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Step-by-step Solution
Step 1: Express the Arithmetic Progression (AP) Conditions
Given that $ \tan\left(\frac{\pi}{9}\right),\, x, \,\tan\left(\frac{7\pi}{18}\right) $ are in AP, it follows by definition of an AP that
$$
x = \frac{\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{7\pi}{18}\right)}{2}.
$$
Similarly, $ \tan\left(\frac{\pi}{9}\right), \, y, \, \tan\left(\frac{5\pi}{18}\right) $ are also in AP, implying
$$
y = \frac{\tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{5\pi}{18}\right)}{2}.
$$
Step 2: Rewrite Angles in Degrees for Simplicity (Optional Clarification)
Note that $ \frac{\pi}{9} = 20^\circ, \frac{5\pi}{18} = 50^\circ, \frac{7\pi}{18} = 70^\circ $.
So one may rewrite:
$$
x = \frac{\tan 20^\circ + \tan 70^\circ}{2},
\quad
2y = \tan 20^\circ + \tan 50^\circ
\;\Big(\text{since } y = \tfrac{1}{2}(\tan 20^\circ + \tan 50^\circ)\Big).
$$
Step 3: Compute $|x - 2y|$
From the above, let us find $|x - 2y|$:
$$
|x - 2y|
= \left|\frac{\tan 20^\circ + \tan 70^\circ}{2} \;-\; \bigl(\tan 20^\circ + \tan 50^\circ\bigr)\right|.
$$
Simplify inside the absolute value:
$$
|x - 2y|
= \left|\frac{\tan 20^\circ + \tan 70^\circ - 2 \tan 20^\circ - 2 \tan 50^\circ}{2}\right|
= \left|\frac{\tan 70^\circ - \tan 20^\circ - 2 \tan 50^\circ}{2}\right|.
$$
Step 4: Use Trigonometric Identities
We use the identity for $\tan(70^\circ)$ in terms of $\tan(20^\circ)$ and $\tan(50^\circ)$:
$$
\tan 70^\circ = \frac{\tan 20^\circ + \tan 50^\circ}{1 - \tan 20^\circ \, \tan 50^\circ}.
$$
A known result (or derived via transformations) is that
$$
\tan 70^\circ - \tan 20^\circ - 2\,\tan 50^\circ = 0.
$$
Hence the numerator in our expression becomes $0$, so:
$$
\tan 70^\circ - \tan 20^\circ - 2 \tan 50^\circ = 0.
$$
Step 5: Conclude the Value of $|x - 2y|$
Substituting this back, we get
$$
|x - 2y| = \left|\frac{0}{2}\right| = 0.
$$
Therefore, the required absolute value is $0$.
Final Answer
$|x - 2y| = 0$.
