MySQL Join Where Not Exists

I'd probably use a LEFT JOIN, which will return rows even if there's no match, and then you can select only the rows with no match by checking for NULLs.

So, something like:

SELECT V.*
FROM voter V LEFT JOIN elimination E ON V.id = E.voter_id
WHERE E.voter_id IS NULL

Whether that's more or less efficient than using a subquery depends on optimization, indexes, whether its possible to have more than one elimination per voter, etc.

I'd use a 'where not exists' -- exactly as you suggest in your title:

SELECT `voter`.`ID`, `voter`.`Last_Name`, `voter`.`First_Name`,
       `voter`.`Middle_Name`, `voter`.`Age`, `voter`.`Sex`,
       `voter`.`Party`, `voter`.`Demo`, `voter`.`PV`,
       `household`.`Address`, `household`.`City`, `household`.`Zip`
FROM (`voter`)
JOIN `household` ON `voter`.`House_ID`=`household`.`id`
WHERE `CT` = '5'
AND `Precnum` = 'CTY3'
AND  `Last_Name`  LIKE '%Cumbee%'
AND  `First_Name`  LIKE '%John%'

AND NOT EXISTS (
  SELECT * FROM `elimination`
   WHERE `elimination`.`voter_id` = `voter`.`ID`
)

ORDER BY `Last_Name` ASC
LIMIT 30

That may be marginally faster than doing a left join (of course, depending on your indexes, cardinality of your tables, etc), and is almost certainly much faster than using IN.