Задача 50. Решите уравнение \(\left| {\,\left| {\,2x + 5\,} \right|-1\,} \right| = 2x + \left| {x-5} \right|\)
ОТВЕТ: -11/3; 1; 9.
\(\left| {\,\left| {\,2x + 5\,} \right|-1\,} \right| = 2x + \left| {x-5} \right|.\) Рассмотрим случаи, когда \(x < 5\) и \(x \ge 5\): \(\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left| {\left| {2x + 5} \right|-1} \right| = 2x-x + 5,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left| {\left| {2x + 5} \right|-1} \right| = 2x + x-5\,\,}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left| {\left| {2x + 5} \right|-1} \right| = x + 5,\,\,\,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left| {\left| {2x + 5} \right|-1} \right| = 3x-5.}\end{array}} \right.}\end{array}} \right.\) Рассмотрим первую систему последней совокупности: \(\left\{ {\begin{array}{*{20}{c}}{x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left| {\left| {2x + 5} \right|-1} \right| = x + 5}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{x + 5 \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x + 5} \right|-1 = x + 5,\,\,}\\{\left| {2x + 5} \right|-1 = -x-5}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{-5 \le x < 5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x + 5} \right| = x + 6\,\,\,}\\{\left| {2x + 5} \right| = -x-4}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{-5 \le x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{x + 6 \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{2x + 5 = x + 6,\,\,\,}\\{2x + 5 = -x-6,}\end{array}} \right.}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{-5 \le x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-x-4 \ge 0,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{2x + 5 = -x-4,}\\{2x + 5 = x + 4\,\,\,\,}\end{array}} \right.}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{-5 \le x < 5}\\{\left[ {\begin{array}{*{20}{c}}{x = 1,}\\{x = -\dfrac{{11}}{3}}\end{array}} \right.}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{-5 \le x-4,}\\{\left[ {\begin{array}{*{20}{c}}{x = -3,}\\{x = -1}\end{array}\,\,\,\,\,\,} \right.}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 1,\,\,\,\,\,\,\,\,}\\{x = -\dfrac{{11}}{3}.}\end{array}} \right.\) Рассмотрим вторую систему: \(\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left| {\left| {2x + 5} \right|-1} \right| = 3x-5}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x + 5} \right|-1 = 3x-5,\,\,}\\{\left| {2x + 5} \right|-1 = -3x + 5}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \) \( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x + 5} \right| = 3x-4,}\\{\left| {2x + 5} \right| = 6-3x\,\,}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{2x + 5 = 3x-4,\,\,}\\{2x + 5 = -3x + 4}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{x \ge 5,\,\,\,\,\,\,}\\{\left[ {\begin{array}{*{20}{c}}{x = 9,}\\{x = -\dfrac{1}{5}}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = 9.\) Таким образом, решением исходного уравнения \(x = -\dfrac{{11}}{3},\,\,\,\,x = 1,\,\,\,x = 9.\) Ответ: \(-\dfrac{{11}}{3};\;\;1;\;\;9.\)