Задача 2С. Решите уравнение \(\left| {\,\left| {\,\left| {\,\left| {\,x + 1\,} \right|-5\,} \right| + 1\,} \right|-2\,} \right| = 2\)
ОТВЕТ: -9; -3; 1; 7.
\(\left| {\,\left| {\,\left| {\,\left| {\,x + 1\,} \right|-5\,} \right| + 1\,} \right|-2\,} \right| = 2.\) Уравнение вида \(\left| {f\left( x \right)} \right| = a\), где \(a \ge 0\), равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = a,\,\,\,}\\{f\left( x \right) = -a.}\end{array}} \right.\) \(\left| {\left| {\left| {\left| {x + 1} \right|-5} \right| + 1} \right|-2} \right| = 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {\left| {x + 1} \right|-5} \right| + 1} \right|-2 = 2,}\\{\left| {\left| {\left| {x + 1} \right|-5} \right| + 1} \right|-2 = -2}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {\left| {x + 1} \right|-5} \right| + 1} \right| = 4,}\\{\left| {\left| {\left| {x + 1} \right|-5} \right| + 1} \right| = 0\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {x + 1} \right|-5} \right| + 1 = 4,\,\,\,}\\{\left| {\left| {x + 1} \right|-5} \right| + 1 = -4,}\\{\left| {\left| {x + 1} \right|-5} \right| + 1 = 0\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {x + 1} \right|-5} \right| = 3,\,\,\,}\\{\left| {\left| {x + 1} \right|-5} \right| = -5,}\\{\left| {\left| {x + 1} \right|-5} \right| = -1\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {x + 1} \right|-5 = 3,\,\,}\\{\left| {x + 1} \right|-5 = -3}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {x + 1} \right| = 8,}\\{\left| {x + 1} \right| = 2\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x + 1 = 8,\,\,\,}\\{x + 1 = -8,}\\{x + 1 = 2,\,\,\,}\\{x + 1 = -2\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 7,\,\,\,}\\{x = -9,}\\{x = 1,\,\,\,\,}\\{x = -3.}\end{array}} \right.\) Ответ: \(-9;\;\;-3;\;\;1;\;\;7.\)