$$(x - 1\frac{8}{9}) + 3\frac{7}{9} = 4\frac{4}{9}$$
$$x - 1\frac{8}{9} = 4\frac{4}{9} - 3\frac{7}{9}$$
$$x - 1\frac{8}{9} = 3\frac{13}{9} - 3\frac{7}{9}$$
$$x - 1\frac{8}{9} = \frac{6}{9} = \frac{2}{3}$$
$$x = \frac{2}{3} + 1\frac{8}{9} = \frac{6}{9} + 1\frac{8}{9} = 1\frac{14}{9} = 1 + 1\frac{5}{9} = 2\frac{5}{9}$$
$$x = 2\frac{5}{9}$$