Linear Inequalities

Class 11 · Mathematics

Linear Inequalities

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Linear Inequalities

Jab do quantities barabar nahi hoti, to unhe \(<,\ >,\ \le,\ \ge\) se compare karte hain — ye inequalities kehlate hain. Real life me bahut saari problems "kam se kam" ya "zyada se zyada" jaisi conditions wali hoti hain, jinhe inequalities se likhte hain.

1. Inequality kya hai

  • \(a < b\): \(a\), \(b\) se chhota.
  • \(a > b\): \(a\), \(b\) se bada.
  • \(a \le b\): \(a\), \(b\) se chhota ya barabar.
  • \(a \ge b\): \(a\), \(b\) se bada ya barabar.

Strict inequalities \(<, >\) hoti hain; slack (ya non-strict) \(\le, \ge\).

2. Linear Inequality Solve karne ke Rules

Inequality ko equation ki tarah hi solve karte hain, par ek zaroori farak: jab dono taraf ko kisi negative number se multiply ya divide karo, to inequality ka sign palat jaata hai.
  • Dono taraf same number jodo/ghatao — sign same rehta hai.
  • Dono taraf positive number se multiply/divide karo — sign same.
  • Dono taraf negative number se multiply/divide karo — sign reverse (\(<\) ban jaata hai \(>\)).
Example

\(3x - 5 < 7\) solve karo. \[ 3x < 12 \;\Rightarrow\; x < 4. \] Solution: \(x \in (-\infty,\ 4)\).

Example (sign flip)

\(-2x + 1 \ge 5\) solve karo. \[ -2x \ge 4 \;\Rightarrow\; x \le -2 \quad (\text{sign palta kyunki } -2 \text{ se divide kiya}). \] Solution: \(x \in (-\infty,\ -2]\).

3. Number Line par Solution

One-variable inequality ka solution number line par dikhate hain:

  • Strict (\(<, >\)) — endpoint par open circle (point shamil nahi).
  • Non-strict (\(\le, \ge\)) — endpoint par filled circle (point shamil).

4. Do Variables ki Linear Inequality

\(ax + by \le c\) jaisi inequality ka solution \(xy\)-plane me ek half-plane hota hai (line ke ek taraf ka poora region).

Graph banane ka tareeka:

  1. Pehle line \(ax + by = c\) banao (boundary).
  2. Strict inequality (\(<, >\)) ho to line dotted; non-strict (\(\le, \ge\)) ho to line solid.
  3. Koi test point (jaise origin \((0,0)\)) inequality me daalo. Satisfy ho to us taraf ka region shade karo, warna doosri taraf.
Example

\(x + y \le 4\) ke liye, origin \((0,0)\) test karo: \(0 + 0 = 0 \le 4\) — true. Toh line \(x + y = 4\) ki origin wali taraf ka half-plane (line samet) solution hai.

5. System of Linear Inequalities

Jab kai inequalities ek saath di ho, to har ek ka half-plane banao; sabka common region (intersection) hi system ka solution hota hai. Yahi idea aage Linear Programming me kaam aata hai.

Key Takeaways

  • Inequalities \(<, >, \le, \ge\) se quantities compare hoti hain.
  • Negative number se multiply/divide karne par sign palat jaata hai — sabse common galti yahi hoti hai.
  • One variable ka solution interval (number line) hota hai; strict me open, non-strict me closed endpoint.
  • Two variables me solution ek half-plane hota hai (test point se decide karo).
  • System ka solution sabhi half-planes ka common region hai.
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