extra x ticks={-8},
extra x tick style={ticklabel style={font=scriptsize, inner xsep=10pt, fill=white}},
extra x tick labels={makebox[0pt][r]{$-$}8},

documentclass{amsart}
usepackage{amsmath}
usepackage{amsfonts}

usepackage{tikz}
usetikzlibrary{calc,intersections}



usepackage{pgfplots}
pgfplotsset{compat=1.11}


begin{document}

begin{tikzpicture}



begin{axis}[width=3.5in, height=3.5in, axis lines=middle, axis on top, clip=false,
    axis lines=middle,
    xmin=-10,xmax=10,
    ymin=-2.5,ymax=10,
    restrict y to domain=-2.5:10,
    xtick={8}, ytick={8},
    ticklabel style={font=scriptsize},
    xticklabels={8},
    extra x ticks={-8},
    extra x tick style={ticklabel style={font=scriptsize, inner xsep=10pt, fill=white}},
    extra x tick labels={makebox[0pt][r]{$-$}8},
    axis line style={latex-latex},
    xlabel=$x$,ylabel=$y$,
    axis line style={shorten >=-12.5pt, shorten <=-12.5pt},
    xlabel style={at={(ticklabel* cs:1)}, xshift=12.5pt, anchor=north west},
    ylabel style={at={(ticklabel* cs:1)}, yshift=12.5pt, anchor=south west}
]

addplot[samples=2, blue, domain=-10:0] {x + 8} node [anchor=south east, pos=0.35, font=footnotesize] {$y= vert x vert + y = 8$};
addplot[samples=2, blue, domain=0:10] {-x + 8};

addplot[dashed, latex-latex, domain=-2.5:10] (6,x) node [pos=0.9, anchor=north, font=footnotesize, sloped] {$x=6$};

coordinate (A) at (6,0);
coordinate (B) at (6,2);
coordinate (C) at (0,8);
coordinate (D) at (-8,0);

end{axis}

fill[gray!50, fill opacity=0.25] (A) -- (B) -- (C) -- (D) -- cycle;

end{tikzpicture}

end{document}