To solve the equation \(\sin\left(\frac{4\pi}{9} + x\right) + \cos\left(\frac{\pi}{18} - x\right) = \sqrt{3}\), we can analyze it using trigonometric properties.

The given equation is:

\[
\sin\left(\frac{4\pi}{9} + x\right) + \cos\left(\frac{\pi}{18} - x\right) = \sqrt{3}
\]

Let's label:

\[
A = \sin\left(\frac{4\pi}{9} + x\right)
\]
\[
B = \cos\left(\frac{\pi}{18} - x\right)
\]

We are given \(A + B = \sqrt{3}\).

First, let's note that the range of both the sine and cosine functions is \([-1, 1]\). Therefore, for their sum to equal \(\sqrt{3}\), each needs to be approaching the extremum of 1 or be evaluated at angles that lead to simplifications that make this equality possible.

We can try evaluating the trigonometric values at certain known angles or use identities to reduce the complexity.

Consider solved using a trigonometric identity for sums:

Reformulating the equation using sine addition formula:

\[
\sin(a + b) = \sin a \cos b + \cos a \sin b
\]

1. \[ \sin\left(\frac{4\pi}{9} + x\right) = \sin\left(\frac{4\pi}{9}\right)\cos x + \cos\left(\frac{4\pi}{9}\right)\sin x \]
2. \[ \cos\left(\frac{\pi}{18} - x\right) = \cos\left(\frac{\pi}{18}\right)\cos x + \sin\left(\frac{\pi}{18}\right)\sin x \]

Combining them would give:

\[ \left(\sin\left(\frac{4\pi}{9}\right) + \sin\left(\frac{\pi}{18}\right)\right)\cos x + \left(\cos\left(\frac{4\pi}{9}\right) + \cos\left(\frac{\pi}{18}\right)\right)\sin x = \sqrt{3} \]

Testing specific cases may simplify further. Given the complexity of specific angles involved, evaluating numerically on known angles or using computational software might yield specific solutions without decomposition. The identity:

\[ \sin A + \cos B = \sqrt{3} \]

implies angles that satisfy:

\[ A + B = \frac{\pi}{3} + 2m\pi \]

Given both are not common angles, evaluation or calculation-based resolution fits here without additional known identities applicable directly.