Postgres inet array contains a single ip
While McNets's solutions would also work, it's possibly the easiest to use ANY with an operator, in this case equality:

SELECT '10.10.10.10' = ANY ('{10.10.10.10, 10.10.10.20}'::inet[]);


The background is that an inet array is no different from any other array type in this regard - ANY will work with any boolean-returning operator that is defined on the base type.

The above applies to equality of single IP addresses.  If you need to match a single IP against a range, you'll need the &lt;&lt;= operator instead of =:

SELECT '10.10.10.10'::inet &lt;&lt;= ANY ('{10.10.10.10, 10.10.10.20}'::inet[]);

 ?column? 
──────────
 t

SELECT '10.10.10.10'::inet &lt;&lt;= ANY ('{10.10.10.0/28, 10.10.10.20}'::inet[]);

 ?column? 
──────────
 t


In your case, the solution would look like 

WHERE '192.168.1.5' &lt;&lt;= ANY ("ipAddresses")


The performance of this can be poor for the lack of really efficient indexing.  To address this, see the suggestion in McNets's answer.