To say, "Open a store" in Korean, do you say, "가계를 벌리다" or "가계를 벌이다"?

ANSWER: You say "가계를 벌이다." 

Both 벌리다 and 벌이다 can mean "to open," but 벌리다 is used in the sense of "spreading something open," like one's mouth or one's legs, while 벌이다 is used in the sense of "to open or start [a business]." The 벌 is both 벌리다 and 벌이다 is pronounced with a long vowel sound, so 벌리다 is pronounced as /버얼리다/, and 벌이다 is pronounced as /버어리다/.

Since 벌리다 and 벌이다 are so similar, Koreans sometimes confuse them, including the Koreans who compiled and edited my 1998 edition of Dong-a's Prime Korean-English Dictionary, where they used the following example sentence under the word 벌리다:

그는 식료품 가계를 벌렸다 (He opened a grocery store).

Yet, in the same dictionary, there is the following example sentence under the word 벌이다:

가계을 벌이다 (Open a store).

 So, in 1998, even Koreans who were compiling dictionaries were confusing the words 벌리다 and 벌이다.

1998 Edition of Dong-a's Prime Korean-English Dictionary

Also, notice in the above dictionary that under the definition for 벌리다, they included the meaning "[늘어놓다] to lay out; arrange; spread; display"; even though that is actually the definition for 벌이다. So, no wonder Koreans were confusing 벌리다 and 벌이다 back in 1998. However, that mistake seems to have been corrected by 2014.

In my 2014 edition of Minjung's Essence Korean-English Dictionary, the meanings of 벌리다 and 벌이다 seem to have been defined correctly. 

2014 Edition of Minjung's Essence Korean-English Dictionary