# Let A and B represent two linear inequalities: A : a 1 </msub> x

Let A and B represent two linear inequalities:
$A:{a}_{1}{x}_{1}+...+{a}_{n}{x}_{n}\beta ₯{k}_{1}$
$B:{b}_{1}{x}_{1}+...+{b}_{n}{x}_{n}\beta ₯{k}_{2}$
If A and B is unsatisfiable (does not have solution), does the following hold in general (the conjunction of two inequalities implies the summation of them )? If so, I am looking for a formal proof?
$A\beta §B\phantom{\rule{thickmathspace}{0ex}}\beta Ή\phantom{\rule{thickmathspace}{0ex}}A+B$
${\pi }_{1}{\pi ₯}_{1}+...+{\pi }_{n}{\pi ₯}_{n}\beta ₯\pi 1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\beta §\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\pi }_{1}{\pi ₯}_{1}+...{\pi }_{n}{\pi ₯}_{n}\beta ₯{\pi }_{2}\phantom{\rule{thickmathspace}{0ex}}\beta Ή\phantom{\rule{thickmathspace}{0ex}}{\pi }_{1}{\pi ₯}_{1}+...+{\pi }_{n}{\pi ₯}_{n}+{\pi }_{1}{\pi ₯}_{1}+...{\pi }_{n}{\pi ₯}_{n}\beta ₯{\pi }_{1}+{\pi }_{2}$
and then I would like to generalize the above theorem to summation of several inequalities.
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Carolyn Farmer
It's still false, even with the unsatisfiability assumption.
Consider the inequalities
$\begin{array}{rl}\beta 2x& >2\\ x& >3\end{array}$
Their sum is
$\beta x>5$
i.e., $x<\beta 5$. But $x<\beta 5$ does not imply that $x>3$.
###### Not exactly what youβre looking for?
quorums15lep
I don't see how the linear combination part is relevant. $A\beta ₯{k}_{1},B\beta ₯{k}_{2}\beta A+B\beta ₯{k}_{1}+{k}_{2}$ regardless of where $A$ and $B$come from. This can be seem by
$A\beta ₯{k}_{2}$
$A\beta {k}_{1}\beta ₯0$
$B+\left(A\beta {k}_{1}\right)\beta ₯B\beta ₯{k}_{2}$
$B+A\beta ₯{k}_{1}+{k}_{2}$