2.1.12
Ek akela inequality jaise x > 3 ek condition describe karta hai number line pe.
Ek compound inequality do conditions ko ek logical word ke saath jodta hai:
AND → number ko dono conditions satisfy karni chahiye ek saath → hum overlap (intersection) lete hain.OR → number ko kam se kam ek condition satisfy karni chahiye → hum union (sab kuch jo cover ho) lete hain.
YE KYUN MATTER KARTA HAI: Almost har real constraint compound hoti hai. "Temperature 20°C aur 25°C ke beech" ek AND hai. "Discount milega agar aap 12 se kum OR 65 se zyada ho" ek OR hai. English word ko ek set operation mein translate karna seekhna hi poora game hai.
LOGIC KAISE PADHEIN:
Word
Logic
Set operation
Number line pe region
AND
dono true
intersection ∩
dono ka overlap
OR
koi bhi ek true
union ∪
dono pieces milke
Maano A = { x : condition 1 true } B = { x : condition 2 true }
"condition 1 AND condition 2" tab hi true hoti hai jab x A aur B definition se, woh set A ∩ B Yeh step kyun? Intersection literally define hota hai "dono mein hai."
"condition 1 OR condition 2" tab true hoti hai jab x A ya B A ∪ B Yeh step kyun? Union literally define hota hai "kam se kam ek mein hai."
Toh set operation koi rule seekhne ke liye nahi hai — yeh AND/OR ka meaning hi hai .
a < f ( x ) < b teeno parts pe EKO SAATH same operation apply karo.
a ≤ 2 x − 1 ≤ 7 1 a + 1 ≤ 2 x ≤ 8 ⇒ 1 ≤ 2 x ≤ 8 a = 0 2 2 1 ≤ x ≤ 4 Yeh step kyun? Jo bhi operation left inequality ko true rakhti hai, woh middle aur right pe bhi karni padti hai, warna chain toot jaati hai.
Common mistake Chain ke sirf ek hisse ka flip karnaGalat idea (sahi lagta hai): − 4 < − 2 x < 6 − 2 ek sign flip karte ho, milta hai 2 > x < − 3 Kyun sahi lagta hai: Yaad hai "negative se divide karne pe sign flip hota hai" toh sirf visible arrow pe apply kar dete ho.
Fix: Negative se divide karne pe dono relation signs ek saath flip hote hain. − 4 < − 2 x < 6 ⇒ 2 > x > − 3 − 3 < x < 2
Worked example Example 1 — AND (intersection)Solve karo x > 1 and x ≤ 5
A = ( 1 , ∞ ) B = ( − ∞ , 5 ] Kyun? Har inequality apni khud ki ray hai.Overlap: numbers jo 1 se bade hain aur zyada se zyada 5 → ( 1 , 5 ]
Compact: 1 < x ≤ 5
Worked example Example 2 — OR (union)Solve karo x < − 2 or x ≥ 3
A = ( − ∞ , − 2 ) B = [ 3 , ∞ ) nahi hota.Union: ( − ∞ , − 2 ) ∪ [ 3 , ∞ ) Kyun? OR ka matlab "kisi mein bhi" hai, toh dono pieces rakho. Koi compact form nahi.
Worked example Example 3 — empty ANDSolve karo x > 4 and x < 1
Koi number > 4 < 1 koi solution nahi ∅
Yeh step kyun? Rays ek doosre se door point kar rahi hain; intersection empty hai.
Worked example Example 4 — OR jo sab cover kar leSolve karo x < 5 or x > 2
Har real number ya toh 5 se kam hai ya 2 se zyada (aur bahut saare dono hain). Union = R
Kyun? Union ke liye sirf ek condition chahiye; dono rays overlap karti hain aur milke poori line fill kar deti hain.
Worked example Example 5 — teen-part inequality solve karnaSolve karo − 1 ≤ 2 3 − x < 4
Sabhi parts ko 2 se multiply karo: − 2 ≤ 3 − x < 8 Kyun? Teeno pe same operation.
3 ghatao: − 5 ≤ − x < 5
− 1 5 ≥ x > − 5 − 5 < x ≤ 5 ( − 5 , 5 ]
Common mistake AND-empty aur OR-everything ko confuse karnaGalat idea: x > 4 x < 1 x > 4 x < 1 Kyun sahi lagta hai: Dono mein do rays opposite direction mein point karti hain, toh "same dikhta hai."
Fix: AND overlap maangta hai → empty . OR union maangta hai → do pieces ( − ∞ , 1 ) ∪ ( 4 , ∞ )
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho ek dark stage pe do spotlights hain. AND = "wahan khado jahan DONO lights tumpe padein" — sirf woh chota sa patch safe hai jahan dono beams cross karti hain. OR = "wahan khado jahan KAM SE KAM EK light tumpe pade" — bahut bada area, dono patches count karte hain. Number line pe numbers stage ke spots ki tarah hain: AND overlap rakhta hai, OR sab kuch lit rakhta hai.
"AND ek Narrow Diamond (∩) hai, OR ek wide Ocean (∪) hai."
AND → intersection ∩ → narrow overlap. OR → union ∪ → ocean zyada cover karta hai.
AND ke saath kaunsa set operation match karta hai? OR ke saath?
x > 2 x ≤ 6 OR compact a < x < b
Solve karo x < 0 x > 3
AND do inequalities ko kis set operation se jodhta hai? Intersection (∩
OR do inequalities ko kis set operation se jodhta hai? Union (∪
Compact form a < x < b AND (dono a < x x < b
Kya OR ko a < x < b Nahi; OR do alag pieces deta hai, isliye do alag inequalities likhni padti hain.
a < f ( x ) < b Use TEENO parts pe simultaneously apply karo.
Teen-part inequality ko negative number se divide karne par kya hota hai? DONO relation signs ek saath flip ho jaate hain.
Solve karo x > 4 x < 1 Koi solution nahi (∅
Solve karo x < 5 x > 2 Saare reals R
Solve karo x < − 2 x ≥ 3 ( − ∞ , − 2 ) ∪ [ 3 , ∞ )
AND=intersection koi memorise karne wala rule kyun nahi hai? Kyunki intersection define hi hota hai "dono sets mein," jo exactly AND ka matlab hai.
Linear inequalities in one variable — ek compound inequality ka har aadha hissa inhi mein se ek hota hai.Set operations — union and intersection — yahan ka logical backbone.Interval notation — hum ( a , b ) [ a , b ] Absolute value inequalities — ∣ x ∣ < a ∣ x ∣ > a Number line representation — overlap vs union ka visual tool.
Two separate inequalities
