Let z be a complex number such that $$|z| + z = 3 + i$$ (where $$i = \sqrt{-1}$$). Then $$|z|$$ is equal to:

We are told that the complex number $$z$$ satisfies the relation $$|z| + z = 3 + i$$, where $$i = \sqrt{-1}$$. In order to unravel this condition, we first express $$z$$ in the usual Cartesian form $$z = x + iy$$ with real numbers $$x$$ and $$y$$.

Using the definition of the modulus of a complex number, we have the well-known identity $$|z| = \sqrt{x^{2} + y^{2}}$$. Substituting $$z = x + iy$$ and $$|z| = \sqrt{x^{2} + y^{2}}$$ into the given equation, we obtain

$$\sqrt{x^{2} + y^{2}} + (x + iy) = 3 + i.$$

Now we separate the real and imaginary parts on both sides. The real part on the left is $$\sqrt{x^{2} + y^{2}} + x$$, and the imaginary part is $$y$$. The right-hand side has real part $$3$$ and imaginary part $$1$$. Therefore we must have

$$\sqrt{x^{2} + y^{2}} + x = 3 \qquad\text{and}\qquad y = 1.$$

From the second equation we immediately obtain $$y = 1$$. We now insert this value into the first equation. Doing so gives

$$\sqrt{x^{2} + 1^{2}} + x = 3,$$

which simplifies to

$$\sqrt{x^{2} + 1} = 3 - x.$$

Because the square-root expression is always non-negative, we must have $$3 - x \ge 0$$, implying $$x \le 3$$, which will be automatically checked once we find $$x$$. Next, to eliminate the square root, we square both sides. The algebraic step is

$$x^{2} + 1 = (3 - x)^{2}.$$

Expanding the right-hand side gives

$$x^{2} + 1 = 9 - 6x + x^{2}.$$

Both sides contain the term $$x^{2}$$, so we subtract $$x^{2}$$ from each side, leaving

$$1 = 9 - 6x.$$

Rearranging for $$x$$, we first subtract $$9$$ from both sides to obtain $$1 - 9 = -6x$$, i.e.

$$-8 = -6x.$$

Dividing by $$-6$$, we get

$$x = \frac{-8}{-6} = \frac{4}{3}.$$

This value satisfies $$x \le 3$$, so it is consistent. We now compute the modulus $$|z|$$ itself. Using $$x = \dfrac{4}{3}$$ and $$y = 1$$, we have

$$|z| = \sqrt{x^{2} + y^{2}} = \sqrt{\left(\frac{4}{3}\right)^{2} + 1^{2}} = \sqrt{\frac{16}{9} + \frac{9}{9}} = \sqrt{\frac{25}{9}} = \frac{5}{3}.$$

Hence, the correct answer is Option B.