The solution of (7x + 12)/(x-9) <4; x ≠ 9 is:

The correct answer is Option (3) → $\{x:-16 < x < 9;x ∈ R\}$

Given inequality:

$\frac{7x+12}{x-9} < 4, \ x \ne 9$

Bring all terms to one side:

$\frac{7x+12}{x-9} - 4 < 0 \Rightarrow \frac{7x+12 - 4(x-9)}{x-9} < 0$

$\frac{7x+12 - 4x + 36}{x-9} < 0 \Rightarrow \frac{3x + 48}{x-9} < 0 \Rightarrow \frac{3(x + 16)}{x-9} < 0$

Critical points: $x = -16$ (numerator), $x = 9$ (denominator)

Check intervals:

1. $x < -16$: numerator negative, denominator negative → fraction positive ❌

2. $-16 < x < 9$: numerator positive, denominator negative → fraction negative ✅

3. $x > 9$: numerator positive, denominator positive → fraction positive ❌

$\text{Answer: } -16 < x < 9$