To solve the inequality $2 \cos(x) - 1 > 0$ for $x$ in the interval $[0, 2\pi]$, follow these steps:

Step 1: Solve the inequality

$2 \cos(x) - 1 > 0$ Add 1 to both sides: $2 \cos(x) > 1$ Divide by 2: $\cos(x) > \frac{1}{2}$

Step 2: Determine the values of $x$ where $\cos(x) > \frac{1}{2}$

The cosine function is positive in the first and fourth quadrants. Specifically, $\cos(x) > \frac{1}{2}$ when: $x \in \left( -\frac{\pi}{3}, \frac{\pi}{3} \right) \cup \left( \frac{5\pi}{3}, \frac{7\pi}{3} \right)$

Step 3: Adjust the intervals to match the given interval $[0, 2\pi]$

We need to find the $x$ values within the interval $[0, 2\pi]$. The relevant portions of the solution are: $x \in \left[ 0, \frac{\pi}{3} \right) \cup \left( \frac{5\pi}{3}, 2\pi \right]$

Final Answer

The values of $x$ that satisfy $2 \cos(x) - 1 > 0$ in the interval $[0, 2\pi]$ are: $x \in \left[ 0, \frac{\pi}{3} \right) \cup \left( \frac{5\pi}{3}, 2\pi \right]$

Would you like more details or have any questions?

Here are some related questions:

1. How do you solve the inequality $\sin(x) > \frac{1}{2}$ over the interval $[0, 2\pi]$?
2. What is the general solution to the equation $\cos(x) = \frac{1}{2}$?
3. How do you find the maximum and minimum values of the function $f(x) = 2 \cos(x) - 1$?
4. What is the graphical interpretation of $\cos(x) > \frac{1}{2}$?
5. How do the solutions of $\cos(x) = \frac{1}{2}$ change if the interval is $[0, 4\pi]$?

Tip: When solving trigonometric inequalities, it's useful to first determine the critical points where the inequality changes sign and then consider the behavior within each interval defined by these points.