Решение уравнений:

1. $$4x = 24 + x$$
   $$4x - x = 24$$
   $$3x = 24$$
   $$x = \frac{24}{3}$$
   $$\boxed{x = 8}$$
2. $$8x - 8 = 20 - 6x$$
   $$8x + 6x = 20 + 8$$
   $$14x = 28$$
   $$x = \frac{28}{14}$$
   $$\boxed{x = 2}$$
3. $$9 - 4x = 3x - 40$$
   $$-4x - 3x = -40 - 9$$
   $$-7x = -49$$
   $$x = \frac{-49}{-7}$$
   $$\boxed{x = 7}$$
4. $$0,6x - 5,4 = -0,8x + 5,8$$
   $$0,6x + 0,8x = 5,8 + 5,4$$
   $$1,4x = 11,2$$
   $$x = \frac{11,2}{1,4}$$
   $$\boxed{x = 8}$$
5. $$4,7 - 1,1x = 0,5x - 3,3$$
   $$-1,1x - 0,5x = -3,3 - 4,7$$
   $$-1,6x = -8$$
   $$x = \frac{-8}{-1,6}$$
   $$\boxed{x = 5}$$
6. $$\frac{5}{6}x + 16 = \frac{4}{9}x + 9$$
   $$\frac{5}{6}x - \frac{4}{9}x = 9 - 16$$
   $$\frac{15}{18}x - \frac{8}{18}x = -7$$
   $$\frac{7}{18}x = -7$$
   $$x = -7 : \frac{7}{18}$$
   $$x = -7 \cdot \frac{18}{7}$$
   $$\boxed{x = -18}$$