a) $\frac{4}{9}\times x = \frac{1}{9}$.

$\Leftrightarrow x = \frac{1}{9}:\frac{4}{9}$

$\Leftrightarrow x = \frac{1}{9}\times \frac{9}{4}$

$\Leftrightarrow x = \frac{1}{4}$.

b) $x:\frac{-1}{12} = \frac{7}{12}$;

$\Leftrightarrow x = \frac{7}{12}\times \frac{-1}{12}$

$\Leftrightarrow x = \frac{-7}{144}$.

c) $\frac{-5}{14}:x = \frac{-3}{10}$.

$\Leftrightarrow x = \frac{-5}{14}:\frac{-3}{10}$

$\Leftrightarrow x = \frac{-5}{14}\times \frac{-10}{3}$

$\Leftrightarrow x = \frac{25}{21}$

d) $\frac{7}{18}\times x - \frac{2}{3} = \frac{5}{18}$;

$\Leftrightarrow \frac{7}{18}\times x = \frac{5}{18} + \frac{2}{3}$;

$\Leftrightarrow \frac{7}{18}\times x = \frac{5}{18} + \frac{12}{18}$;

$\Leftrightarrow \frac{7}{18}\times x = \frac{17}{18}$;

$\Leftrightarrow x = \frac{17}{18} : \frac{7}{18}$

$\Leftrightarrow x = \frac{17}{18} \times \frac{18}{7}$

$\Leftrightarrow x = \frac{17}{7}$

e) $\frac{4}{9} - \frac{7}{8}\times x = \frac{-2}{3}$.

$\Leftrightarrow \frac{7}{8}\times x = \frac{4}{9} + \frac{2}{3}$

$\Leftrightarrow \frac{7}{8}\times x = \frac{4}{9} + \frac{6}{9}$

$\Leftrightarrow \frac{7}{8}\times x = \frac{10}{9}$

$\Leftrightarrow x = \frac{10}{9}:\frac{7}{8}$

$\Leftrightarrow x = \frac{10}{9}\times \frac{8}{7}$.

$\Leftrightarrow x = \frac{80}{63}$.

f) $\frac{1}{6}+\frac{5}{7}:x = \frac{-7}{18}$.

$\Leftrightarrow \frac{5}{7}:x = \frac{-7}{18} - \frac{1}{6}$.

$\Leftrightarrow \frac{5}{7}:x = \frac{-7}{18} - \frac{3}{18}$.

$\Leftrightarrow \frac{5}{7}:x = \frac{-10}{18}$.

$\Leftrightarrow \frac{5}{7}:x = \frac{-5}{9}$.

$\Leftrightarrow x = \frac{5}{7}:\frac{-5}{9}$.

$\Leftrightarrow x = \frac{5}{7}\times \frac{-9}{5}$.

$\Leftrightarrow x = \frac{-9}{7}$.