A numerical approach.

Suppose the search is for integer solutions. Examining ABC = 4 would imply that the possible values of A, B, and C would factors of 4, that is 1, 2, or 4. Further, either all 3 values are positive or exactly two of them are negative.

Consider values of ABC = 4 and evaluate 3A +2B - C to see if it is equal to 3.

A B C 3A + 2B - C

1 1 4 1 1 -1 -4 5 -1 1 -4 3 -1 -1 4 -1Continue the table for other sets of values where ABC = 4. There should be 24 of them for integer solutions.

Of course, we could explore non-integer values numerically but there would not be a table as above with a finite number of rows.