Which of the following signs is associated with difficult to understand speech as an indicator of depressant use?

The sign associated with difficult-to-understand speech as an indicator of depressant use is thick, slurred speech. This characteristic is often observed in individuals who have consumed depressants, which can affect the central nervous system, leading to impaired motor function and coordination. Slurred speech occurs because depressants slow down brain activity and muscle control involved in speech production, making it difficult for individuals to articulate words clearly. This symptom, along with other indicators of depressant use, can help differentiate someone who is under the influence of these substances.

Other signs, like droopy eyelids and drowsiness, may suggest depressant use as well, but they do not specifically indicate speech issues. Unsteady, staggering movements can also be associated with depressants but are more related to physical coordination rather than speech. Thus, thick, slurred speech stands out as the correct answer in the context of speech-related symptoms of depressant use.