Homework inequality -- Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

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Homework Help Overview

The problem involves proving the inequality (a+1)(b+1)(c+1)(d+1) < 8(abcd+1) for variables a, b, c, and d, all greater than 1. The context is within the subject area of inequalities in algebra.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various approaches, including the use of inequalities like Cauchy-Schwarz (C-S) and the Arithmetic Mean-Geometric Mean (AM-GM) inequalities. There is also a suggestion to explore induction as a potential method.

Discussion Status

The discussion is ongoing, with participants providing leads and partial results. Some have attempted to break down the problem into smaller parts, while others are exploring different methods and transformations. There is a collaborative effort to guide each other towards a solution without reaching a consensus yet.

Contextual Notes

Participants note the constraints of the problem, specifically that all variables must be greater than 1, which may influence the approaches taken. There is also mention of the problem being a test question for the original poster's daughter.

ssd
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Homework Statement


For a,b,c,d >1, Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)

Homework Equations


How to show this?

The Attempt at a Solution


I could show for two variables, (a+1)(b+1)<2(ab+1). Tried C-S, AM-GM inequalities in different form and variable transformations. But still no result. It's my daughters test question. Any help is appreciated.
 
Last edited:
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Can this be done by induction?
 
(a+1)(b+1)<2(ab+1)
(d+1)(c+1)<2(dc+1)

(a+1)(b+1)(d+1)(c+1)< 4(ab+1)(dc+1)

Can you complete now ?
 
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Likes   Reactions: Vassili sansford
ssd, Buffu has given a nice lead. Recognize that ##ab>1## and ##cd>1##. So replace these in your expressions for ##a## and ##b##
 
Thanks a lot to both of you.
 

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