Which of the following is not a linear inequation?

1. 3x - 8 > 5 + 2x

2. $\dfrac{5}{2}x - 3 \le \dfrac{9}{4}x + 12$

3. $4x - \dfrac{7}{3} \ge 13x + 8$

4. $6x + 13 \lt \dfrac{4}{x} - 3$

Solving, option 4 :

$\Rightarrow 6x + 13 \lt \dfrac{4}{x} - 3 \\[1em] \Rightarrow 6x + 13 \lt \dfrac{4 - 3x}{x} \\[1em]$

Case 1 : If x is positive,

$\Rightarrow x (6x + 13) \lt 4 - 3x \\[1em] \Rightarrow 6x^2+ 13x \lt 4 - 3x \\[1em] \Rightarrow 6x^2+ 13x + 3x \lt 4 \\[1em] \Rightarrow 6x^2+ 16x \lt 4 \\[1em]$

Case 2 : If x is negative,

$\Rightarrow x(6x + 13) \gt 4 - 3x \\[1em] \Rightarrow 6x^2 + 13x \gt 4 - 3x \\[1em] \Rightarrow 6x^2 + 13x + 3x \gt 4 \\[1em] \Rightarrow 6x^2 + 16x \gt 4 \\[1em]$

Since, the highest power of x is 2.

$\therefore 6x + 13 \lt \dfrac{4}{x} - 3$ is not linear.

Hence, Option 4 is the correct option.