Start with the inequality \(2(3^{1-2x}) < 5^x\).

Take the natural logarithm of both sides: \(\ln(2) + \ln(3^{1-2x}) < x \ln(5)\).

Apply the logarithm power rule: \(\ln(2) + (1-2x)\ln(3) < x \ln(5)\).

Rearrange the inequality: \(\ln(2) + \ln(3) - 2x \ln(3) < x \ln(5)\).

Combine like terms: \(\ln(6) < x(\ln(5) + 2\ln(3))\).

Solve for \(x\): \(x > \frac{\ln(6)}{\ln(5) + 2\ln(3)}\).

Simplify the denominator: \(x > \frac{\ln(6)}{\ln(45)}\).