A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $1/2$. Then the length of the semi-major axis is:

Let the focus be $S(0,0)$ and the directrix be $x=4$.The distance from the focus to the directrix is given by $\frac{a}{e}-ae$.Given $e=\frac{1}{2}$ and distance = $4$:
$$\frac{a}{e}-ae=4$$
$$a\left(\frac{1}{1/2}-\frac{1}{2}\right)=4$$
$$a\left(2-\frac{1}{2}\right)=4$$
$$a\left(\frac{3}{2}\right)=4$$
$$a=\frac{8}{3}$$
Thus, the length of the semi-major axis is $\frac{8}{3}$.