Problem
ALG-B2-M01-P005 Products in the right order
#5
★★★☆☆ Level 3 of 5
Let \(u>v>0\) and \(p>q>0\). Prove that \(up+vq>uq+vp\).
Hint 1. Subtract the right side from the left side.
Hint 2. The difference should factor into two positive factors.
We have \(up+vq-uq-vp=u(p-q)-v(p-q)=(u-v)(p-q)\). Both factors are positive, so the difference is positive.
Module training problem. Method tags: ordering, product-estimate.