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

\((2x - 3)^2 > 4^2(x + 1)^2\)

This simplifies to:

\((2x - 3)^2 > 16(x + 1)^2\)

Expanding both sides, we get:

\(4x^2 - 12x + 9 > 16(x^2 + 2x + 1)\)

\(4x^2 - 12x + 9 > 16x^2 + 32x + 16\)

Rearranging terms gives:

\(4x^2 - 12x + 9 - 16x^2 - 32x - 16 < 0\)

\(-12x^2 - 44x - 7 < 0\)

Solving the quadratic inequality \(12x^2 + 44x + 7 < 0\) gives the critical values:

\(x = -\frac{7}{2}\) and \(x = -\frac{1}{6}\)

The solution to the inequality is:

\(\frac{7}{2} < x < -\frac{1}{6}\)