LESS THAN STANDARD SEPARATION BETWEEN 2 ACR. OPERATIONAL ERROR.

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I WAS WORKING THE MEM HI SECTOR; TRAINERS; WHEN I ASKED TO HOLD MEM INBNDS. I HAD ABOUT 4 IN HOLDING AND I HAD AN ACFT LNDG NASHVILLE THAT I NEEDED TO DES. I VECTORED AN ARR (ACR X) TO MEM (THAT WAS NOT YET HOLDING) BEHIND THE NASHVILLE INBND; AND DESCENDED THE NASHVILLE INBND. I RECEIVED WORD THAT THE ARRS TO MEM COULD BE ACCEPTED BY LOW ALT SECTOR. I GAVE ALL MY ATTN TO THE INBNDS TO MEM (HOLDING) AND FORGOT TO TURN BACK ACR X; THE MEM INBND; ON THE VECTOR BEHIND THE NASHVILLE ARR. I THEN WAS TOLD BY MY TRAINER TO DO SOMETHING WITH MY MEM ARR (X) ON A VECTOR; WHO WAS NOW HEADED TOWARD ACR Y; AN OVERFLT DOWN J29. I TURNED BOTH ACFT AND DESCENDED ACR X. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING: REPORTER GAVE YRS IN ATC. DESCRIBED THE ACFT INVOLVED IN LTSS. SHE TOLD ME THE LOCATION OF CONFLICT. FAC DID REVIEW AS OPERROR AND CALLED IT A 'MINOR/MINOR' WITH 4.8 LAT AND 200' VERT SEP. FLT CREWS DID NOT RPT SEEING OTHER ACFT. AFTER CONFLICT ALERT ACTIVATED; RPTR TURNED BOTH ACFT AND GAVE ACR X DES. HAD TO CALL ACFT ABOUT 3 TIMES BEFORE ACR X RECEIVED AVOIDANCE INSTRUCTIONS. THINKS FLT CONDITIONS WERE VMC AT FL310.