I made a video some years ago that shows this. It's in a copyright-protected part of my website, so you have to login as "biostudent" and use the password "science". The URL is .http://www2.mtroyal.ca/~tnickle/Animations/Lac-operon.html and you can see this process for yourself.
For the purpose of this exercise, you can ignore positive regulation (anything involving the CAP/cAMP complex). Your task is to figure out whether B-galactosidase and permease (proteins from the lacZ and lacY genes, respectively) are likely to be produced by a cell in an environment where lactose is present or not present. This question mirrors the one in the Sanders & Bowman Genetic Analysis: an integrated approach (1st edition) chapter 14, question 18. Sadly, the question at the back of the textbook has an incorrect example. The one below has been corrected, and I'll explain why this one is correct when I show the answer.
Thanks to my super-amazing daughter, I was able to reproduce this table from the one that we handed out in lecture earlier (she did the typing 'cause my colleague Dr. Bird lost the original file!). Here's an important note: my other tutorials take into account the "polar effect", but this one doesn't. In my other tutorials, having lacZ- means the ribosome falls off before hitting the lacY ORF. In this case, you can assume that you can express permease even if lacZ isn't synthesized.