jmc

We have $$F(x)=\{\begin{array}{ll}-(x+1)-(x-5)& \text{if }x<-1\\ (x+1)-(x-5)& \text{if }-1\le x<5\\ (x+1)+(x-5)& \text{if }x\ge 5\end{array}.$$Simplifying, we have $$F(x)=\{\begin{array}{ll}4-2x& \text{if }x<-1\\ 6& \text{if }-1\le x<5\\ 2x-4& \text{if }x\ge 5\end{array}.$$For $x<-1,$ the function $4-2x$ achieves all values in $(6,\infty ),$ and for $x\ge 5,$ the function $2x-4$ achieves all values in $[6,\infty ).$Thus, the range of $F(x)$ is $\boxed{[6,\infty )}.$