$7\frac{1}{2}-\Big[2\frac{1}{4}\div\Big\{1\frac{1}{4}-\frac{1}{2}\Big(\frac{3}{2}-\overline{\frac{1}{3}-\frac{1}{6}}\Big)\Big\}\Big]$
$=7\frac{1}{2}-\Big[2\frac{1}{4}\div\Big\{1\frac{1}{4}-\frac{1}{2}\Big(\frac{3}{2}-\overline{\overline{\frac{1}{3}}-\frac{1}{6}}\Big)\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\Big\{\frac{5}{4}-\frac{1}{2}\Big(\frac{3}{2}-\overline{\overline{\frac{1}{3}}-\frac{1}{6}}\Big)\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\Big\{\frac{5}{4}-\frac{1}{2}\Big(\frac{3}{2}-\frac{1}{6}\Big)\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\Big\{\frac{5}{4}-\frac{1}{2}\Big(\frac{9-1}{6}\Big)\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\Big\{\frac{5}{4}-\frac{1}{2}\times\frac{4}{3}\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\Big\{\frac{5}{4}-\frac{2}{3}\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\Big\{\frac{15-8}{12}\Big\}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\div\frac{7}{12}\Big]$
$=\frac{15}{2}-\Big[\frac{9}{4}\times\frac{12}{7}\Big]$
$=\frac{15}{2}-\frac{27}{7}$
$\frac{105-54}{14}=\frac{51}{14}=3\frac{9}{14}$