The possible values of $$x$$ in the set $$\{1, 5, 13\}$$ for which the mean of eight observations $$5, 8, 3x + 2, 15, 27, 29, 36, 5x - 2$$ equals their median are

Eight observations: $$5, 8, 3x+2, 15, 27, 29, 36, 5x-2$$.

Mean $$= \dfrac{122 + 8x}{8}$$.

$$x = 1$$: values are $$5, 8, 5, 15, 27, 29, 36, 3$$; sorted: $$3, 5, 5, 8, 15, 27, 29, 36$$. Mean $$= 16$$, median $$= 11.5$$. Unequal.

$$x = 5$$: values $$5, 8, 17, 15, 27, 29, 36, 23$$; sorted: $$5, 8, 15, 17, 23, 27, 29, 36$$. Mean $$= 20$$, median $$= 20$$. ✓

$$x = 13$$: values $$5, 8, 41, 15, 27, 29, 36, 63$$; sorted: $$5, 8, 15, 27, 29, 36, 41, 63$$. Mean $$= 28$$, median $$= 28$$. ✓

So $$x \in \{5, 13\}$$.