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If I have 1/A > B, then should A > 1/B or A < 1/B? I feel that it should be swapped b/c of the fraction/reciprocal, but I can't quite exactly recall the reason.

Thanks in advance for your help.

2 Answers 2

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It depends on whether $A$ and $B$ are positive or negative. If they are both positive then you are multiplying or dividing by a positive number, but then in a sense the direction reverses when you swop the two sides:

$\frac{1}{|A|} \gt |B| \iff 1 \gt |A|\,|B| \iff \frac{1}{|B|} \gt |A| \iff |A| \lt \frac{1}{|B|}.$

If they are both negative the same thing happens though with a slight difference in the middle

$\frac{1}{-|A|} \gt -|B| \iff 1 \lt (-|A|)\,(-|B|) \iff \frac{1}{-|B|} \gt -|A| \iff -|A| \lt \frac{1}{-|B|}.$

If $A$ is positive and $B$ is negative then you must have the unreversed:

$\frac{1}{|A|} \gt 0 \gt -|B| \text{ and } |A| \gt 0 \gt \frac{1}{-|B|},$

and similarly if $A$ is negative and $B$ is positive you must have

$\frac{1}{-|A|} \lt 0 \lt |B| \text{ and } -|A| \lt 0 \lt \frac{1}{|B|}.$

If $A$ or $B$ are zero, then you face an expression with division by zero.

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It is $A < 1/B$, this is because you swap $A$ and $B$ and the inequality sign stays intact. All this is valid only when both $A,B >0$.