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**Deveno** a direction is not a number. it could be a vector (like say, a unit vector).

your set S is a subset of the first quadrant. it is bounded below by $\displaystyle x_2 = 1$, bounded on the left by $\displaystyle x_1 = 0$ and bounded above by

$\displaystyle x_2 = x_1 + 2$ and $\displaystyle x_2 = \dfrac{1}{2}x_1 + 3$.

of these two lines, the second one has the lesser slope, and eventually is the more restrictive.

so for $\displaystyle (x_1,x_2) + \lambda(d_1,d_2) \in S$, whenever $\displaystyle (x_1,x_2) \in S$ for all $\displaystyle \lambda > 0$ we have to have:

$\displaystyle d_1 \geq 0,\ d_2 \leq \dfrac{1}{2}d_1$

that is, $\displaystyle (d_1,d_2)$ must lie between the rays$\displaystyle (\alpha,0)$ and $\displaystyle (\alpha,\alpha/2), \alpha \geq 0$.