Boolean variable that captures whether an inequality holds
Add the inequalities

$$\begin{align*}
a_1 x_1 + \dots + a_n x_n &amp;\ge b - M (1-y)\\
a_1 x_1 + \dots + a_n x_n &amp;&lt; b + M y,
\end{align*}$$

where $M$ is chosen sufficiently large.

Why does this work?  If $y=1$, then the first inequality forces $a_1 x_1 + \dots + a_n x_n \ge b$ to hold.  If $y=0$, then the second inequality forces $a_1 x_1 + \dots + a_n x_n &lt; b$ to hold, i.e., $a_1 x_1 + \dots + a_n x_n \ge b$ to be false.

How large does $M$ need to be?  If the $x_i$'s are zero-or-one variables, then it suffices to take $M$ to be at least $\max(b,a_1+\dots+a_n-b+1)$.  In general, if you have upper and lower bounds for each $x_i$, you can derive how large $M$ needs to be.