The lac operon is an example of an inducible operon that is also subject to activation in the absence of glucose. The lac operon encodes three structural genes necessary to acquire and process the disaccharide lactose from the environment, breaking it down into the simple sugars glucose and galactose. For the lac operon to be expressed, lactose must be present. This makes sense for the cell because it would be energetically wasteful to create the enzymes to process lactose if lactose was not available.
In the absence of lactose, the lac repressor is bound to the operator region of the lac operon, physically preventing RNA polymerase from transcribing the structural genes. However, when lactose is present, the lactose inside the cell is converted to allolactose. Allolactose serves as an inducer molecule, binding to the repressor and changing its shape so that it is no longer able to bind to the operator DNA. Removal of the repressor in the presence of lactose allows RNA polymerase to move through the operator region and begin transcription of the lac structural genes.
Parker, N., Schneegurt, M., Thi Tu, A.-H., Forster, B. M., & Lister, P. (n.d.). Microbiology. Houston, Texas: OpenStax. Access for free at: https://openstax.org/details/books/microbiology