Solve {l}{|x+y|leq1}{|x-y|leq2}

$$\left. \begin{array} { l } { | x + y | \leq 1 } \\ { | x - y | \leq 2 } \end{array} \right.$$

Evaluate (complex solution)

$$|x+y|\leq 1,\ |x-y|\leq 2$$

Solve for x

$$\left\{\begin{matrix}x\in \begin{bmatrix}y-2,1-y\end{bmatrix}\text{, }&y>\frac{1}{2}\text{ and }y<\frac{3}{2}\\x=y+2\text{, }&y=-\frac{3}{2}\\x\in \begin{bmatrix}-\left(y+1\right),y+2\end{bmatrix}\text{, }&y>-\frac{3}{2}\text{ and }y<-\frac{1}{2}\\x\in \begin{bmatrix}-\left(y+1\right),1-y\end{bmatrix}\text{, }&|y|\leq \frac{1}{2}\\x=1-y\text{, }&y=\frac{3}{2}\end{matrix}\right.$$

Solve for y

$$\left\{\begin{matrix}y\in \begin{bmatrix}-\left(x+1\right),x+2\end{bmatrix}\text{, }&x>-\frac{3}{2}\text{ and }x<-\frac{1}{2}\\y=1-x\text{, }&x=\frac{3}{2}\\y\in \begin{bmatrix}x-2,1-x\end{bmatrix}\text{, }&x>\frac{1}{2}\text{ and }x<\frac{3}{2}\\y\in \begin{bmatrix}-\left(x+1\right),1-x\end{bmatrix}\text{, }&|x|\leq \frac{1}{2}\\y=x+2\text{, }&x=-\frac{3}{2}\end{matrix}\right.$$