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Clearly, one can solve a system of linear equations by adding the constituent equations together or subtracting them from one another (via the so-called "elimination" method), but can one do the same with a system of linear inequalities. In other words, is an operation of the following form valid?

   x > y +  a > b     Therefore, x + a > y + b. 

Many thanks!

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Yes, it is valid. You can prove it's validness in two steps: First, if $a then $a+x.
Secondly, if $x then $0, by adding $-x$ to both sides. Hence $b+x.
Finally, $a+x

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