MHB What happens when dividing by a negative in an inequality?

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Dividing by a negative in an inequality requires flipping the inequality signs, which is a crucial step in solving such expressions. The discussion highlights the example of the inequality -10 < 2x + 3 < -1, which simplifies to -13 < 2x < -4, and ultimately to -13/2 < x < -2. Participants emphasize the importance of recognizing this sign change when rearranging inequalities, particularly when dividing by a negative coefficient. One method is noted as more efficient, but the preference for clarity in presentation is also acknowledged. Understanding the implications of dividing by a negative is essential for accurate inequality manipulation.
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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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