$|\frac{12}{13}x-\frac{11}{12}|-1=\frac{15}{12}$
$\Leftrightarrow |\frac{12}{13}x-\frac{11}{12}|=\frac{9}{4}$
$\Leftrightarrow \left[ \begin{array}{l}\frac{12}{13}x-\frac{11}{12}=\frac{9}{4}\\\frac{12}{13}-\frac{11}{12}=-\frac{9}{4}\end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{l}x=\frac{247}{72}\\x=-\frac{13}{9}\end{array} \right.$