# Proof of an Inequality

Hello,

I am given that β > α, which can be written as β - α > 0. What justification would I have to use in order to conclude that α - 2β < 0, given that the preceding propositions are true? Could someone possibly help me?

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Ray Vickson
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Dearly Missed
Hello,

I am given that β > α, which can be written as β - α > 0. What justification would I have to use in order to conclude that α - 2β < 0, given that the preceding propositions are true? Could someone possibly help me?
This is not necessarily true: if α = -3 and β = -2 we have α < β but 2β < α.

Anyway, you are supposed to show your work.

This isn't actually a homework problem. It was something I came across and found curious, when I was drawing the slopes fields of a differential equation.

I also forgot to mention that alpha and beta are both positive.

Note: I can not provide my attempt at solving this problem, for I never did attempt at solving it; the reason being, that I do not know enough about inequalities to solve this problem. As I said, this is not a homework problem, it is a side-tracking.

PeroK
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Gold Member
So, to re-phrase the question. You have two positive numbers. You start with the smaller one and subtract twice the larger one and you want to prove that the answer is less than zero?

PeroK, if I read your sentences correctly, yes that is what I would like to prove, rather, I would like to know how to prove, as I have never done any proves involving inequalities.

α > 0 and β > 0, and β > α, and I would like to know if we can conclude that α - 2β < 0.

PeroK
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Gold Member
PeroK, if I read your sentences correctly, yes that is what I would like to prove, rather, I would like to know how to prove, as I have never done any proves involving inequalities.

α > 0 and β > 0, and β > α, and I would like to know if we can conclude that α - 2β < 0.
Well, if you had $α in the bank. And you withdrew$β. Then you withdrew \$β again. Do you think you might be overdrawn? If β > α.

I think there might be a misunderstanding. I understand that statement α - 2β < 0 is true; I am looking for a little more mathematically rigorous proof of the fact.

PeroK
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Gold Member
I think there might be a misunderstanding. I understand that statement α - 2β < 0 is true; I am looking for a little more mathematically rigorous proof of the fact.

α - 2β = α - β - β < α - β (as β > 0) < 0 (as α < β)

I think if you take a step back and examine PeroK's last post (edit: the one talking about bank accounts), you might find that it is VERY suggestive of a proof. Very suggestive.

PeroK
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Gold Member
Actually, I like this proof better:

α < β => α < 2β => α - 2β < 0

Ray Vickson
Homework Helper
Dearly Missed
Actually, I like this proof better:

α < β => α < 2β => α - 2β < 0
You have assumed what he is trying to prove: he want to show that for a,b >0, a < b => a < 2b.

PeroK