How do I check for linearity for the following piecewise-defined system? The problem at hand:

Where I'm currently stuck:

I'm not entirely sure about how to move on from this point, I'm trying to find the superposition of the responses of the two individual signals so I can later check if that's equal to the response of their (the two signals') superpositions. The intervals in each response involve different signals and thus I'm unable to find the superposition of responses.
First, you are not trying to find the linearity of a signal, but the linearity of a system defined by the transform above between input and output signals.
If you suspect your system is not linear (the different expressions depending on the sign of $x(t)$ is a big red flag), then you only need a counterexample rather than a general proof of linearity.
In this case, if you scale your input by a negative number, the output is not scaled accordingly. That is: $T{\alpha x} \neq \alpha T{x}$ if $\alpha$ is negative.