change of variables, region with infinity

Question

I've change of variables integration problem, given the integrals in variables $x_1$ and $x_2$ I need to change the variables to $y_1$ and $y_2$ with transformation equations: $y_1 = (x_1−x_2)/2\;, y_2 = x_2$ or, equivalently, $x_1 = 2y_1+y_2\;, x_2=y_2$. The region or set using $x_1$, and $x_2$ coordinates is therefore, $S = \{(x_1, x_2) : 0 < x_1 < \infty, 0 < x_2 < \infty\} $ The book says that with the above given transformation equations, $S$ is mapped to the new set: $T = \{(y_1, y_2) : −2y_1 < y_2\;, 0 < y_2 < \infty, −\infty < y_1 < \infty\}$. I understand the $−2y_1 < y_2$ and $0 < y_2 < \infty$ part but couldn't figure out why and how $y_1$ is $−\infty < y_1 < \infty$. Is there a simple way to work with inequalities?

Answer 1

$x_1,x_2$ can be made arbitrarily far apart from each other. For instance, let $x_1=1$ and $x_2=2\cdot 10^{10}+1$; then $y_1=-10^{10}$. Do you see the point? In this way we can make $y_1$ arbitrarily large in either negative or positive direction, so yes, $-\infty\lt y_1\lt\infty$ is the correct region.