The range of $\left|x-2\right|+\left|x-5\right|$  is

$f\left(x\right)=\left|x-2\right|+\left|x-5\right|$  and domain f = R

For $x<2,f\left(x\right)=2-x+5-x=7-2x>3;$

For $2<x<5f\left(x\right)=x-2+5-x=3$

For $x>5,f\left(x\right)=x-2+x-5=2x-7>3$

Range  $f=\left[3,\infty \right)$

$\left(0R\right) Range of \left|x-a\right|+\left|x-b\right| is [\left|b-a\right|,\infty )$ $=[\left|5-2\right|,\infty )=[3,\infty )$