\[\boxed{\mathbf{4.}\mathbf{65}}\]
\[\textbf{а)}\ | - 7| - | - 6| = 7 - 6 = 1\]
\[\textbf{б)}\ | - 710| + | - 290| =\]
\[= 710 + 290 = 1000\]
\[\textbf{в)}\ | - 4,2| + |5,9| = 4,2 + 5,9 =\]
\[= 10,1\]
\[\textbf{г)}\ | - 3,6| - | - 2,7| = 3,6 - 2,7 =\]
\[= 0,9\]
\[\textbf{д)}\ \left| - \frac{8}{9} \right| - \left| - \frac{3}{4} \right| = \frac{8}{9} - \frac{3}{4} =\]
\[= \frac{32}{36} - \frac{27}{36} = \frac{5}{36}\]
\[\textbf{е)}\ \ \left| 4\frac{1}{9} \right| - \left| - 2\frac{7}{18} \right| =\]
\[= 4\frac{1}{9} - 2\frac{7}{18} = 4\frac{2}{18} - 2\frac{7}{18} =\]
\[= 3\frac{20}{18} - 2\frac{7}{18} = 1\frac{13}{18}\]