# Consider the system { <mtable columnalign="left left" rowspacing=".2em" columnspacin

Consider the system
$\left\{\begin{array}{l}1=A+B=C+D\\ B\ge C\end{array}$
with $A,B,C,D$ positive.
Does the system imply that $A\le D$?
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Elijah Benjamin
Yes.
The $1=A+B$ has nothing to do with it, but if $A+B=C+D$, and $B\ge C$, then $C-B\le 0$, and so
$A=C+D-B=D+C-B\le D+0=D$

Esmeralda Lane
Yes. Suppose for the sake of contradiction that $A>D$. We know $B\ge C$, so add these two inequalities, giving $A+B>C+D$, a contradiction.
Hence $A\le D$