Problem
ALG-B2-M04-P001 One incorrect swap
#1
★★☆☆☆ Level 2 of 5
Let \(a\le b\) and \(x\le y\). Prove that \(ax+by\ge ay+bx\).
Hint. Move everything to one side and factor two terms.
We have \(ax+by-ay-bx=(b-a)(y-x)\). Both factors are nonnegative, so the difference is nonnegative.
Basic one-swap problem. Emphasize that this swap is the local engine of the rearrangement inequality.