1) [m] -1 ≤ \frac{1}{x} ≤ 1[/m] ⇔ [m]\left\{\begin{matrix}
\frac{1}{x} ≤ 1\\\frac{1}{x} ≥- 1 \end{matrix}\right.[/m] ; [m]\left\{\begin{matrix}
\frac{1-x}{x} ≤ 0\\\frac{1+x}{x} ≥0 \end{matrix}\right.[/m]
Решаем каждое методом интервалов:
__________-____ (0) __-__ [1]_____
__+__[-1] _____ (0) ___+___
О т в е т. (0;1)
Остальные аналогично.
2)
[m] -1 ≤ \frac{1}{x-2} ≤ 1[/m] ⇔ [m]\left\{\begin{matrix}
\frac{1}{x-2} ≤ 1\\\frac{1}{x-2} ≥- 1 \end{matrix}\right.[/m] ;[m]\left\{\begin{matrix}
\frac{1-x+2}{x-2} ≤ 0\\\frac{1+x-2}{x-2} ≥0 \end{matrix}\right.[/m] ;
3)[m] -1 ≤ \frac{2}{x+2} ≤ 1[/m] ⇔ [m]\left\{\begin{matrix}
\frac{1-x-2}{x+2} ≤ 1\\\frac{2+x+2}{x+2} ≥- 1 \end{matrix}\right.[/m]
4) [m] -1 ≤ \frac{1}{x-1} ≤ 1[/m] ⇔ [m]\left\{\begin{matrix}
\frac{1-x+1}{x-1} ≤ 1\\\frac{1+x-1}{x-1} ≥- 1 \end{matrix}\right.[/m]