To solve the given equation [tex]\(4 \sqrt{2} \tan x - \sqrt{2} = 3 \sqrt{2} \tan x\)[/tex], we start by isolating the term involving [tex]\(\tan x\)[/tex].
1. Begin with the equation:
[tex]\[
4 \sqrt{2} \tan x - \sqrt{2} = 3 \sqrt{2} \tan x
\][/tex]
2. Subtract [tex]\(3 \sqrt{2} \tan x\)[/tex] from both sides:
[tex]\[
4 \sqrt{2} \tan x - 3 \sqrt{2} \tan x - \sqrt{2} = 0
\][/tex]
3. Simplify the equation by combining like terms:
[tex]\[
\sqrt{2} \tan x - \sqrt{2} = 0
\][/tex]
4. Factor out [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\sqrt{2} (\tan x - 1) = 0
\][/tex]
5. Divide both sides by [tex]\(\sqrt{2}\)[/tex] (since [tex]\(\sqrt{2} \neq 0\)[/tex]):
[tex]\[
\tan x - 1 = 0
\][/tex]
6. Solve for [tex]\(\tan x\)[/tex]:
[tex]\[
\tan x = 1
\][/tex]
The tangent function [tex]\(\tan x\)[/tex] is equal to 1 at angles where [tex]\(x = \frac{\pi}{4} + n\pi\)[/tex] for any integer [tex]\(n\)[/tex]. Now, we need to find the solutions within the interval [tex]\(0 \leq x \leq 2\pi\)[/tex].
7. Determine the specific values of [tex]\(x\)[/tex] within the interval:
- For [tex]\(n = 0\)[/tex]:
[tex]\[
x = \frac{\pi}{4}
\][/tex]
- For [tex]\(n = 1\)[/tex]:
[tex]\[
x = \frac{\pi}{4} + \pi = \frac{\pi}{4} + \frac{4\pi}{4} = \frac{5\pi}{4}
\][/tex]
8. Ensure these [tex]\(x\)[/tex] values are within the given interval [tex]\(0 \leq x \leq 2\pi\)[/tex]:
[tex]\[
0 \leq \frac{\pi}{4} \leq 2\pi \quad \text{and} \quad 0 \leq \frac{5\pi}{4} \leq 2\pi
\][/tex]
Thus, the solutions are:
[tex]\[
x = \frac{\pi}{4} \quad \text{and} \quad x = \frac{5\pi}{4}
\][/tex]
Converting these values into decimal form:
[tex]\[
x \approx 0.7853981633974483 \quad \text{and} \quad x \approx 3.9269908169872414
\][/tex]
The final answers, in ascending order, are:
[tex]\(\boxed{0.7853981633974483}\)[/tex] and [tex]\(\boxed{3.9269908169872414}\)[/tex]
