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[M2] Please help inequality proof

Please help inequality proof

Could someone please help me with this proof?

I did the part (a) and (b) and half of the part (c) (I attached my working out for the half I did), but not sure how to prove the half which I circled.
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The working out for half of part (c):

[ 本帖最後由 妙妙貓 於 2022-4-3 07:46 PM 編輯 ]

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Yeah, it needs some kind of intuition (and basically bang you head against the wall until you find it lol). Although this one isn't too hard to figure it out.

Here's my thinking process:
The 9 on the LHS looks familiar, isn't it? We've seen that in (b)!
So maybe we want to use that inequality somehow, i.e. replace a,b,c with something else.
If you get confused, maybe try rewriting the second inequality in (b) with x,y,z instead of a,b,c, and then replace the x,y,z with something.

Solution, albeit a _very_ short one: https://www.docdroid.net/KqcrGNC/ineq-pdf

Note: Beware! Your work doesn't end here.

Mild de-brief: I think inequality problems are just hard because "it needs intuition", but I think often times the proof will work itself out when you (i) work backwards; and (ii) review what you are given/proved, and spot similarities. If you look at HKALE Pure Math inequality problems, the substitutions are wild, but often times you can work it out just by doing the two things I've mentioned.

[ 本帖最後由 風之男 於 2022-4-3 08:57 PM 編輯 ]

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Thank you very much for your sincere help and tips. I was able to successfully solve the inequality with your prompt.

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