To solve the inequality \(3|2x - 1| > |x + 4|\), we first consider the non-modular inequality \(3^2(2x - 1)^2 > (x + 4)^2\).

This simplifies to \(35x^2 - 44x - 7 = 0\).

We solve the quadratic equation \(35x^2 - 44x - 7 = 0\) by factoring or using the quadratic formula.

Factoring gives \((5x - 7)(7x + 1) = 0\), leading to critical values \(x = \frac{7}{5}\) and \(x = -\frac{1}{7}\).

The solution to the inequality is \(x > \frac{7}{5}\) or \(x < -\frac{1}{7}\).