Thread: triple inequalities

i really dontknow this questoin any help would be great

prove that if real numbers a,b,c satisfy

a + b + c>0, ab +ac + bc>0 , abc>0

then each of a,b,c is positive


thank you in advance

Originally Posted by shosho

i really dontknow this questoin any help would be great

prove that if real numbers a,b,c satisfy

a + b + c>0, ab +ac + bc>0 , abc>0

then each of a,b,c is positive


thank you in advance

It ain't elegant but it works:





1) if
or
2) if
or
3) if

Now:
3) is the easiest case. and are contradictory. Thus .

So consider case 1):
1) if

We know that and a is positive, so . This means that either both b, c are positive or both b, c are negative.

If both b, c are positive then is trivial since the RHS is negative, and we have no contradiction.

If both b, c are negative then








Now if then

is a contradiction because and the RHS is negative.

If and we know that c is negative then cannot be true because , a contradiction.

Thus we are left with .
Thus



another contradiction.

Thus both b and c cannot be negative.

Thus case 1) says that if a is positive, so are both b and c.

Case 2) works in a similar fashion to show that, at best, all three of a, b, and c must be negative. (I leave this for you to show.) This again contradicts .

-Dan

hey shosho seems to understand this problem.. but im not quite sure i do.
i dont understand the contradiction bit.. could you explain more? it seems like a reeli hard question
thanks

Originally Posted by ieatfood

hey shosho seems to understand this problem.. but im not quite sure i do.
i dont understand the contradiction bit.. could you explain more? it seems like a reeli hard question
thanks

The basic idea behind what I was doing is this:
Given a real number a, we know that only one of the following possibilities is true:
a < 0
a = 0
a > 0

I used this idea in two places in the proof of case 1): First in showing that if a is positive then b,c must both be either positive or negative. I showed that there is no contradiction if both b, c are positive. Then I showed that if b,c are both negative that a contradiction exists by looking at a + b in reference to >,=,<. There was a contradiction in all three cases, so I showed that a + b, a real number, is neither >, =, or < 0, an impossibility.

-Dan