Уравнения с модулями. Задача 41math100admin44242024-06-18T22:03:13+03:00
Задача 41. Решите уравнение \(\left| {\,{{\left( {x + 4} \right)}^3} + 49\,} \right| = 76\)
Решение
\(\left| {{{\left( {x + 4} \right)}^3} + 49} \right| = 76.\)
Уравнение вида \(\left| {f\left( x \right)} \right| = a,\) где \(a \ge 0,\) равносильно совокупности: \(\left[ {\begin{array}{*{20}{c}}{f\left( x \right) = a,\;\,}\\{f\left( x \right) = -a.}\end{array}} \right.\)
\(\left| {{{\left( {x + 4} \right)}^3} + 49} \right| = 76\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{{{\left( {x + 4} \right)}^3} + 49 = 76,\;}\\{{{\left( {x + 4} \right)}^3} + 49 = -76}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{{{\left( {x + 4} \right)}^3} = 27,\;\;\;}\\{{{\left( {x + 4} \right)}^3} = -125}\end{array}} \right.\;\;\;\; \Leftrightarrow \)
\( \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{{{\left( {x + 4} \right)}^3} = {3^3},\;\;\;\;}\\{{{\left( {x + 4} \right)}^3} = {{\left( {-5} \right)}^3}}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{x + 4 = 3,\;}\\{x + 4 = -5}\end{array}} \right.\;\;\;\; \Leftrightarrow \;\;\;\;\left[ {\begin{array}{*{20}{c}}{x = -1,}\\{x = -9.}\end{array}} \right.\)
Ответ: \(-9;\;\;\;-1.\)