Задача 1С. Решите уравнение \(\left| {\,\left| {\,\left| {\,\left| {\,x-3\,} \right|-1\,} \right| + 2\,} \right|-3\,} \right| = 1\)
ОТВЕТ: 0; 2; 4; 6.
\(\left| {\,\left| {\,\left| {\,\left| {\,x-3\,} \right|-1\,} \right| + 2\,} \right|-3\,} \right| = 1.\) Уравнение вида \(\left| {f\left( x \right)} \right| = a\), где \(a \ge 0\), равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = a,\,\,\,}\\{f\left( x \right) = -a.}\end{array}} \right.\) \(\left| {\left| {\left| {\left| {x-3} \right|-1} \right| + 2} \right|-3} \right| = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {\left| {x-3} \right|-1} \right| + 2} \right|-3 = 1,}\\{\left| {\left| {\left| {x-3} \right|-1} \right| + 2} \right|-3 = -1}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {\left| {x-3} \right|-1} \right| + 2} \right| = 4,}\\{\left| {\left| {\left| {x-3} \right|-1} \right| + 2} \right| = 2\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {x-3} \right|-1} \right| + 2 = 4,\,\,\,}\\{\left| {\left| {x-3} \right|-1} \right| + 2 = -4,}\\{\left| {\left| {x-3} \right|-1} \right| + 2 = 2,\,\,\,}\\{\left| {\left| {x-3} \right|-1} \right| + 2 = -2\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {\left| {x-3} \right|-1} \right| = 2,\,\,\,}\\{\left| {\left| {x-3} \right|-1} \right| = -6,}\\{\left| {\left| {x-3} \right|-1} \right| = 0,\,\,\,}\\{\left| {\left| {x-3} \right|-1\,} \right| = -4}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {x-3} \right|-1 = 2,\,\,\,}\\{\left| {x-3} \right|-1 = -2,}\\{\left| {x-3} \right|-1 = 0\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {x-3} \right| = 3,\,\,\,}\\{\left| {x-3} \right| = -1,}\\{\left| {x-3} \right| = 1\,\,\,\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x-3 = 3,\,\,\,}\\{x-3 = -3,}\\{x-3 = 1,\,\,\,\,}\\{x-3 = -1\,\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 6,}\\{x = 0,}\\{x = 4,}\\{x = 2.}\end{array}} \right.\) Ответ: \(0;\;\;2;\;\;4;\;\;6.\)