To solve \(|4x - 1| = |x - 3|\), consider the cases for the absolute values.

Case 1: \(4x - 1 = x - 3\)

Solve for \(x\):

\(4x - x = -3 + 1\)

\(3x = -2\)

\(x = -\frac{2}{3}\)

Case 2: \(4x - 1 = -(x - 3)\)

Solve for \(x\):

\(4x - 1 = -x + 3\)

\(4x + x = 3 + 1\)

\(5x = 4\)

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

Thus, the solutions are \(x = -\frac{2}{3}\) and \(x = \frac{4}{5}\).