Inequalities — linear, solving, number line representation

2.1.11 · Maths › Algebra — Introduction & Intermediate

Overview

Inequalities un relationships ko describe karte hain jahan ek quantity doosri se greater than ya less than hoti hai, equal nahi. Ye optimization, constraint problems, aur solutions ki ranges samajhne ki foundation hain.

Core Concepts

Linear Inequality Kya Hai?

Linear kyun? Kyunki variable $x$ sirf first power par aata hai — koi $x^{2}$ nahi, koi $x$ nahi, koi $\frac{1}{x}$ nahi. Agar hum ise equation mein convert karte, toh graph ek straight line hoti.

Chaar symbols:

$<$ : less than (strict, boundary include NAHI karta)
$\leq$ : less than or equal to (boundary include karta hai)
$>$ : greater than (strict)
$\geq$ : greater than or equal to (boundary include karta hai)

Linear Inequalities Solve Karna: Rules

Add/Subtract freely: $x + 3 < 5 ⟹ x < 5 - 3 ⟹ x < 2$
- WHY? Dono sides mein same number add karne se order preserve hota hai (agar $a < b$ hai, toh $a + k < b + k$ )
Multiply/Divide by positive: $2 x < 6 ⟹ x < 3$
- WHY? Dono sides ko positive factor se scale karne par order preserve hota hai
Multiply/Divide by negative → inequality sign FLIP karo: $- 2 x < 6 ⟹ x > - 3$
- WHY? Jab tum negative se multiply karte ho, tum number line ko reverse kar dete ho. Agar $2 < 5$ hai, toh $- 2 > - 5$ . Yeh critical rule hai jo most students bhool jaate hain.

Flip rule ki DERIVATION: $a < b$ se start karo. Dono sides ko $- 1$ se multiply karo: $- a ? - b$ $a = 2, b = 5$ lo: hamare paas $2 < 5$ hai, toh $- 2 ? - 5$ . Number line par, $- 2$ $- 5$ ke right mein hai (zero ke kareeb), toh $- 2 > - 5$ . Isliye: negative se multiply karne par inequality reverse ho jaati hai.

Solution: $3 x - 7 \leq 11$ Dono sides mein 7 add karo: $3 x \leq 18 (WHY? Inequality preserve hoti hai)$ 3 se divide karo (positive): $x \leq 6 (WHY? Positive se scale karne par order preserve hota hai)$

Answer: $x \in (- \infty, 6]$ (6 tak aur 6 including saari real numbers)

Solution: $- 4 x + 5 > 17$ 5 subtract karo: $- 4 x > 12 (WHY? Standard subtraction rule)$ $- 4$ se divide karo → sign FLIP karo: $x < - 3 (WHY? Negative se divide karne par order reverse hota hai)$

Answer: $x \in (- \infty, - 3)$ ( $- 3$ se chhoti saari numbers)

Common trap: Students $x > - 3$ likhte hain aur flip karna bhool jaate hain. Check karo: agar $x = - 2$ hai, toh $- 4 (- 2) + 5 = 8 + 5 = 13 \neq > 17$ . Kaam nahi karta! Lekin $x = - 4$ deta hai $- 4 (- 4) + 5 = 16 + 5 = 21 > 17$ . ✓

Solution: $\frac{2 - 3 x}{5} \geq 4$ Dono sides ko 5 se multiply karo (positive, koi flip nahi): $2 - 3 x \geq 20 (WHY? Denominator clear ho raha hai)$ 2 subtract karo: $- 3 x \geq 18 (WHY? x wale term ko isolate kar rahe hain)$ $- 3$ se divide karo → FLIP karo: $x \leq - 6 (WHY? Negative division reverse karta hai)$

Answer: $x \in (- \infty, - 6]$

Number Line Representation

Number line kyun use karo? Kyunki yeh ek hi baar mein saare solutions ka visual map deta hai. Tum turant range aur boundaries dekh sakte ho.

$x \geq - 1$ ke liye:

$- 1$ par closed circle
Right ki taraf arrow
Interval notation: $[- 1, \infty)$

Interval Notation Summary

Inequality	Number Line	Interval Notation	Set Builder
$x < a$	$a$ par open circle, left arrow	$(- \infty, a)$	${x : x < a}$
$x \leq a$	$a$ par closed circle, left arrow	$(- \infty, a]$	${x : x \leq a}$
$x > a$	$a$ par open circle, right arrow	$(a, \infty)$	${x : x > a}$
$x \geq a$	$a$ par closed circle, right arrow	$[a, \infty)$	${x : x \geq a}$

Kyun sahi lagta hai: Hum equations ke itne aadat hain jahan signs flip nahi hote. Hamara brain division ko "neutral" operation treat karta hai.

