Решите неравенство: (x^2-13x+30)/(x^2+7x+10)<0.

Ответ:

\[\frac{x^{2} - 13x + 30}{x^{2} + 7x + 10} < 0\]

\[1)\ x^{2} - 13x + 30 =\]

\[= (x - 3)(x - 10)\]

\[x_{1} + x_{2} = 13;\ \ x_{1} \cdot x_{2} = 30\]

\[x_{1} = 10;\ \ x_{2} = 3.\]

\[2)\ x^{2} + 7x + 10 =\]

\[= (x + 5)(x + 2)\]

\[x_{1} + x_{2} = - 7;\ \ \ x_{1} \cdot x_{2} = 10\]

\[x_{1} = - 2;\ \ \ x_{2} = - 5.\ \]

\[\frac{(x - 3)(x - 10)}{(x + 5)(x + 2)} < 0\]

\[- 5 < x < - 2;\ \ 3 < x < 10.\]