Two alternative approaches:

1. Rewrite the primal problem in standard form by replacing equality with two inequalities and multiplying both sides of $\le$ inequalities by $-1$ to reverse the sense to $\ge$.
2. For equality constraints in the primal, use free variables in the dual, and for $\le$ constraints in the primal, use $\le 0$ variables. The second approach is simpler.

Two alternative approaches:

1. Rewrite the primal problem in standard form by replacing equality with two inequalities and multiplying both sides of $\le$ inequalities by $-1$ to reverse the sense to $\ge$.
2. For equality constraints in the primal, use free variables in the dual, and for $\le$ constraints in the primal, use $\le 0$ variables. The second approach is simpler. See this paper.