Multiply both sides of the equation by the denominator (11 - 7x).

Move all terms to one side to set the equation to zero.

$$0 = 7x^2 - 11x - 3x + 8$$ $$7x^2 - 14x + 8 = 0$$

For the equation $7x^2 - 14x + 8 = 0$, we have $a = 7$, $b = -14$, and $c = 8$.

First, calculate the discriminant $\Delta = b^2 - 4ac$.

$$\Delta = (-14)^2 - 4(7)(8)$$ $$\Delta = 196 - 224$$ $$\Delta = -28$$

Since the discriminant is negative, there are no real solutions. The solutions are complex numbers.

$$x = \frac{-(-14) \pm \sqrt{-28}}{2(7)}$$ $$x = \frac{14 \pm \sqrt{28}i}{14}$$

We can simplify $\sqrt{28}$ as $\sqrt{4 \times 7} = 2\sqrt{7}$.

$$x = \frac{14 \pm 2\sqrt{7}i}{14}$$

Divide both terms in the numerator by 14.

$$x = \frac{14}{14} \pm \frac{2\sqrt{7}i}{14}$$ $$x = 1 \pm \frac{\sqrt{7}}{7}i$$

The solutions for $x$ are:

$$\boxed{x = 1 + \frac{\sqrt{7}}{7}i, \quad x = 1 - \frac{\sqrt{7}}{7}i}$$