Solve each equation for exact solutions.
2 arccos (x/3 - π/3) = 2π

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Start by isolating the arccosine expression. Given the equation \(2 \arccos\left(\frac{x}{3} - \frac{\pi}{3}\right) = 2\pi\), divide both sides by 2 to get \(\arccos\left(\frac{x}{3} - \frac{\pi}{3}\right) = \pi\).

Recall the definition and range of the arccosine function: \(\arccos(y)\) returns an angle \(\theta\) such that \(\cos(\theta) = y\) and \(\theta \in [0, \pi]\). Here, \(\arccos\left(\frac{x}{3} - \frac{\pi}{3}\right) = \pi\) means the angle whose cosine is \(\frac{x}{3} - \frac{\pi}{3}\) is \(\pi\).

Use the property of cosine: \(\cos(\pi) = -1\). Set the inside of the arccos equal to \(\cos(\pi)\), so \(\frac{x}{3} - \frac{\pi}{3} = -1\).

Solve the resulting linear equation for \(x\): multiply both sides by 3 to clear the denominator, then isolate \(x\).

Check the solution to ensure it lies within the domain of the arccos function, which requires the argument \(\frac{x}{3} - \frac{\pi}{3}\) to be in the interval \([-1, 1]\).

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