Linear Inequalities ⚖️

Equations mein = tha, ab <, >, ≤, ≥ aate hain — ek value nahi, ek range milti hai answer mein!

🛒 Shopping Budget

Tum ek mela mein gaye ho aur tumhare paas ₹500 hain. Ek ride ki ticket ₹30 ki hai. Toh tum kitni rides le sakte ho? Agar x = rides ki sankhya ho, toh: 30x ≤ 500, yaani x ≤ 16.66, yaani x ≤ 16 (kyunki x integer hona chahiye). Yeh ek linear inequality hai! Equations se yeh is tarah alag hai ki ek fixed answer ki jagah answers ka ek pura range milta hai.

1. Inequality ke Symbols

Inequalities mein 4 symbols hote hain, inhe clearly samajh lo:

Open interval (a,b): a aur b included nahi (strict inequality). Closed interval [a,b]: a aur b bhi included (≤ ya ≥).

2. Rules for Solving Inequalities

Equations ki tarah hi solve karte hain, lekin ek important difference hai — negative number se multiply/divide karte waqt inequality ka sign palat jaata hai!

✅ Addition/Subtraction

Dono sides par same number add/subtract karo — inequality sign nahi badlta.

x+3 > 7 ⟹ x > 4

✅ Multiply/Divide (Positive)

Positive number se multiply/divide karo — sign nahi badlta.

2x < 8 ⟹ x < 4

⚠️ Multiply/Divide (NEGATIVE)

Negative number se multiply/divide karte waqt inequality sign PALAT JATA HAI!

−2x < 8 ⟹ x > −4 ⚠️

🔗 Adding Inequalities

Same direction ki do inequalities add kar sakte ho.

a>b aur c>d ⟹ a+c > b+d

3. Solving Linear Inequalities — Step by Step

✅ Solve: 3x − 2 > 4x + 1

Step 1: x terms ek side le jao → 3x − 4x > 1 + 2
Step 2: −x > 3
Step 3: −1 se divide karo → sign palte ga! → x < −3
Solution: x ∈ (−∞, −3) — yaani x ki value −3 se strictly choti

Number Line: x < −3 ka solution

−5 −4 −3 −2 −1 x < −3 (open circle = −3 included nahi)

✅ Solve: −3 ≤ 4x + 1 ≤ 13 (Double Inequality)

Sabse pehle sab parts mein se 1 ghataao: −3−1 ≤ 4x ≤ 13−1
−4 ≤ 4x ≤ 12
4 se divide karo (positive, sign nahi badlega): −1 ≤ x ≤ 3
Solution: x ∈ [−1, 3]

Number Line: −1 ≤ x ≤ 3

−2 −1 0 1 3 filled circle = endpoint included (≤)

4. Inequalities aur Interval Notation

Solution ko interval notation mein likhna sikhna zaroori hai — exams mein frequently aata hai:

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

( ) = open = endpoint NOT included. [ ] = closed = endpoint included.

5. Word Problems — Real Life Inequalities

✅ Riya ke final exams mein 5 subjects hain. Pehle 4 mein 72, 68, 85, 90 marks aaye. 5th mein kitne marks chahiye ki average 80 se zyada ho?

Let 5th subject mein marks = x
Average > 80 → (72+68+85+90+x)/5 > 80
(315+x)/5 > 80
315+x > 400
x > 85
Riya ko 5th subject mein 85 se zyada marks chahiye

6. Practice Questions

Q1. 5x − 3 ≥ 3x + 7 solve karo aur interval notation mein likho.

Solution

2x ≥ 10 → x ≥ 5 → x ∈ [5, ∞)

Q2. −8 ≤ 3x + 1 < 16 solve karo.

Solution

−9 ≤ 3x < 15 → −3 ≤ x < 5 → x ∈ [−3, 5)

Q3. (2x−1)/3 > (3x+2)/5 solve karo.

Solution

5(2x−1) > 3(3x+2) → 10x−5 > 9x+6 → x > 11 → x ∈ (11,∞)

Q4. |x−3| < 5 solve karo.

Solution

−5 < x−3 < 5 → −2 < x < 8 → x ∈ (−2, 8)

Q5. Ek number x aisa ho ki uska teen guna uske char gune se 5 zyada ho. Inequality banao aur solve karo.

Solution

3x > 4x + 5 nahi... sahi padhna: "teen guna, char gune se 5 zyada" → 3x > 4x+5 → −x > 5 → x < −5

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