Finding the impulse response given response to another signal
This exercise is meant to help the student appreciate the fact that if the response $y_1(t)$ of an LTI system to an input $x_1(t)$ is known, then the response to an input
$$x_2(t)=\sum_{k=1}^Ka_kx_1(t-t_k)\tag{1}$$
is given by
$$y_2(t)=\sum_{k=1}^Ka_ky_1(t-t_k)\tag{2}$$
which is a direct consequence of linearity and time-invariance.
Consequently, if an input signal can be expressed in the form given by $(1)$, it is not necessary to compute the system's impulse response in order to determine the output signal.
Also note that this exercise is almost a copy of problem P3.8 in this MIT Signals and Systems problem set. Only the input signals have been changed.
It now appears that the given signal $x_2(t)$ can't be expressed in the form $(1)$, whereas signal $x_3(t)$ can. I believe that this a mistake in the exercise, because as far as I can see, any other method to determine the solution is beyond what can be expected from a student at this level.