Start by considering the inequality \(|x - 4| < 2|3x + 1|\).

Rewrite it as \((x - 4)^2 < (2(3x + 1))^2\).

This simplifies to \((x - 4)^2 < 4(3x + 1)^2\).

Expand both sides: \(x^2 - 8x + 16 < 36x^2 + 24x + 4\).

Rearrange to form a quadratic inequality: \(0 < 35x^2 + 32x - 12\).

Find the critical points by solving \(35x^2 + 32x - 12 = 0\).

Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 35\), \(b = 32\), \(c = -12\).

Calculate the discriminant: \(b^2 - 4ac = 32^2 - 4 \times 35 \times (-12) = 1024 + 1680 = 2704\).

\(x = \frac{-32 \pm \sqrt{2704}}{70}\).

\(\sqrt{2704} = 52\), so \(x = \frac{-32 \pm 52}{70}\).

This gives critical points \(x = \frac{20}{70} = \frac{2}{7}\) and \(x = \frac{-84}{70} = -\frac{6}{5}\).

The solution to the inequality is \(x < -\frac{6}{5}\) or \(x > \frac{2}{7}\).