Задача 44. Решите уравнение \(\left| {\,\left| {2x-8} \right|-x\,} \right| = 7-x\)
ОТВЕТ: 0,5; 3,75.
\(\left| {\,\left| {2x-8} \right|-x\,} \right| = 7-x.\) Уравнение вида \(\left| {f\left( x \right)} \right| = g\left( x \right)\) равносильно системе: \(\left\{ {\begin{array}{*{20}{c}}{g\left( x \right) \ge 0,\;\;\;\;\;\;\;\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = g\left( x \right),\;\,}\\{f\left( x \right) = -g\left( x \right).}\end{array}} \right.}\end{array}} \right.\) \(\left| {\,\left| {2x-8} \right|-x\,} \right| = 7-x\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{7-x \ge 0,\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x-8} \right|-x = 7-x,}\\{\left| {2x-8} \right|-x = x-7\;}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \le 7,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x-8} \right| = 7,\,\;\;\;\;\;\;\,}\\{\left| {2x-8} \right| = 2x-7.}\end{array}} \right.}\end{array}} \right.\) Рассмотрим первое уравнение. Уравнение вида \(\left| {f\left( x \right)} \right| = a,\) где \(a \ge 0,\) равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = a,\;\,}\\{f\left( x \right) = -a.}\end{array}} \right.\) \(\left| {2x-8} \right| = 7\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{2x-8 = 7,\;}\\{2x-8 = -7}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{x = \dfrac{{15}}{2},}\\{x = \dfrac{1}{2}.\;\,}\end{array}} \right.\) Рассмотрим второе уравнение: \(\left| {2x-8} \right| = 2x-7\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{2x-7 \ge 0,\;\;\;\;\;\;\;\;\,\;}\\{\left[ {\begin{array}{*{20}{c}}{2x-8 = 2x-7,\;}\\{2x-8 = -2x + 7}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \ge \dfrac{7}{2},\;\;\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{-8 = -7,}\\{4x = 15\;}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \ge \dfrac{7}{2},\,\;\;}\\{\left[ {\begin{array}{*{20}{c}}{x \notin R,\;\,}\\{x = \dfrac{{15}}{4}\;}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;x = \dfrac{{15}}{4}.\) \(\left\{ {\begin{array}{*{20}{c}}{x \le 7,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\{\left[ {\begin{array}{*{20}{c}}{\left| {2x-8} \right| = 7,\,\;\;\;\;\;\,}\\{\left| {2x-8} \right| = 2x-7}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left\{ {\begin{array}{*{20}{c}}{x \le 7,\;\;\;\;}\\{\left[ {\begin{array}{*{20}{c}}{x = \dfrac{{15}}{2},}\\{x = \dfrac{1}{2},\;\,}\\{x = \dfrac{{15}}{4}\;\,}\end{array}} \right.}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{x = \dfrac{1}{2},\;}\\{x = \dfrac{{15}}{4}.}\end{array}} \right.\) Ответ: \(\dfrac{1}{2};\;\;\;\dfrac{{15}}{4}.\)