How to find the domain of this double integral when figuring out the CDF of $\min(X,Y)$?

Question

We have two random variables such that $$f(x,y)=\begin{cases} e^{-2x-y}, & y\geq -x \\ 0, & \text{otherwise} \end{cases}$$ Now if $t\geq0$ then it's easy to see that the CDF is: $$P(\min(X,Y)\leq t)=1-P(X>t,Y>t)=1-\int_t^\infty\int_t^\infty e^{-2x-y}\,dx\,dy$$ However I don't understand the case of $t<0$ where the double integral is found to be: $$P(\min(X,Y)\leq t)=\int_{-t}^\infty\int_{-x}^te^{-2x-y}\,dy\,dx$$ How did we arrive at these boundaries? Why is $x\geq-t$ and $t\geq y\geq -x$ (the second part is obvious, $y\geq -x$ is always true but why is $t\geq y$?)

Answer 1

$y \geq -x$ is same as $x+y \geq 0$ or $x \geq -y$.

$x\leq t, x \leq t $ and $y \geq -x$ gives $x \geq -y \geq -t$. I hope the rest is clear.