Flight Instructor with student reported a near miss during training maneuvers. Flight Instructor took evasive action to avoid a collision.

Narrative

While flying eastbound and after determining that the area was clear; at about 3;000 ft MSL I pulled the student pilot's throttle; notifying of a simulated engine failure. Student pilot began engine out procedures; identifying a point to simulate an engine out landing. After completing all checklist and memory items; the student pilot set himself on a left downwind for the area of simulated landing. After turning left and establishing the plane on a westbound final approach toward the area of simulated landing the traffic alert system notified of traffic in the area. I acknowledged the alert immediately to identify the location of the aircraft on the moving map. After seeing the aircraft on our left on the moving map; I looked outside our left window to see the aircraft approaching us northbound just off of our left wing; directly perpendicular to our position. The distance shown on the map was 500 ft vertical distance and the aircraft was about 700 ft horizontal. As the instructor; I immediately verbalized 'My controls' and took airplane controls. I decided to maintain a westbound course as the other aircraft did not seem to initiate a right turn or any collision avoidance maneuvers. I raised the left wing to maintain visual; then I initiated climbs immediately after determining that it was safe to do so; in order to maintain a safe amount of separation from the terrain. I used my best assessment of the situation to determine that a southwest course was the safest. The training flight was conducted normally and soon after we proceeded back to the home airport. After reviewing the ADS-B track log it appears the other aircraft made a left turn toward our position.Key takeaways from the situation; visual scanning for traffic is a good way to help ensure the area is clear; but cross-referencing with ADS-B on avionics map can help further situation awareness. Additionally; aircraft maneuvering without ADS-B can increase the chances of an incident; this mid-air collision was able to be evaded due to both aircraft having ADB-S.