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I want to find the value of $x'$ such that all of the following constraints are satisfied:

$x' * o' < x * o$

$ o' < o $

$ x \le x'$

These are all variables but I always know the values of $x$, $o$ and $o'$. I'm sure that this is easy to solve but I have been completely unable to!




For clarification, what I'm actually trying to do is place two bets on a betting exchange such that I make a profit regardless of the outcome of an event.

I place a back bet of £x at odds of $back_odds$, if that wins then my payout is $£x*back_odds$, I have to place a lay bet at lower odds but with a higher stake in order to make a profit on all outcomes.

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It depends on if the values $x$, $o$ and $o'$ are positive or negative. Further try to use symbol other than $o$ since it looks like $0$. – user17762 Mar 20 '11 at 23:40
Are $x'$ and $x$ related in some way, and $o$ and $o'$, too? If so, please describe. If not, please use four different characters (and not $o$). What range of variables are allowable? Is it $\mathbb{R}? – Ross Millikan Mar 20 '11 at 23:44
up vote 1 down vote accepted

I'll assume that $x$, $o$, and $o'$ are all positive; you should be able to work out solutions in other cases by a similar method.

In this case, we need $x \le x' < x(o/o')$ as our sole requirement, which can be determined by dividing the first expression by $o'$.

Keep in mind for the other cases that when multiplying/dividing by negative numbers, inequalities are reversed.

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