Understanding Inequalities: Division by Negative Numbers

AI Thread Summary
The discussion focuses on the implications of dividing inequalities by negative numbers, specifically examining the inequality ty <= 1 - sx. It confirms that when t > 0, the inequality translates to y <= 1/t - s/tx, and when t < 0, it becomes y >= 1/t - s/tx. This is based on the principle that multiplying or dividing by a positive number maintains the inequality direction, while doing so with a negative number reverses it. The participants emphasize the importance of understanding these nuances to avoid common mistakes in inequality manipulation. Overall, clarity on these rules is essential for accurate mathematical reasoning.
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So, say, we have the inequality ty <= 1-sx

Does this inequality always translate to y <= 1/t - s/tx (for t > 0)
y >= 1/t - s/tx (for t < 0)? (due to division by divided signs?)

I'm sure this is true, as I've tested it for xy <= 1. But I just want to be sure, since this nuance is definitely easy to miss (and requires a step that I'm usually not accustomed to taking so far).
 
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Yes, If a\le b and c> 0 then ac\le bc while if c< 0, ac\ge bc. If t> 0 then so is 1/t and if t< 0 so is 1/t: dividing by t is the same as multiplying by 1/t.
 

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