The set of values of $x$ for which $\left|f\right(x\left)\right|+\left|g\right(x\left)\right|>\left|f\right(x)+g(x\left)\right|\text{ if }f\left(x\right)=x-3$ and $g\left(x\right)=4-x$

Here we have,

$\left|f\right(x\left)\right|+\left|g\right(x\left)\right|=7-2x\text{ if }x<3;$ and is equal to $1$ if  $3\le x\le 4$ and is equal to $2x-7$ if $x>4.$ Also 

$f\left(x\right)+g\left(x\right)=1\text{ so }\left|f\right(x)+g(x\left)\right|=1.$

We need those points for which LHS is greater than $1.$ Clearly, we can exclude values of $x$ between $3$ and $4$. For $x < 3, 7 – 2x > 1$ and for $x > 4, 2x – 7 > 1.$ Therefore, the given inequality is true for values if $x\in \mathbf{R}~[3,4].$