Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 714

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

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

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

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

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

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

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

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

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

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

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

\[y < - 1\ \ и\ \ y > \frac{5}{2};\]

\[Первое\ значение:\]

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

\[Второе\ значение:\]

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

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

\[x < - 1;\]

\[Ответ:\ \ x < - 1.\]

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

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

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

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

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

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

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

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

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

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

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

\[- 2 < x < 1.\]

\[Ответ:\ \ - 2 < x < 1.\]

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

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

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

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

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

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

\[x > 1;\]

\[Неравенство\ всегда\ верно\ при:\]

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

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

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

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

\[x < 0.\]

\[Ответ:\ \ x < 0;\ \ x > 1.\]

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

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

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

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

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

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

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

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

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

\[= 4 + 96 = 100\]

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

\[= - \frac{4}{3} = - 1\frac{1}{3};\]

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

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

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

\[Ответ:\ \ - 1\frac{1}{3} < x < 2.\]