Решебник по алгебре 11 класс Никольский Параграф 7. Равносильность уравнений и неравенств Задание 21

\[\boxed{\mathbf{21.}}\]

\[\textbf{а)}\ 4^{x} + 2^{x} + x^{2} < x^{2} + 6\]

\[\left( 2^{2} \right)^{x} + 2^{x} - 6 < 0\]

\[\left( 2^{x} \right)^{2} + 2^{x} - 6 < 0\]

\[t = 2^{x}:\]

\[t^{2} + t - 6 = 0\]

\[t_{1} + t_{2} = - 1;\ \ t_{1} \cdot t_{2} = - 6\]

\[t_{1} = - 3;\ \ t_{2} = 2.\]

\[- 3 < t < 2\]

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

\[2^{x} < 2\]

\[x < 1.\]

\[\textbf{б)}\ 27^{x} + 9^{x} > 3^{x} + 6 + 27^{x}\]

\[9^{x} - 3^{x} - 6 > 0\]

\[\left( 3^{2} \right)^{x} - 3^{x} - 6 > 0\]

\[\left( 3^{x} \right)^{2} - 3^{x} - 6 > 0\]

\[t = 3^{x}:\]

\[t^{2} - t - 6 = 0\]

\[t_{1} + t_{2} = 1;\ \ t_{1} \cdot t_{2} = - 6\]

\[t_{1} = 3;\ \ t_{2} = - 2.\]

\[t < - 2;\ \ t > 3.\]

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

\[нет\ решения.\]

\[3^{x} > 3\]

\[x > 1.\]