Start with the equation:

\(\frac{2e^x + e^{-x}}{e^x - e^{-x}} = 4\)

Multiply both sides by \(e^x - e^{-x}\):

\(2e^x + e^{-x} = 4(e^x - e^{-x})\)

Expand the right side:

\(2e^x + e^{-x} = 4e^x - 4e^{-x}\)

Rearrange terms:

\(2e^x + e^{-x} + 4e^{-x} = 4e^x\)

\(2e^x + 5e^{-x} = 4e^x\)

Rearrange to isolate terms with \(e^x\):

\(2e^x - 4e^x = -5e^{-x}\)

\(-2e^x = -5e^{-x}\)

Divide both sides by \(-2\):

\(e^x = \frac{5}{2}e^{-x}\)

Multiply both sides by \(e^x\):

\(e^{2x} = \frac{5}{2}\)

Take the natural logarithm of both sides:

\(2x = \ln\left(\frac{5}{2}\right)\)

Solve for \(x\):

\(x = \frac{1}{2} \ln\left(\frac{5}{2}\right)\)

Calculate \(x\) to 2 decimal places:

\(x \approx 0.46\)