I want to linearize the following disjunctive form.
$$\left[\begin{gathered}w_{1}\\x \geq a\end{gathered}\right] \vee \left[\begin{gathered}w_{2}\\x \geq b\end{gathered}\right]$$
where $w_1$ and $w_2$ are binary and $x$ is a free variable $\in \{ -2, \cdots, 10\}$ and $\{ -2 \lt a \lt b \lt 10 \}$. I think one possible way is as follows:
$$\left[\begin{gathered}(w_{1}=1) \land (x \geq a) \end{gathered}\right] \vee \left[\begin{gathered}(w_{2}=1) \land (x \geq b)\end{gathered}\right]$$
Then:
$$(w_{1} \lor w_{2}) \bigwedge (w_{1} \implies x \geq a) \bigwedge (w_{2} \implies x \geq b)$$
I would like to know if, is there a way to linearize such a disjunctive form by introducing the new auxiliary variable $z$ for each disjunct? (Also, the answer by Rob is a perfect one).