Re: Giải bất phương trình $(x+2)(\sqrt{2x+3}-2\sqrt{x+1})+\sqrt{2x^2+5x+3} \geq 1$ $\begin{gathered} dk:x \geqslant - 1 \hfill \\ bpt(1) \Leftrightarrow \left[ {{{\left( {\sqrt {2x + 3} } \right)}^2} - {{\left( {\sqrt {x + 1} } \right)}^2}} \right].\left( {\sqrt {2x + 3} - 2\sqrt {x + 1} } \right) + \sqrt {2x + 3} .\sqrt {x + 1} \geqslant {\left( {\sqrt {2x + 3} } \right)^2} - 2{\left( {\sqrt {x + 1} } \right)^2} \hfill \\ \Leftrightarrow \left[ {{{\left( {\sqrt {2x + 3} } \right)}^2} - {{\left( {\sqrt {x + 1} } \right)}^2}} \right].\left( {\sqrt {2x + 3} - 2\sqrt {x + 1} } \right) \geqslant {\left( {\sqrt {2x + 3} } \right)^2} - \sqrt {2x + 3} .\sqrt {x + 1} - 2{\left( {\sqrt {x + 1} } \right)^2} \hfill \\ \Leftrightarrow \left( {\sqrt {2x + 3} - \sqrt {x + 1} } \right).\left( {\sqrt {2x + 3} + \sqrt {x + 1} } \right).\left( {\sqrt {2x + 3} - 2\sqrt {x + 1} } \right) \geqslant \left( {\sqrt {2x + 3} + \sqrt {x + 1} } \right).\left( {\sqrt {2x + 3} - 2\sqrt {x + 1} } \right) \hfill \\ \Leftrightarrow \left( {\sqrt {2x + 3} + \sqrt {x + 1} } \right).\left( {\sqrt {2x + 3} - 2\sqrt {x + 1} } \right)\left( {\sqrt {2x + 3} - \sqrt {x + 1} - 1} \right) \geqslant 0 \hfill \\ \Leftrightarrow \left( {\sqrt {2x + 3} - 2\sqrt {x + 1} } \right)\left( {\sqrt {2x + 3} - \sqrt {x + 1} - 1} \right) \geqslant 0 \hfill \\ \hfill \\ \end{gathered}$ |