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

Expanding both sides, we have:

\((3(x - 1))^2 = 9(x^2 - 2x + 1) = 9x^2 - 18x + 9\)

\((2x + 1)^2 = 4x^2 + 4x + 1\)

Setting the inequality:

\(9x^2 - 18x + 9 < 4x^2 + 4x + 1\)

Simplifying, we get:

\(5x^2 - 22x + 8 < 0\)

Solving the quadratic inequality \(5x^2 - 22x + 8 = 0\) gives the critical points:

\(x = \frac{2}{5}\) and \(x = 4\)

Testing intervals, we find the solution to the inequality is:

\(\frac{2}{5} < x < 4\)