What happens when dividing by a negative in an inequality?

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Discussion Overview

The discussion revolves around the implications of dividing by a negative number in the context of inequalities. Participants explore different methods of rearranging inequalities and the effects on the direction of the inequality signs.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents an example of an inequality and its transformation through steps, leading to a final expression for x.
  • Another participant introduces a general form of an inequality and discusses the rearrangement process, emphasizing the necessity to flip the inequality signs when dividing by a negative.
  • There is a discussion about the efficiency of different methods for handling inequalities, with one participant expressing a preference for a particular format of inequalities over another.

Areas of Agreement / Disagreement

Participants express differing preferences for methods of rearranging inequalities and the presentation of results, indicating that multiple views remain on the best approach.

Contextual Notes

Some steps in the mathematical transformations may depend on specific assumptions about the variables involved, and the discussion does not resolve the efficiency of the proposed methods.

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-10 < 2x + 3 < -1
-13 < 2x < -4
-13/2 < x < -2
 
-c < ax + b < -d

Easier to work with (after re-arranging):
c > -(ax + b) > d
 
Wilmer said:
-c < ax + b < -d

Easier to work with (after re-arranging):
c > -(ax + b) > d
But when you get to the step c + b > -ax > d + b you have to divide by -a anyway and the >s flip again. I think it's good in that you get that extra step (to stress the point of what happens when you divide by a negative) but greg1313's method is slightly more efficient.

-Dan
 
topsquark said:
But when you get to the step c + b > -ax > d + b you have to divide by -a anyway and the >s flip again. I think it's good in that you get that extra step (to stress the point of what happens when you divide by a negative) but greg1313's method is slightly more efficient.
-Dan
Ya...agree...BUT li'l ole me prefers ? > ? > ? to ? < ? < ?
A bit like reading left to right instead of backwards!
Anyway, not important...
 

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