The LinearMultivariateSystem command solves a system of linear inequalities with respect to the given variables.

The LinearMultivariateSystem command returns a region describing the values of the variables or a piecewise function of regions depending on parameters.

A region is a union (set) of subregions, each of which is represented by a list. A subregion is a list of intervals. Each interval is a set of inequalities for each variable to solve for. First interval is an interval for the first variable, possibly depending on parameters, second interval is an interval for the second variable, depending on parameters and the first variable, third interval is an interval for the third variable depending on parameters and the first two variables etc. Note that the order of variables is determined by their order in var.

$\mathrm{with}\left(\mathrm{SolveTools}\left[\mathrm{Inequality}\right]\right)\:$

$\mathrm{LinearMultivariateSystem}\left(\left\{y<x-1\&comma;1-x<y\right\}\&comma;\left[x\&comma;y\right]\right)$

$\left\{\left[\left\{1<x\right\}\&comma;\left\{y<x-1\&comma;1-x<y\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{y<x-1\&comma;1-x<y\right\}\&comma;\left[y\&comma;x\right]\right)$

$\left\{\left[\left\{y\le 0\right\}\&comma;\left\{-y+1<x\right\}\right]\&comma;\left[\left\{0<y\right\}\&comma;\left\{1+y<x\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{y<1-x\&comma;x-1<y\&comma;-3x-1<y\right\}\&comma;\left[x\&comma;y\right]\right)$

$\left\{\left[\left\{x\le 0\&comma;\mathrm{-1}<x\right\}\&comma;\left\{y<1-x\&comma;-3x-1<y\right\}\right]\&comma;\left[\left\{0<x\&comma;x<1\right\}\&comma;\left\{y<1-x\&comma;x-1<y\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{y<1-x\&comma;x-1<y\&comma;-3x-1<y\right\}\&comma;\left[y\&comma;x\right]\right)$

$\left\{\left[\left\{y\le 0\&comma;\mathrm{-1}<y\right\}\&comma;\left\{x<1+y\&comma;-\frac{1}{3}-\frac{y}{3}<x\right\}\right]\&comma;\left[\left\{0<y\&comma;y<2\right\}\&comma;\left\{x<-y+1\&comma;-\frac{1}{3}-\frac{y}{3}<x\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{x<3\&comma;y<3\&comma;y+x<4\&comma;-x<3\&comma;-y<3\&comma;x-y<4\&comma;y-x<4\&comma;-x-y<4\right\}\&comma;\left[x\&comma;y\right]\right)$

$\left\{\left[\left\{x=0\right\}\&comma;\left\{\mathrm{-3}<y\&comma;y<3\right\}\right]\&comma;\left[\left\{1\le x\&comma;x<3\right\}\&comma;\left\{y<4-x\&comma;-4+x<y\right\}\right]\&comma;\left[\left\{x\le \mathrm{-1}\&comma;\mathrm{-3}<x\right\}\&comma;\left\{y<4+x\&comma;-4-x<y\right\}\right]\&comma;\left[\left\{\mathrm{-1}<x\&comma;x<0\right\}\&comma;\left\{\mathrm{-3}<y\&comma;y<3\right\}\right]\&comma;\left[\left\{0<x\&comma;x<1\right\}\&comma;\left\{\mathrm{-3}<y\&comma;y<3\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{x+y<1\right\}\&comma;\left[x\&comma;y\right]\right)$

$\left\{\left[\left\{x\right\}\&comma;\left\{y<1-x\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{x+y+z<1\&comma;x-y+z<2\right\}\&comma;\left[y\&comma;x\&comma;z\right]\right)$

$\left\{\left[\left\{y\le -\frac{1}{2}\right\}\&comma;\left\{x\right\}\&comma;\left\{z<2-x+y\right\}\right]\&comma;\left[\left\{-\frac{1}{2}<y\right\}\&comma;\left\{x\right\}\&comma;\left\{z<1-y-x\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{0<z\&comma;x+y+z<1\&comma;-x+y+z<1\&comma;x-y+z<1\&comma;-x-y+z<1\right\}\&comma;\left[x\&comma;y\&comma;z\right]\right)$

$\left\{\left[\left\{\frac{1}{2}\le x\&comma;x<1\right\}\&comma;\left\{y<0\&comma;x-1<y\right\}\&comma;\left\{0<z\&comma;z<1-x+y\right\}\right]\&comma;\left[\left\{x\le 0\&comma;\mathrm{-1}<x\right\}\&comma;\left\{0\le y\&comma;y<x+1\right\}\&comma;\left\{0<z\&comma;z<1+x-y\right\}\right]\&comma;\left[\left\{x\le 0\&comma;\mathrm{-1}<x\right\}\&comma;\left\{y<0\&comma;-1-x<y\right\}\&comma;\left\{0<z\&comma;z<1+y+x\right\}\right]\&comma;\left[\left\{0<x\&comma;x<1\right\}\&comma;\left\{0\le y\&comma;y<1-x\right\}\&comma;\left\{0<z\&comma;z<1-y-x\right\}\right]\&comma;\left[\left\{0<x\&comma;x<\frac{1}{2}\right\}\&comma;\left\{y\le -x\&comma;x-1<y\right\}\&comma;\left\{0<z\&comma;z<1-x+y\right\}\right]\&comma;\left[\left\{0<x\&comma;x<\frac{1}{2}\right\}\&comma;\left\{y<0\&comma;-x<y\right\}\&comma;\left\{0<z\&comma;z<1-x+y\right\}\right]\right\}$

$\mathrm{LinearMultivariateSystem}\left(\left\{y<1-x\&comma;ax+y<1\right\}\&comma;\left[x\&comma;y\right]\right)$

$\left\{\begin{array}{cc}\left\{\left[\left\{0\le x\right\}\&comma;\left\{y<-ax+1\right\}\right]\&comma;\left[\left\{x<0\right\}\&comma;\left\{y<1-x\right\}\right]\right\}& 1<a\\ \left\{\left[\left\{x\le 0\right\}\&comma;\left\{y<-ax+1\right\}\right]\&comma;\left[\left\{0<x\right\}\&comma;\left\{y<1-x\right\}\right]\right\}& a<1\\ \left\{\left[\left\{x\right\}\&comma;\left\{y<1-x\right\}\right]\right\}& a=1\end{array}\right.$