Задание 6. Найдите корень уравнения.

Решение:

1. \( x + \frac{x}{9} = -\frac{10}{3} \)
   \( \frac{9x + x}{9} = -\frac{10}{3} \)
   \( \frac{10x}{9} = -\frac{10}{3} \)
   \( 10x \cdot 3 = -10 \cdot 9 \)
   \( 30x = -90 \)
   \( x = \frac{-90}{30} \)
   \( x = -3 \)
2. \( x - \frac{x}{7} = 6 \)
   \( \frac{7x - x}{7} = 6 \)
   \( \frac{6x}{7} = 6 \)
   \( 6x = 6 \cdot 7 \)
   \( 6x = 42 \)
   \( x = \frac{42}{6} \)
   \( x = 7 \)
3. \( x + \frac{x}{5} = -\frac{12}{5} \)
   \( \frac{5x + x}{5} = -\frac{12}{5} \)
   \( \frac{6x}{5} = -\frac{12}{5} \)
   \( 6x = -12 \)
   \( x = \frac{-12}{6} \)
   \( x = -2 \)
4. \( x - \frac{x}{12} = \frac{11}{3} \)
   \( \frac{12x - x}{12} = \frac{11}{3} \)
   \( \frac{11x}{12} = \frac{11}{3} \)
   \( 11x \cdot 3 = 11 \cdot 12 \)
   \( 33x = 132 \)
   \( x = \frac{132}{33} \)
   \( x = 4 \)
5. \( x + \frac{x}{2} = -9 \)
   \( \frac{2x + x}{2} = -9 \)
   \( \frac{3x}{2} = -9 \)
   \( 3x = -9 \cdot 2 \)
   \( 3x = -18 \)
   \( x = \frac{-18}{3} \)
   \( x = -6 \)
6. \( x - \frac{x}{11} = \frac{50}{11} \)
   \( \frac{11x - x}{11} = \frac{50}{11} \)
   \( \frac{10x}{11} = \frac{50}{11} \)
   \( 10x = 50 \)
   \( x = \frac{50}{10} \)
   \( x = 5 \)
7. \( 6 + \frac{x}{2} = \frac{x+3}{5} \)
   \( \frac{30 + 5x}{5} = \frac{x+3}{5} \)
   \( 30 + 5x = x+3 \)
   \( 5x - x = 3 - 30 \)
   \( 4x = -27 \)
   \( x = -\frac{27}{4} \)
8. \( -4 + \frac{x}{5} = \frac{x+4}{2} \)
   \( \frac{-20 + x}{5} = \frac{x+4}{2} \)
   \( 2(-20 + x) = 5(x+4) \)
   \( -40 + 2x = 5x + 20 \)
   \( 2x - 5x = 20 + 40 \)
   \( -3x = 60 \)
   \( x = \frac{60}{-3} \)
   \( x = -20 \)
9. \( 1 + \frac{x}{5} = \frac{x+9}{7} \)
   \( \frac{5 + x}{5} = \frac{x+9}{7} \)
   \( 7(5+x) = 5(x+9) \)
   \( 35 + 7x = 5x + 45 \)
   \( 7x - 5x = 45 - 35 \)
   \( 2x = 10 \)
   \( x = \frac{10}{2} \)
   \( x = 5 \)

Ответ: 1) -3; 2) 7; 3) -2; 4) 4; 5) -6; 6) 5; 7) -27/4; 8) -20; 9) 5.