Translation:I live in the suburbs of Cairo in Egypt.
ضاحية = suburb (singular)
ضواحي =suburbs (plural)
It is an Idaafa construction:
showing the suburbs belong to Cairo...suburbs of Cairo, and therefore definite. Cairo is a proper name, so it would make the idaafa definite even if it were a city name that did not include the definite article as part of its name.
ضواحي نو يورك = the suburbs of new york