Start with the equation:

\(4 \sin\left(\frac{1}{2}x - 30^\circ\right) = 2\sqrt{2}\)

Divide both sides by 4:

\(\sin\left(\frac{1}{2}x - 30^\circ\right) = \frac{\sqrt{2}}{2}\)

Recognize that \(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45^\circ\) or \(135^\circ\).

Set up the equations:

\(\frac{1}{2}x - 30^\circ = 45^\circ\)

\(\frac{1}{2}x - 30^\circ = 135^\circ\)

Solve each equation for \(x\):

For \(45^\circ\):

\(\frac{1}{2}x = 45^\circ + 30^\circ\)

\(\frac{1}{2}x = 75^\circ\)

\(x = 150^\circ\)

For \(135^\circ\):

\(\frac{1}{2}x = 135^\circ + 30^\circ\)

\(\frac{1}{2}x = 165^\circ\)

\(x = 330^\circ\)

Thus, the solutions are \(x = 150^\circ\) and \(x = 330^\circ\).