Small Aircraft; High Wing; 1 Eng; Fixed Gear
Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 75; Flight Crew Total 400; Flight Crew Type 400
Communication Breakdown; Situational Awareness
ATC Issue All Types; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Environment - Non Weather Related; Human Factors
We were climbing via the North Town Three Departure. Around 3;000 feet we contacted departure. ATC (Las Vegas Departure) gave us a new altimeter setting from 30.03 to 30.00 and then proceeded with the following clearance. Aircraft X; fly straight out; climb maintain 9;000 feet. Our current altitude was approximately 3;800 feet before reaching the LAS R-313 so we were still heading westerly towards terrain and not established on the LAS R313 outbound. Student noted how strange it was for ATC to vector us this low and close to terrain. Moment later; ATC gave us a low altitude alert and suggested a right turn to 090. Then ATC proceeded about a possible ATC deviation and to call a number.Confusion between what ATC meant to say and what pilots actually did. The clearance 'fly straight out' was filled with ambiguity (was still flying westerly and not heading northwest on published procedure and was perceived as a vector for climb. Before pilots had a chance to request clarification; low altitude alert was issued.
A pilot and instructor were confused by the updated clearance provided by ATC that appeared to direct them towards rising terrain. Before the crew could question the clearance; ATC provided an low altitude alert along with notice of a possible pilot deviation.