I want to know how is inorganic phosphate able to inhibit alkaline phosphatase.
When alkaline phosphatase dephosphorylates a substrate one of the intermediates in the reaction pathway is a phosphorylated form of the enzyme in which the phosphate from the substrate has been transferred to the enzyme. This phosphate is subsequently lost by hydrolysis. Under certain conditions alkaline phosphatase will actually catalyse a reverse reaction - the addition of inorganic phosphate on to alcohol substrates, and this also proceeds via the phospho-enzyme intermediate, but proceeds in the opposite direction.
This means that phosphate will inhibit the enzyme as a phosphatase via the formation of the phospho-enzyme, thus blocking the active site to substrates for dephosphorylation.
Reaction mechanism of alkaline phosphatase based on crystal structures: Two-metal ion catalysis Kim, EE & Harold W. Wyckoff, HW 1991 Reaction mechanism of alkaline phosphatase based on crystal structures: Two-metal ion catalysis. J Mol Biol. 218: 449-464