\({\left( {\dfrac{3}{7}} \right)^{2x + 1}} \ge \dfrac{2}{7}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\left( {\dfrac{3}{7}} \right)^{2x + 1}} \ge {\left( {\dfrac{3}{7}} \right)^{{{\log }_{\dfrac{3}{7}}}\dfrac{2}{7}}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2x + 1 \le {\log _{\dfrac{3}{7}}}\dfrac{2}{7}\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,2x \le {\log _{\dfrac{3}{7}}}\dfrac{2}{7}-1\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x \le \dfrac{1}{2}{\log _{\dfrac{3}{7}}}\dfrac{2}{7}-\dfrac{1}{2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \in \,\left( {-\infty ;\dfrac{1}{2}{{\log }_{\dfrac{3}{7}}}\dfrac{2}{7}-\dfrac{1}{2}} \right].\)
Ответ: \(\,\left( {-\infty ;\dfrac{1}{2}{{\log }_{\dfrac{3}{7}}}\dfrac{2}{7}-\dfrac{1}{2}} \right].\)