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I’m trying to prove that $F=0.15w+0.5x+0.15y+0.2z$ is greater than or equal to $G=0.25w+0.25x+0.25y+0.25z$. I also know some additional information:

$w\ge x\ge y\ge z$

$x=0.6w+0.4z$

$y=0.2w+0.8z$

So far, using some substitution, I’ve managed to work the problem down to this: $F=0.48w+0.52z$ and $G=0.45w+0.55z$. This is where I am stuck. I’m not sure how to prove $F\ge G$ from here. Any help would be appreciated, and links to resources where I could learn more about how to solve this particular problem are encouraged. Thanks!

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  • $\begingroup$ Do we know if $w \ge 0$ or $z \ge 0$? $\endgroup$
    – Daniel P
    Commented Sep 7, 2022 at 21:39
  • $\begingroup$ We don't know if either is greater than or equal to zero in this case. $\endgroup$
    – Stochastis
    Commented Sep 7, 2022 at 21:42

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Based on what you've already done, $F-G=0.03(w-z)$, which is non-negative because $w \geq z$.

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  • $\begingroup$ Aha! That makes sense. Thank you. Does this technique have a name, or do you have any links to resources about this method? I'd like to look it up and try some practice problems so I can remember this for future problems. $\endgroup$
    – Stochastis
    Commented Sep 7, 2022 at 21:47
  • $\begingroup$ No name that I know of. It's a pretty standard technique to prove one quantity is greater than another by proving that the difference between them is positive. $\endgroup$
    – Robert Shore
    Commented Sep 7, 2022 at 21:48
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    $\begingroup$ Oh yeah! I didn't even think to try that. Thanks for helping out! $\endgroup$
    – Stochastis
    Commented Sep 7, 2022 at 21:49

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