I’m revising for my upcoming Integer Programming exam and I don’t understand an aspect of generating logical inequalities.
In the photo below I don’t understand how the inequalities listed are obtained from constraint $3$ because personally I would multiply constraint $3$ by $-1$ to obtain $2x_{2}+3x_{3}+6x_{4}\geq5$. Then I would set $x_{4}=1$ and rearrange to obtain $2x_{2}+3x_{3}\geq-1$ and then by setting $x_{2}=1$ I obtain $3x_{3}\geq-3$ which to me is feasible and therefore $x_{4}\geq x_{2}$ which is wrong. Could someone explain this mistake to me?
You show that $x_2$ may be 1 when $x_4$ is 1. However, $x_4 \geq x_2$ implies that $x_2=0$ whenever $x_4=0$. You have not shown that implication.