Решебник по алгебре 11 класс Никольский Параграф 9. Равносильность уравнений и неравенств системам Задание 60

\[\boxed{\mathbf{60.}}\]

\[\textbf{а)}\ |x| + \frac{1}{3x - 7} \leq \frac{9x - 20}{3x - 7}\]

\[|x| \leq \frac{9x - 20}{3x - 7} - \frac{1}{3x - 7}\]

\[|x| \leq \frac{9x - 20 - 1}{3x - 7}\]

\[|x| \leq \frac{9x - 21}{3x - 7}\]

\[|x| \leq \frac{3 \cdot (3x - 7)}{3x - 7}\]

\[|x| \leq 3.\]

\[ОДЗ:\]

\[3x - 7 \neq 0\]

\[3x \neq 7\]

\[x \neq \frac{7}{3}.\]

\[|x| \leq 3\]

\[- 3 \leq x \leq 3.\]

\[Ответ:x \in \lbrack - 3;3\rbrack.\]

\[\textbf{б)}\ |x| + \frac{2}{2x + 5} \leq \frac{6x + 17}{2x + 5}\]

\[|x| \leq \frac{6x + 17}{2x + 5} - \frac{2}{2x + 5}\]

\[|x| \leq \frac{6x + 17 - 2}{2x + 5}\]

\[|x| \leq \frac{6x + 15}{2x + 5}\]

\[|x| \leq \frac{3 \cdot (2x + 5)}{2x + 5}\]

\[|x| \leq 3.\]

\[ОДЗ:\]

\[2x + 5 \neq 0\]

\[x \neq - \frac{5}{2}\]

\[x \neq - 2,5.\]

\[|x| \leq 3\]

\[- 3 \leq x \leq 3.\]

\[Ответ:\]

\[x \in \lbrack - 3; - 2,5) \cup ( - 2,5;3\rbrack.\]

\[\textbf{в)}\ |x| - \frac{3}{5x - 4} \leq \frac{10x - 11}{5x - 4}\]

\[|x| \leq \frac{10x - 11}{5x - 4} + \frac{3}{5x - 4}\]

\[|x| \leq \frac{10x - 11 + 3}{5x - 4}\]

\[|x| \leq \frac{10x - 8}{5x - 4}\]

\[|x| \leq \frac{2(5x - 4)}{5x - 4}\]

\[|x| \leq 2.\]

\[ОДЗ:\]

\[5x - 4 \neq 0\]

\[5x \neq 4\]

\[x \neq 0,8.\]

\[|x| \leq 2\]

\[- 2 \leq x \leq 2.\]

\[Ответ:x \in \lbrack - 2;0,8) \cup (0,8;2\rbrack.\]

\[\textbf{г)}\ |x| - \frac{4}{4x - 5} \leq \frac{8x - 14}{4x - 5}\]

\[|x| \leq \frac{8x - 14}{4x - 5} + \frac{4}{4x - 5}\]

\[|x| \leq \frac{8x - 10}{4x - 5}\]

\[|x| \leq \frac{2 \cdot (4x - 5)}{4x - 5}\]

\[|x| \leq 2\]

\[- 2 \leq x \leq 2.\]

\[ОДЗ:\]

\[4x - 5 \neq 0\]

\[4x \neq 5\]

\[x \neq 1,25.\]

\[Ответ:x \in \lbrack - 2;1,25) \cup (1,25;2\rbrack.\]