-4\frac{1}{6} : x = 11\frac{1}{9}

Convert the mixed numbers to improper fractions: $$-\frac{25}{6} : x = \frac{100}{9}$$

Rewrite the division as multiplication: $$-\frac{25}{6} \times \frac{1}{x} = \frac{100}{9}$$

Solve for x: $$x = -\frac{25}{6} \times \frac{9}{100} = -\frac{25 \times 9}{6 \times 100} = -\frac{225}{600} = -\frac{3}{8}$$