As opposed to "V1て V2" presented above (Lesson 20-6), "V1ないで、V2" and "V1(ない)ずに、V2" are used when the action represented by V2 takes place without the action represented by V1. These are also used to indicate that V2 is performed instead of V1. ずに is preceded by the "Vない minus ない" form (Vない). Note that the verb する becomes せずに.
I went to bed without straightening up my room.
My grandfather is watching TV without wearing his glasses.
That person is walking without using an umbrella.
Last night, Ali didn't sleep, but studied.