1. The problem is to solve for $x$ given the equation $\cos x = -\frac{2}{7}$.\n\n2. Recall that the cosine function, $\cos x$, gives the ratio of the adjacent side to the hypotenuse in a right triangle, and its values range between $-1$ and $1$. Since $-\frac{2}{7}$ is within this range, the equation has real solutions.\n\n3. To find $x$, use the inverse cosine function: $$x = \cos^{-1}\left(-\frac{2}{7}\right).$$\n\n4. The principal value of $x$ is $$x_1 = \cos^{-1}\left(-\frac{2}{7}\right).$$\n\n5. Because cosine is negative, $x$ lies in the second and third quadrants. The second solution in $[0,2\pi)$ is $$x_2 = 2\pi - x_1.$$\n\n6. Therefore, the general solutions are $$x = x_1 + 2k\pi \quad \text{and} \quad x = x_2 + 2k\pi, \quad k \in \mathbb{Z}.$$\n\n7. Numerically, $x_1 \approx \cos^{-1}(-0.2857) \approx 1.860$ radians, and $x_2 \approx 2\pi - 1.860 = 4.423$ radians.\n\nFinal answer: $$x \approx 1.860 + 2k\pi \quad \text{or} \quad x \approx 4.423 + 2k\pi, \quad k \in \mathbb{Z}.$$