"Teacher Li is not happy today."
You only need to use 是 when you are describing a noun as another noun. When you are using the structure Noun is Adjective, do not use 是. I’m this case, the adjective is sort of a verb itself. So think of 高兴 as “to be happy” instead of just “happy” in sentences like this.
Here's what the course Tips and Notes have to say on the matter:
When we want to describe something we don’t use the verb “to be”, we have to put something else before an adjective. The most common word used to do this is 很 hěn, which literally means very. It goes between the subject (who or what we are describing) and the adjective (what we are describing it as). Since 很 is used so often like this, it oftentimes doesn’t mean “very”, just a link between a noun and an adjective.
(I'm guessing maybe they're talking about a certain level of formality when they say "we have to put something else before an adjective" and colloquially it's ok to just not put anything?)
Another thing to note about 不 is that when we want to say “I am not happy” or “I am not tall”, we don’t need the 很 anymore. 我很高兴 -> 我不高兴 (not 我不很高兴)
I think, then, that the reason there's no 很 here is because it's a negative sentence.
Why can't it be 李老师今天不很高兴 ? is it to do with the 很 ?
I get that in a lot of situations it replaces the 是 and expresses [very; a lot] in addition to [is; are; am].
So why can't it have 很 as it does in previous sentences? does it make more sense or is this another problem?
if it's gotta do with the 很 can I not just express that as 真 if it really wants to express the sentence as [very; a lot]