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

The correct answer is Option (3) → \(\{x: -16 \lt x \lt 9;\ x \in \mathrm{R}\}\)

With everything on one side, a single fraction is compared with 0:

    \( \displaystyle \frac{7x + 12}{x - 9} - 4 \lt 0 \)

    \( \displaystyle \frac{7x + 12 - 4x + 36}{x - 9} \lt 0 \)

    \( \displaystyle \frac{3(x + 16)}{x - 9} \lt 0 \)

A fraction is negative when the numerator and denominator have opposite signs. The critical points are −16 and 9:

    \(x \lt -16: \text{ both negative, fraction positive}\)
    \(-16 \lt x \lt 9: \text{ numerator positive, denominator negative}\)
    \(x \gt 9: \text{ both positive, fraction positive}\)

So the solution is \(-16 \lt x \lt 9\).

Tip: Do not multiply both sides by x − 9, because its sign is unknown. Doing so gives x < −16, which is wrong.