Решебник по алгебре 11 класс Никольский Параграф 10. Равносильность уравнений на множествах Задание 2

\[\boxed{\mathbf{2.}}\]

\[\textbf{а)}\ \frac{x^{2} + x - 2}{x + 2} = 0;\]

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

\[x + 2 \neq 0\]

\[x \neq - 2.\]

\[M = ( - \infty;\ - 2) \cup ( - 2; + \infty).\]

\[\textbf{б)}\ \sqrt{x} = 1;\ \ x^{2} = 1;\]

\[x \geq 0;\]

\[M = \lbrack 0; + \infty).\]

\[\textbf{в)}\ x^{3} + 2x^{2} - 1 = 0;\]

\[\sqrt{x}\left( x^{3} + 2x^{2} - 1 \right) = 0;\]

\[x > 0;\]

\[M = (0; + \infty).\]

\[\textbf{г)}\ x^{2} + 5x + \sqrt{x} = \sqrt{x} - 4;\]

\[x^{2} + 5x + 4 = 0\]

\[x \geq 0;\]

\[M = \lbrack 0; + \infty).\]

\[\textbf{д)}\lg\left( x^{2} - 1 \right) = \lg x;\]

\[x^{2} - 1 = x;\]

\[x^{2} - 1 > 0\]

\[x^{2} > 1.\]

\[x > 0.\]

\[M = (1; + \infty).\]

\[\textbf{е)}\log_{2}x + \log_{2}(x + 2) = 3;\]

\[\log_{2}\left( x(x + 2) \right) = 3;\]

\[x(x + 2) > 0\]

\[x < - 2;\ \ \ x > 0.\]

\[M = (0; + \infty).\]

\[\textbf{ж)}\log_{2}x - \log_{2}(x - 3) = 2;\]

\[\log_{2}\left( \frac{x}{x - 3} \right) = 2;\]

\[x - 3 > 0\]

\[x > 3.\]

\[x < 0.\]

\[M = (3; + \infty).\]

\[\textbf{з)}\ \frac{\sqrt{2x - 3}}{\sqrt{x - 2}} = 1;\]

\[\sqrt{\frac{2x - 3}{x - 2}} = 1\]

\[\frac{2x - 3}{x - 2} \geq 0;\ \ \ x \neq 2\]

\[x \leq \frac{3}{2};\ \ x > 2.\]

\[M = (2; + \infty).\]