(i) Sketch the graphs of \(y = 2 - x\) and \(y = \ln x\). The intersection point of these graphs represents the root of the equation \(2 - x = \ln x\). Since \(y = 2 - x\) is a straight line with a negative slope and \(y = \ln x\) is a logarithmic curve, they intersect at only one point, indicating one root.

(ii) Evaluate \(2 - x - \ln x\) at \(x = 1.4\) and \(x = 1.7\):

For \(x = 1.4\), \(2 - 1.4 - \ln(1.4) \approx 0.246\).

For \(x = 1.7\), \(2 - 1.7 - \ln(1.7) \approx -0.036\).

Since the sign changes between \(x = 1.4\) and \(x = 1.7\), the root lies in this interval.

(iii) Rearrange the equation \(x = \frac{1}{3}(4 + x - 2 \ln x)\) to \(2 - x = \ln x\), confirming that the root satisfies both equations.

(iv) Use the iterative formula:

\(x_1 = 1.5\)

\(x_2 = \frac{1}{3}(4 + 1.5 - 2 \ln(1.5)) \approx 1.5581\)

\(x_3 = \frac{1}{3}(4 + 1.5581 - 2 \ln(1.5581)) \approx 1.5560\)

\(x_4 = \frac{1}{3}(4 + 1.5560 - 2 \ln(1.5560)) \approx 1.5563\)

\(x_5 = \frac{1}{3}(4 + 1.5563 - 2 \ln(1.5563)) \approx 1.5562\)

The root correct to 2 decimal places is 1.56.