\(\sin \dfrac{{\pi \left( {x + 1} \right)}}{3} = -\dfrac{1}{2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\dfrac{{\pi \left( {x + 1} \right)}}{3} = -\dfrac{\pi }{6} + 2\pi k\left| { \cdot 3} \right.\,\,\,\,}\\{\dfrac{{\pi \left( {x + 1} \right)}}{3} = -\dfrac{{5\pi }}{6} + 2\pi k\left| { \cdot 3} \right.}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\pi \left( {x + 1} \right) = -\dfrac{\pi }{2} + 6\pi k\left| {:\pi \,\,\,} \right.}\\{\pi \left( {x + 1} \right) = -\dfrac{{5\pi }}{2} + 6\pi k\left| {:\pi } \right.}\end{array}} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x + 1 = -\dfrac{1}{2} + 6k,}\\{x + 1 = -\dfrac{5}{2} + 6k\,}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = -1,5 + 6k,}\\{x = -3,5 + 6k,}\end{array}} \right.\,\,\,\,\,\,k \in Z.\)
Рассмотрим \(x = -1,5 + 6k,\,\,\,k\, \in \,Z\). Если \(k = 1\), то \(x = 4,5\); если \(k = 0\), то \(x = -1,5.\)
Рассмотрим \(x = -3,5 + 6k,\,\,\,k\, \in \,Z\). Если \(k = 1\), то \(x = 2,5\); если \(k = 0\), то \(x = -3,5.\)
Следовательно, наименьший положительный корень \(x = 2,5.\)
Ответ: 2,5.