This is the SDE:
$$dX_t=\operatorname{sign}(X_t+1)dt + dB_t$$
This is a $\mathbb{Q}$-Brownian motion:
$$W_t=B_t -\int_{0}^{t}\operatorname{sign}(B_s+1)ds$$
I've already shown that $B_t$ under measure $\mathbb{Q}$ solves the above SDE.
However, I don't know how to go about proving that this weak solution is unique. The only hint I've received is to transform $X_t$ into a $\mathbb{Q}$-Brownian motion, but I'm a bit lost as to how to do this and where to go from there.