Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 239

\[\boxed{\mathbf{239}\mathbf{.}}\]

\[1)\ {0,4}^{x} - {2,5}^{x + 1} > 1,5\]

\[\left( \frac{4}{10} \right)^{x} - 2,5 \bullet \left( \frac{25}{10} \right)^{x} - 1,5 > 0\]

\[\left( \frac{2}{5} \right)^{x} - 2,5 \bullet \left( \frac{5}{2} \right)^{x} - 1,5 > 0\]

\[Пусть\ y = \left( \frac{2}{5} \right)^{x}:\]

\[y - \frac{2,5}{y} - 1,5 > 0\ \ \ \ \ | \bullet y\]

\[y^{2} - 2,5 - 1,5y > 0\]

\[2y^{2} - 3y - 5 > 0\]

\[D = 3^{2} + 4 \bullet 2 \bullet 5 = 9 + 40 = 49\]

\[y_{1} = \frac{3 - 7}{2 \bullet 2} = - \frac{4}{4} = - 1;\ \]

\[y_{2} = \frac{3 + 7}{2 \bullet 2} = \frac{10}{4} = \frac{5}{2}.\]

\[(y + 1)\left( y - \frac{5}{2} \right) > 0\]

\[y < - 1;\ \text{\ \ }y > \frac{5}{2}.\]

\[1)\ \left( \frac{2}{5} \right)^{x} < - 1\]

\[нет\ корней.\]

\[2)\ \left( \frac{2}{5} \right)^{x} > \frac{5}{2}\]

\[\left( \frac{2}{5} \right)^{x} > \left( \frac{2}{5} \right)^{- 1}\ \]

\[x < - 1\]

\[Ответ:\ \ x < - 1.\]

\[2)\ 25 \bullet {0,04}^{2x} > {0,2}^{x(3 - x)}\]

\[\frac{{0,04}^{2x}}{{0,2}^{x(3 - x)}} > \frac{1}{25}\]

\[\frac{{0,2}^{2 \bullet 2x}}{{0,2}^{3x - x^{2}}} > \left( \frac{1}{5} \right)^{2}\]

\[{0,2}^{4x - \left( 3x - x^{2} \right)} > (0,2)^{2}\]

\[(0,2)^{x + x^{2}} > (0,2)^{2}\]

\[x + x^{2} < 2\]

\[x^{2} + x - 2 < 0\]

\[D = 1^{2} + 4 \bullet 2 = 1 + 8 = 9\]

\[x_{1} = \frac{- 1 - 3}{2} = - 2;\ \text{\ \ }\]

\[x_{2} = \frac{- 1 + 3}{2} = 1.\]

\[(x + 2)(x - 1) < 0\]

\[- 2 < x < 1\]

\[Ответ:\ \ - 2 < x < 1.\]

\[3)\ \frac{4^{x}}{4^{x} - 3^{x}} < 4\]

\[\frac{1}{1 - \frac{3^{x}}{4^{x}}} < 4\]

\[1 < 4\left( 1 - \left( \frac{3}{4} \right)^{x} \right)\]

\[1 < 4 - 4 \bullet \left( \frac{3}{4} \right)^{x}\]

\[4 \bullet \left( \frac{3}{4} \right)^{x} < 3\]

\[\left( \frac{3}{4} \right)^{x} < \left( \frac{3}{4} \right)^{1}\ \]

\[x > 1.\]

\[Неравенство\ всегда\ верно:\]

\[4^{x} - 3^{x} < 0\]

\[4^{x} < 3^{x}\]

\[\frac{4^{x}}{3^{x}} < 1\]

\[\left( \frac{4}{3} \right)^{x} < \left( \frac{4}{3} \right)^{0}\]

\[x < 0.\]

\[Ответ:\ \ x < 0;\ \ \ x > 1.\]

\[4)\ \left( \frac{1}{4} \right)^{x} - 32 \bullet \left( \frac{1}{8} \right)^{x^{2} - 1} < 0\]

\[\left( \frac{1}{2} \right)^{2x} - 2^{5} \bullet \left( \frac{1}{2} \right)^{3\left( x^{2} - 1 \right)} < 0\]

\[2^{- 2x} - 2^{5} \bullet 2^{- 3\left( x^{2} - 1 \right)} < 0\]

\[2^{- 2x} - 2^{5 - 3\left( x^{2} - 1 \right)} < 0\]

\[2^{- 2x} < 2^{5 - 3\left( x^{2} - 1 \right)}\]

\[- 2x < 5 - 3\left( x^{2} - 1 \right)\]

\[- 2x < 5 - 3x^{2} + 3\]

\[3x^{2} - 2x - 8 < 0\]

\[D = 2^{2} + 4 \bullet 3 \bullet 8 =\]

\[= 4 + 96 = 100\]

\[x_{1} = \frac{2 - 10}{2 \bullet 3} = - \frac{8}{6} = - \frac{4}{3} =\]

\[= - 1\frac{1}{3};\ \]

\[x_{2} = \frac{2 + 10}{2 \bullet 3} = \frac{12}{6} = 2.\]

\[\left( x + 1\frac{1}{3} \right)(x - 2) < 0\]

\[- 1\frac{1}{3} < x < 2\]

\[Ответ:\ \ - 1\frac{1}{3} < x < 2.\]