We are given the inequality:

$4 \cos(x) + 2 > \pi$ and the interval for $x$ is $0 \leq x \leq 2\pi$.

Step 1: Simplify the inequality

First, subtract 2 from both sides of the inequality:

$4 \cos(x) > \pi - 2$

Now compute $\pi - 2$:

$\pi - 2 \approx 3.14 - 2 = 1.14$

So the inequality becomes:

$4 \cos(x) > 1.14$

Step 2: Solve for $\cos(x)$

Now divide both sides of the inequality by 4:

$\cos(x) > \frac{1.14}{4}$

$\cos(x) > 0.285$

Step 3: Determine the range of $x$

We need to find the values of $x$ in the interval $0 \leq x \leq 2\pi$ such that $\cos(x) > 0.285$.

The cosine function is positive in the first and fourth quadrants, so we will find the reference angle where $\cos(x) = 0.285$.

Using the inverse cosine function:

$x = \cos^{-1}(0.285) \approx 1.28 \text{ radians}$

Thus, in the first quadrant, $x = 1.28$.

In the fourth quadrant, the corresponding solution is:

$x = 2\pi - 1.28 \approx 6.28 - 1.28 = 5.00 \text{ radians}$

Step 4: Solution intervals

Since $\cos(x)$ is greater than 0.285 between these values, the solution for $x$ is:

[ x \in [0, 1.28] \cup [5.00, 2\pi] ]

In decimal form:

[ x \in [0, 1.28] \cup [5.00, 6.28] ]

Thus, the values of $x$ that satisfy the inequality $4 \cos(x) + 2 > \pi$ are in the intervals $[0, 1.28] \cup [5.00, 6.28]$.

Do you have any further questions or want more details?

Here are 5 related questions to expand on this topic:

1. How do you find the cosine of an angle in different quadrants?
2. What are the properties of the cosine function over the interval $[0, 2\pi]$?
3. Can you explain the significance of the inverse trigonometric functions?
4. How does the cosine graph look in the range $[0, 2\pi]$?
5. How would the solution change if the inequality was $4 \cos(x) + 2 < \pi$?

Tip: Always ensure your calculator is in the correct mode (radians or degrees) based on the given problem context.