For a substance to be called amphoteric, it should be able to react as a base as well as an acid. Hence, when it is reacted with a base, it should act as an acid, and when it is reacted with an acid, it should behave as a base.

The reactions for amphoteric nature of Al₂O₃. x H₂O(s) are

${\text{Al}}_2{\text{O}}_3.x{\text{H}}_2\text{O}\left(\text{s}\right)\stackrel{{\text{OH}}_{\text{aq}}^{-}}{\to }\text{Al}{\left(\text{OH}\right)}_4^{-}aq$ 
                                  Aluminate anion

${\text{Al}}_2{\text{O}}_3.x{\text{H}}_2\text{O}\left(\text{s}\right)\stackrel{{\text{H}}_{\text{aq}}^{+}}{\to }{\left[\text{Al}{\left({\text{H}}_2\text{O}\right)}_6\right]}_{\text{aq}}^{3+}$.

[Al(H₂O)₆]³⁺ is octahedral complex ion

[Al(OH)₄]^-1 tetrahedral at pH = 13