$$\begin{array}{ll} \text{maximize} & \frac{1}{2} x_1 + x_2\\ \text{subject to} & x_1 + 2x_2 \leq 12\\ & x_1 \leq 4\\ & x_1, x_2 \geq 0\end{array}$$
I know the definition for it to be inconsistent is
$$(Ax \leq b) \vdash (0^{T}x \leq -1)$$
It's consistent otherwise. But I don't see how to apply the definition to the pratical problem.