First, consider the equation \(3|2^x - 1| = 2^x\). This can be split into two cases:

Case 1: \(2^x - 1 \geq 0\), so \(|2^x - 1| = 2^x - 1\).

Then the equation becomes \(3(2^x - 1) = 2^x\).

Rearranging gives \(3 \cdot 2^x - 3 = 2^x\).

Thus, \(2^x = \frac{3}{2}\).

Taking logarithms, \(x = \log_2\left(\frac{3}{2}\right)\).

Case 2: \(2^x - 1 < 0\), so \(|2^x - 1| = 1 - 2^x\).

Then the equation becomes \(3(1 - 2^x) = 2^x\).

Rearranging gives \(3 - 3 \cdot 2^x = 2^x\).

Thus, \(2^x = \frac{3}{4}\).

Taking logarithms, \(x = \log_2\left(\frac{3}{4}\right)\).

Calculating these values gives:

\(x = \log_2\left(\frac{3}{2}\right) \approx 0.585\)

\(x = \log_2\left(\frac{3}{4}\right) \approx -0.415\)

Therefore, the solutions are \(x = 0.585\) and \(x = -0.415\).