Steel-man: Student sahi se equation rule apply kar raha hai lekin bhool raha hai ki inequalities ki ek direction hoti hai jo number line reflect hone par reverse ho jaati hai.

Fix: Hamesha poochho: "Kya main negative se multiply/divide kar raha hoon?" Agar haan, toh sign flip karo. Sahi solution: $- 2 x < 8 ⟹ x > - 4$ .

Verification: $x = 0$ lo (hamare answer mein, kyunki $0 > - 4$ ): $- 2 (0) = 0$ , aur $0 < 8$ ✓. $x = - 5$ lo (hamare answer ke bahar, kyunki $- 5 < - 4$ ): $- 2 (- 5) = 10$ , aur $10 < 8$ galat hai ✗ (sahi se exclude hua). Confirm karta hai ki $x > - 4$ sahi hai.

Kyun sahi lagta hai: Students $<$ aur $\leq$ mein confuse ho jaate hain.

Fix: $\leq$ aur $\geq$ boundary ko include karte hain → closed circle. $<$ aur $>$ use exclude karte hain → open circle. Mnemonic: symbol mein "equal" = "filled in" circle.

Recall Feynman Box (Ek 12-saal ke bachche ko explain karo)

Socho tumhare paas ₹100 hain aur tum chocolates khareedna chahte ho jo ₹15 ki ek hain. Tum exact number nahi jaanna chahte — tum jaanna chahte ho ki tum kitni khareed sakte ho.

Agar tum $x$ chocolates khareedoge, tum $15 x$ rupaye kharach karoge. Tumhe $15 x \leq 100$ chahiye (apne budget se kam ya barabar kharach karo). Dono sides ko 15 se divide karo: $x \leq 6.67$ . Kyunki tum chocolate ka koi hissa nahi khareed sakte, $x \leq 6$ (tum 0, 1, 2, 3, 4, 5, ya 6 chocolates khareed sakte ho).

Ab socho tumpar ₹50 ka debt hai aur tum ₹10 per week wapas kar rahe ho. $x$ weeks baad, tumne $10 x$ wapas kar diya, toh tumhara debt $50 - 10 x$ hai. Tumhara debt kab khatam hoga? Jab $50 - 10 x \leq 0$ . Solve karo: $50 \leq 10 x ⟹ 5 \leq x$ . Tumhe kam se kam 5 weeks chahiye.

"Flip rule" aisi hai: agar tum kisi ke paisa dete ho (negative), aur tum multiply karo kitna dete ho, toh jitna zyada dete ho, utni buri situation — direction reverse ho jaati hai!

Connections

Linear Equations — inequalities equations hain jisme "wiggle room" hoti hai
Absolute Value Equations and Inequalities — inequality solving ko distance concepts ke saath combine karta hai
Systems of Inequalities — multiple constraints ko ek saath satisfy karne wale regions dhundhna
Quadratic Inequalities — curved boundaries tak extend hota hai, similar sign-analysis use karta hai
Linear Programming — linear inequality constraints ke under optimization
Number Line and Real Numbers — inequality representation ki underlying geometric structure

#flashcards/maths

Ek variable mein linear inequality kya hoti hai? :: $a x + b < 0$ (ya $\leq, >, \geq$ ) form ki inequality jahan $a \neq = 0$ ho, jiska solution values ka ek set hota hai jo ek interval banata hai

Jab tum dono sides ko negative number se multiply ya divide karte ho toh inequality sign ka kya hota hai?

Inequality sign flip (reverse) ho jaati hai:

<

ban jaata hai

>

\leq

ban jaata hai

\geq

, etc.

Number line par

x < 3

ko kaise represent karte hain?

3 par open circle aur left ki taraf arrow

Number line par

x \geq - 2

ko kaise represent karte hain?

- 2

par closed (filled) circle aur right ki taraf arrow

5 se chhoti saari real numbers ke liye interval notation kya hai?

(- \infty, 5)

- 1

se badi ya barabar saari real numbers ke liye interval notation kya hai?

[- 1, \infty)

Agar

- 3 x + 7 < 16

hai, toh

x

kya hai?

x > - 3

(7 subtract karo toh

- 3 x < 9

milta hai, phir

- 3

se divide karo aur flip karo toh

x > - 3

milta hai)

Sahi ya Galat:

2 x - 5 > 3

solve karte waqt inequality sign flip karna padega.

Galat — tum tabhi flip karte ho jab negative number se multiply/divide karo

x

ke liye solve karo:

\frac{x - 4}{2} \leq 3

x \leq 10

(dono sides ko 2 se multiply karo toh

x - 4 \leq 6

milta hai, phir 4 add karo)

Number line par closed circle kya indicate karta hai?

Boundary value solution mein included hai (use hota hai

\leq

\geq

ke liye)