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Take $ \frac{x-a}{b-a}(b) + \frac{b-x}{b-a}(a) = x.$
I need an interpretation of this, using concrete example.

well i don't have difficulty doing it and is not HW. I'm ust drawing a blank right now trying to interpret this. example would be on a number like 2-9 i want 6 then applying this $(4/7)(9)+(3/7)2 =6$. I'm TRYING to see what's going on in , not making sense to me but it works

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    It's the linear interpolant passing through $(a,a)$ and $(b,b)$. I'm not sure why you think there's something deeper into this?2011-05-03

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Building on Shai's interpretation, consider the following more general question: suppose $l \leq x \leq r$. Since $x$ is inside the interval $[l,r]$, it can be represented as a weighted average of $l$ and $r$. Let $a = x-l$ be the distance from $x$ to $l$, and $b = r-x$ be the distance from $x$ to $r$. Then clearly $x = \frac{a}{a+b} l + \frac{b}{a+b} r.$ The left endpoint $l$ can be recovered by starting with $x$ and going $a$ backwards, so $l = x-a$. Similarly, the right endpoint $r$ can be recovered by starting with $x$ and going $b$ forwards, so $r = x + b$. So $x = \frac{a}{a+b} (x-a) + \frac{b}{a+b} (x+b).$ Your formula replaces $b$ with $-b$ for some reason, but this formula is actually more intuitive.

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    this is really good yuval thank u2011-02-07