Решение:
$$5\frac{2}{9} - (3\frac{1}{9} - 2\frac{4}{9}) + \frac{5}{9} = 5\frac{2}{9} - (2\frac{10}{9} - 2\frac{4}{9}) + \frac{5}{9} = 5\frac{2}{9} - \frac{6}{9} + \frac{5}{9} = 5\frac{2}{9} + \frac{5}{9} - \frac{6}{9} = 5\frac{7}{9} - \frac{6}{9} = 5\frac{1}{9}$$Ответ: $$5\frac{1}{9}$$