Under what conditions is the identity |a-c| = |a-b| + |b-c| true?

Question

As the title suggests, I need to find out under what conditions the identity |a-c| = |a-b| + |b-c| is true.

I really have no clue as to where to start it. I know that I must know under what conditions the two sides of a triangle are equal to the remaining one. However, I really can't figure out when is that true. Would anyone care to enlighten me as to how would I go about doing this?

Answer 1

Think in terms of distances!

Put $a$ and $c$ on a line. Note that $|a-c|$ is the distance between $a$ and $c$. Now if you put a third point $b$ somewhere between $a$ and $c$, it is clear that $|a-b|+|b-c|=|a-c|$. If you move $b$ so it is no longer between $a$ and $c$, you will see that it is no longer true.

A nice thing with this geometric approach is that it provides a generalisation to higher dimensions. If you consider $a,b,c$ as points in the plane (or in space), the same conclusion holds: $|a-c|=|a-b|+|b-c|$ if and only if $b$ lies on the line segment between $a$ and $c$.

Answer 2

if $a<c \implies a<b<c$ and if $c<a \implies c<b<a$ with those condition that identify is always true. So for any b in the interval $(a,c)$ or $(c,a)$ the identity $|a-c| = |a-b| + |b-c|$ is true."