Задание 8. Найдите корень уравнения. 1) \(\frac{12}{x+5} = -\frac{12}{5}\); 2) \(\frac{6}{x+8} = -\frac{3}{4}\); 3) \(\frac{1}{x+2} = -\frac{1}{2}\); 7) \(\frac{7}{x-5}=2\); 8) \(\frac{4}{x-4}=-5\); 9) \(\frac{11}{x-9}=-10\); 13) \(\frac{3}{x-19} = \frac{19}{x-3}\); 14) \(\frac{13}{x-5} = \frac{5}{x-13}\); 15) \(\frac{6}{x-8} = \frac{8}{x-6}\)

Решение:

Решаем уравнения, используя основное свойство пропорции (произведение крайних членов равно произведению средних) или приводя к общему знаменателю.

1. \(\frac{12}{x+5} = -\frac{12}{5}\)
   
   • \(12 \cdot 5 = -12 \cdot (x+5)\)
   • \(60 = -12x - 60\)
   • \(12x = -120\)
   • \(x = -10\)


2. \(\frac{6}{x+8} = -\frac{3}{4}\)
   
   • \(6 \cdot 4 = -3 \cdot (x+8)\)
   • \(24 = -3x - 24\)
   • \(3x = -48\)
   • \(x = -16\)


3. \(\frac{1}{x+2} = -\frac{1}{2}\)
   
   • \(1 \cdot 2 = -1 \cdot (x+2)\)
   • \(2 = -x - 2\)
   • \(x = -4\)


4. \(\frac{7}{x-5}=2\)
   
   • \(7 = 2 \cdot (x-5)\)
   • \(7 = 2x - 10\)
   • \(2x = 17\)
   • \(x = \frac{17}{2} = 8.5\)


5. \(\frac{4}{x-4}=-5\)
   
   • \(4 = -5 \cdot (x-4)\)
   • \(4 = -5x + 20\)
   • \(5x = 16\)
   • \(x = \frac{16}{5} = 3.2\)


6. \(\frac{11}{x-9}=-10\)
   
   • \(11 = -10 \cdot (x-9)\)
   • \(11 = -10x + 90\)
   • \(10x = 79\)
   • \(x = \frac{79}{10} = 7.9\)


7. \(\frac{3}{x-19} = \frac{19}{x-3}\)
   
   • \(3 \cdot (x-3) = 19 \cdot (x-19)\)
   • \(3x - 9 = 19x - 361\)
   • \(16x = 352\)
   • \(x = 22\)


8. \(\frac{13}{x-5} = \frac{5}{x-13}\)
   
   • \(13 \cdot (x-13) = 5 \cdot (x-5)\)
   • \(13x - 169 = 5x - 25\)
   • \(8x = 144\)
   • \(x = 18\)


9. \(\frac{6}{x-8} = \frac{8}{x-6}\)
   
   • \(6 \cdot (x-6) = 8 \cdot (x-8)\)
   • \(6x - 36 = 8x - 64\)
   • \(2x = 28\)
   • \(x = 14\)

Ответ: 1) -10; 2) -16; 3) -4; 7) 8.5; 8) 3.2; 9) 7.9; 13) 22; 14) 18; 15) 14.