I get the same Xs on the first pass.

From clue 8 we know that Jimenez is 2 less than Caldwell. That means 39 has to be one of those two, and nobody else can be 39.

So we put the Xs in the 39 row for Gilmore and Owen.

Then look at the intersections between Pre-Calculus and some others. Since Pre-Calculus can only be 33 or 35, then it cannot be 129, Caldwell, or Jimenez. Put the Xs in.

Then start looking at clues 1, 2 and 4.

The right half of clue 1 can only be 37 or Owen. 37 can never be Caldwell, and Owen can never be 39 (we just filled in that X).

So the left side can never be Caldwell or 39. Fill in the Xs that exclude 114 from being 39 or Caldwell and that exclude Biology from being 39 or Caldwell.

Now clue 2 tells us that Herman cannot be 41. Looking at the intersection of Herman and 41, we see that Herman cannot be 39 (clue 5) and 41 cannot be Jimenez (clue 8). This means that 114 cannot be 39 or Jimenez.

We're working on that column under 114, and we're almost there.

Now clue 4 tells us that Jimenez cannot be 120. Looking at the intersection of Jimenez and 120, we see that Jimenez cannot be 114 (we just filled that X) and 120 cannot be Caldwell (clue 3). This means that 37 cannot be 114 or Caldwell. That's just what we needed.

Now look at clue 1. 114 can no longer be 37, so it has to be Owen. Biology has to be 37. Bingo. Two positives out of thin air.

Keep working on that 114 column and revisit all the clues that mention teacher's names. Everything should start falling into place.

If I got that right, it should be a lot easier from here. If I didn't, that's what the reset button is for.