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## Homework Statement

Hi! Please check my solution.

If x, y, z in R and x<=z, show that x<=y<=z IFF |x-y|+|y-z|=|x-z|

Interpret geometrically.

## The Attempt at a Solution

1) Assume that x<=y<=z. We need to prove that |x-y|+|y-z|=|x-z|.

Knowing that x<=y<=z,

-(x-y)+(-(y-z))= -(x-z)

y-x-y+z=-x+z

-x+z=-x+z

2) Assume that |x-y|+|y-z|=|x-z|. We need to prove that x<=y<=z.

a) Assume y<=z<=x.

(x-y)-(y-z)=(x-z)

x-2y+z=x-z

z-2y=z

contradiction

b) Assume z<=x<=y

-(x-y)+(y-z)=(x-z)

-x-z=x-z

contradiction

c) Assume x<=y<=z

-(x-y)+(-(y-z))= -(x-z)

y-x-y+z=-x+z

-x+z=-x+z

Correcto!

Thanks!