Уравнения с модулями. Задача 58

ОТВЕТ: 1;  12.

\(\left| {9-{3^x}} \right| + \left| {x-6} \right| = {3^x}-x + 9.\)

Решим исходное уравнение методом интервалов:

\(\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 2,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{9-{3^x}-x + 6 = {3^x}-x + 9,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{2 \le x < 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-9 + {3^x}-x + 6 = {3^x}-x + 9,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x > 6,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{-9 + {3^x} + x-6 = {3^x}-x + 9}\end{array}\,} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 2,\,\,\,\,\,\,\,\,}\\{2 \cdot {3^x} = 6,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{2 \le x < 6}\\{-3 = 9\,\,\,\,\,\,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x > 6,\,\,\,\,}\\{2x = 24}\end{array}\,} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x < 2,}\\{x = 1,}\end{array}} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x > 6,}\\{x = 12}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 1,\,\,\,\,}\\{x = 12.}\end{array}} \right.\)

Ответ: \(1;\;\;12.\)