Are they opposite, or do they mean the same thing?
My teacher gave me an example like...
but I am still having trouble understanding this.
Chinese Language Stack Exchange is a question and answer site for students, teachers, and linguists wanting to discuss the finer points of the Chinese language. It only takes a minute to sign up.Sign up to join this community
Put simply, they have opposite meanings.
你忙得過來嗎？ Can you manage your work?
我忙得過來。 I can manage my work.
我忙不過來。 I have more work than can manage.
2. The construction Vmain +得/不+ Vaux
得 indicates ability in the positive sense (Vmain +得+ Vaux), and 不 in the negative sense (Vmain +不+ Vaux).
Let's consider a simpler example where 得 and 不 are placed within another compound verb, 吃完 (to finish eating). 吃 (to eat) is the main verb, and 完 (to finish) is a verb that is at most auxiliary in nature. Here it helps indicate the extent of completion of the main verb.
我吃完飯。 I finished my meal.
我吃得完飯。 I can/could finish the meal.
我吃不完飯。 I cannot/could not finish the meal.
This compound verb is seldom (dare I say never) used without 得/不 inserted in between, but still one appreciates the invariable structure where the main verb 忙 (to be busy) is followed by the auxiliary verbs 過 and 來 (note nothing is advancing towards the speaker; it is merely a figurative way of saying someone has finished or done being busy. See more explanation of 來 as an auxiliary verb here.)
An example of Vmain +得/不+ 過 is 說得/不過去; if you say something is 說得過去, you mean that it is justifiable, passable (as if you could complete a justification for something).
4. Other Vmain +得/不+ Vaux examples
跟上 (to catch up) 跟得上 跟不上
他跟得上節奏。lit. He can catch up with the rhythm.
寫下 (lit. to write down) 寫得下 寫不下
這裡空間太小了，寫不下。 There is too little space here for me to write.
上, 下, 完, 過去, and 過來 are common auxiliaries.
5. What about 你忙不過來嗎？
The teacher could also have asked in the negative, but in it also lies the assumption or presupposition of the speaker that the student could not finish their work. Asking in the affirmative (which also includes 你忙得過來不？ where 不 acts as a question tag) is in general more polite.