\begin{align*} \min_{x_1,x_2} \quad - 4x_1 & + 2x_2 \\ \text{subject to} \quad - x_1 & + x_2 \geq 2 \\ x_1 & - x_2 \geq 1 \\ & x_1,x_2 \geq 0 \end{align*}
\begin{align*}\text{Introducing four slack variables, $a,b,c,d\geq 0$}\\ L(x_1,x_2,a,b,c,d)=-4x_1+2x_2+a(2+x_1-x_2)+b(1+x_2-x_1)+cx_1-dx_2\\ =2a+b+x_1(-4+a-b+c)+x_2(2-a+b-d)\\ \max_{a,b,c,d}\min_{x_1,x_2}(2a+b+x_1(-4+a-b+c)+x_2(2-a+b-d))\\ \text{Since $x_1 \geq 0, x_2 \geq 0$}\\ a-b+c \geq 4\\ a-b+d \leq 2\\ \end{align*}
So finally we have, \begin{align*} \max_{a,b,c,d} \quad 2a+b \\ \text{subject to} \quad a-b+c\geq4\\ a-b+d \leq 2 \\ a,b,c,d \geq 0 \\ \end{align*}
If it is right, where else should I add to the derivation process in order to make it more formal?