karte hain — ek fark ke saath.
Zaroori Rule:
Agar negative number se multiply ya divide karo — sign palat jaata hai.
< becomes > aur > becomes <
Example 1 — Simple:
2x + 3 > 7 solve karo:
Step 1:
2x > 7 – 3
2x > 4
Step 2:
x > 4/2
x > 2 ✅
Matlab — x ki value 2 se badi hogi — jaise 3, 4, 5, 100…
Example 2 — Negative Number Wala:
-3x > 9 solve karo:
Step 1:
x > 9/(-3)
Par dhyan do — negative se divide kiya — sign palat gaya.
x < -3 ✅
Number Line Pe Dikhana
Inequality ka solution number line pe dikhate hain.
x > 2 ke liye:
2 pe ek hollow circle lagao — 2 include nahi
Daayein taraf arrow — 2 se bada sab include
x ≥ 2 ke liye:
2 pe ek filled circle lagao — 2 include hai
Daayein taraf arrow
Yaad karo:
< ya > — hollow circle — woh number include nahi
≤ ya ≥ — filled circle — woh number include hai
Interval Notation — Naya Concept
Solution ko interval mein likhte hain:
Yaad karo:
( — bracket — number include nahi
[ — square bracket — number include hai
∞ — infinity — hamesha ( ) brackets mein
Double Inequality — Beech Wali Value
Jab variable do numbers ke beech mein ho:
Example:
-3 < 2x + 1 < 7 solve karo:
Step 1 — Teeno hisson pe same kaam karo:
-3 – 1 < 2x < 7 – 1
-4 < 2x < 6
Step 2:
-4/2 < x < 6/2
-2 < x < 3 ✅
Matlab x — -2 aur 3 ke beech hoga.
Two Variables Mein Linear Inequality
Jab do variables hote hain — graph pe solve karte hain.
Example:
2x + 3y ≤ 6 solve karo:
Step 1 — Pehle line kheecho:
2x + 3y = 6
x=0 rakho — y=2 — point (0,2)
y=0 rakho — x=3 — point (3,0)
Dono points milo — line ban gayi.
Step 2 — Kaunsa hissa:
Origin (0,0) daalo — 2(0) + 3(0) = 0 ≤ 6 ✅
Origin wala hissa — solution hai.
Woh area shade karo.
Step 3:
≤ hai — to line bhi include hai — solid line kheecho
< hota — to line include nahi — dotted line kheecho
System of Inequalities
Jab ek saath do ya zyada inequalities hon:
Example:
x > 2 aur x < 5 — dono ek saath.
Solution:
x > 2 — 2 se bada
x < 5 — 5 se chhota
Dono milake — 2 < x < 5 ✅
Matlab x — 2 aur 5 ke beech.
Real Life Mein Linear Inequalities
Linear Inequalities roz ki zindagi mein bahut hain:
Example 1 — Shopping:
Budget hai ₹500
Pen ki keemat ₹10 — book ki keemat ₹50
10x + 50y ≤ 500 — yeh inequality hai.
Example 2 — Health:
Doctor ne kaha — roz 2000 se kam calories khao
c < 2000 — yeh inequality hai.
Example 3 — Driving:
Speed limit 60 km/h
s ≤ 60 — yeh inequality hai.
Example 4 — Exam:
Pass hone ke liye 33% chahiye
marks ≥ 33 — yeh inequality hai.
Common Galtiyan — Jo Nahi Karni
Galti 1 — Sign bhool jaana:
Negative se divide karne pe sign palat jaata hai.
-2x > 6 — to x < -3 — sign palat gaya.
Galti 2 — Hollow aur Filled circle mein confusion:
< ya > — hollow circle
≤ ya ≥ — filled circle
Galti 3 — Interval notation galat likhna:
∞ ke saath hamesha ( ) — kabhi [ ] nahi.
x > 2 — (2, ∞) ✅ — [2, ∞) ❌
Galti 4 — Double inequality mein galti:
Teeno hisson pe same operation karo.
Ek hisse pe karo to galti hogi.
Yaad Rakhne Ka Aasan Tarika
Linear Inequalities ko speed limit ki tarah socho. 🚗
•Speed limit 60 — s ≤ 60 — inequality.
•60 se kam — safe — solution.
•60 se zyada — challan — solution nahi.
•Negative se divide — sign palat — jaise U-turn.
Jab bhi road pe speed limit dekho — Linear Inequality yaad aa jayegi.
Toh dosto, yeh tha Class 11 Linear Inequalities ka poora damdaar guide. Ek ek concept samjho — roz practice karo — board exam mein zaroor kaam aayega. Agar yeh article helpful laga to comment mein batao aur doston ke saath share karo.