What are the possible values of a in the inequality a < 1/a < b < 1/b?

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Homework Help Overview

The problem involves inequalities between two nonzero real numbers, a and b, specifically the condition a < 1/a < b < 1/b. Participants are exploring the implications of this inequality and what it suggests about the possible values of a.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants are considering the implications of the inequality a < 1/a and discussing the possible intervals for a. There is mention of needing to prove that a < 0 and subsequently a < -1. Some participants are questioning the completeness of the inequalities and the assumptions made.

Discussion Status

The discussion is ongoing, with participants providing hints and prompting further exploration of the implications of the inequalities. There is no explicit consensus, but some guidance has been offered regarding the interpretation of the inequalities.

Contextual Notes

There are indications that the problem may have been initially misrepresented or that there are additional complexities in the inequalities that need to be addressed. Participants are encouraged to clarify their assumptions and the definitions involved.

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Homework Statement



I originally made this thread for something else, but I have another problem that I need help with.

Suppose a and b are nonzero real numbers. Prove that if a < 1/a < b < 1/b then a < -1.

Homework Equations



A hint was given for the problem: Assume a < 1/a < b < 1/b. Now prove that a < 0, and then use this fact to prove that a < -1.


Thanks
 
Last edited:
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For b, you're missing the inequality in the triangle inequality.
 
new problem bump
 
It's not like threads go in the landfill and pollute kindergarten playgrounds after we're done with them. No need to recycle. Make a new thread when you have a new problem.

For this problem, first think about what it means when a&lt;\frac1a. What values could a have? You should identify two possibilities (two open intervals where a could be). The second part of the inequality will let you narrow it down to one.
 

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