No, running forward from the back of the plane at 10 MPH while the plane is traveling at a speed just below the speed of sound (765 MPH) will not cause you to break the sound barrier.
The speed of sound, also known as Mach 1, varies with temperature and other atmospheric conditions. However, at typical conditions near the Earth's surface, the speed of sound is approximately 767 MPH or 1,235 kilometers per hour (kph). In this scenario, where the plane is traveling at 765 MPH, it is just below the speed of sound.
When you run forward inside the plane at 10 MPH, your speed relative to the surrounding air is still 10 MPH. It does not get added to the speed of the plane. Therefore, your total speed relative to the ground would be the speed of the plane (765 MPH) plus your running speed (10 MPH), resulting in a total speed of 775 MPH.
However, it's important to note that breaking the sound barrier requires surpassing the speed of sound itself, not just the speed relative to the ground. So, in this case, you would not break the sound barrier since your total speed (775 MPH) is still below the speed of sound (767 MPH in typical conditions).