Let ‘$Z$ ’denote the set of integers, then $\{x\in X:x-3<4\}\cap \{x\in Z:\left[x-4\right)<5\}$  is

$\left|x-4\right|<5,-5<x-4<5,-1<x<9\Rightarrow$ $x\in \left\{0,1,2,3,4,5,6,7,8\right\}$

$\therefore \left\{x\in X:\left|x-3\right|<4\right\}\cap \left\{x\in Z:\left|x-4\right|<5\right\}$

$=\left\{0,1,2,3,4,5,6\right\}\cap \left\{0,1,2,3,4,5,6,7,8\right\}$

$=\left\{0,1,2,3,4,5,6\right\}$