what is meant here?

A

allegansveritatem

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I came across this today and could not quite figure it out. The statement in question is, to paraphrase: Since the squared expression is always going to be positive it can be excluded from the sign diagram because it will have no effect on the sign of the quotient. Here is the text:
11766

It seems to me that the squared term's positive sign has a lot to do with the sign of the quotient. I mean, if the term is the only term in the numerator then the denominator has to be positive to get a positive quotient and it has to be negative to get a negative quotient. So...what is the author talking about here?
 
Yes, so the sign of the fraction depends only on the sign of the quantity in the denominator. Once it is known that the numerator is positive for all x, only the sign of the denominator is relevant.
 
Suppose the numerator of a fraction is positive. So the denominator is either positive or negative. Fair enough?

If we divide a positive by a positive we get a positive, which is the sign of the denominator.
If we divide a positive by a negative we get a negative, which is the sign of the denominator.
So whatever we divide the positive number by the sign of the fraction will always be the sign of the denominator!
 
Jomo said:
Suppose the numerator of a fraction is positive. So the denominator is either positive or negative. Fair enough?

If we divide a positive by a positive we get a positive, which is the sign of the denominator.
If we divide a positive by a negative we get a negative, which is the sign of the denominator.
So whatever we divide the positive number by the sign of the fraction will always be the sign of the denominator!
Click to expand...
Yes, I see that...but why is the squared term excluded from the sign diagram? Or is the sign diagram's only job to ....what? My understanding of it is: The diagram helps give a clear idea of the intervals that are relevant to the quadratic inequality being processed or solved. No?
 
HallsofIvy said:
Yes, so the sign of the fraction depends only on the sign of the quantity in the denominator. Once it is known that the numerator is positive for all x, only the sign of the denominator is relevant.
Click to expand...
Yes, but why is it excluded from the sign diagram? I must have a fuzzy idea of what the function of the sign diagram is. Seems to me it should included all the solutions to the quadratic inequality in question.
 
I guess I have a confused notion of what the sign diagram is supposed to do. I will have to study the concept again. The whole subject of inequalities stands on shaky ground in the dim world of my understanding.
 
The sign diagram is just a tool. It is used to determine the sign of the function on each interval.

You are welcome to include a line for (2x+1)^2; it will show a + in every column, which will have no effect on the final signs, but if you feel better including it, you will not be wrong.

You could even include two separate lines for (2x+1), which would be the same, so that you either have two +'s or two -'s in the same column, and again the final result will not be affected.

Often we teach the most efficient way to do something, forgetting that students won't automatically see what we see and understand why some details can be skipped over. Do what you need to do in order to understand it; understanding is more important than just doing what you're shown.
 
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