\(\left| {27-{3^x}} \right| + \left| {x-5} \right| = {3^x}-x + 14.\)
Решим исходное уравнение методом интервалов:
\(\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 3,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{27-{3^x}-x + 5 = {3^x}-x + 14,}\end{array}\,\,\,} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{3 \le x < 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-27 + {3^x}-x + 5 = {3^x}-x + 14,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x > 5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-27 + {3^x} + x-5 = {3^x}-x + 14}\end{array}\,} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 3,\,\,\,\,\,\,\,\,\,\,\,\,}\\{2 \cdot {3^x} = 18,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{3 \le x < 5,}\\{-22 = 14,}\end{array}\,\,} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x > 5,\,\,\,\,\,}\\{2x = 46}\end{array}\,\,\,\,\,\,} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 3,}\\{x = 2,}\end{array}\,\,} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x > 5,}\\{x = 23}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 2,\,\,\,}\\{x = 23.}\end{array}} \right.\)
Ответ: \(2;\;\;23.\)