Задача 14А. Решите уравнение \(\left| {\,\left| {\,x + 1\,} \right| + 2\,} \right| = 2\)
Решение
\(\left| {\,\left| {\,x + 1\,} \right| + 2\,} \right| = 2.\)
Уравнение вида \(\left| {f\left( x \right)} \right| = a\), где \(a \ge 0\), равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = a,\,\,\,}\\{f\left( x \right) = -a.}\end{array}} \right.\)
\(\left| {\left| {x + 1} \right| + 2} \right| = 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {x + 1} \right| + 2 = 2,\,\,}\\{\left| {x + 1} \right| + 2 = -2}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left| {x + 1} \right| = 0,\,\,\,}\\{\left| {x + 1} \right| = -4}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 1 = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = -1.\)
Ответ: \(-1.\)