1/căn(x+2)-căn (3-x)<=1/căn(5-2x) giải bpt này hộ mk vs ạ

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1. Đáp án:

   \[\left[ \begin{array}{l}
   x < \frac{1}{2}\\
   x \ge 2
   \end{array} \right.\]

   Giải thích các bước giải:

    ĐKXĐ:

   \(\left\{ \begin{array}{l}
   x + 2 \ge 0\\
   3 – x \ge 0\\
   5 – 2x \ge 0\\
   x + 2 \ne 3 – x
   \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
   x \ge  – 2\\
   x \le 3\\
   x \le \frac{5}{2}\\
   x \ne \frac{1}{2}
   \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
    – 2 \le x \le \frac{5}{2}\\
   x \ne \frac{1}{2}
   \end{array} \right.\)

   Ta có:

   \(\begin{array}{l}
   \frac{1}{{\sqrt {x + 2}  – \sqrt {3 – x} }} \le \frac{1}{{\sqrt {5 – 2x} }}\\
    \Leftrightarrow \left[ \begin{array}{l}
   \sqrt {x + 2}  – \sqrt {3 – x}  < 0\\
   \left\{ \begin{array}{l}
   \sqrt {x + 2}  – \sqrt {3 – x}  > 0\\
   \sqrt {x + 2}  – \sqrt {3 – x}  \ge \sqrt {5 – 2x} 
   \end{array} \right.
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x + 2 < 3 – x\\
   \left\{ \begin{array}{l}
   x + 2 > 3 – x\\
   \sqrt {x + 2}  \ge \sqrt {3 – x}  + \sqrt {5 – 2x} 
   \end{array} \right.
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x < \frac{1}{2}\\
   \left\{ \begin{array}{l}
   x > \frac{1}{2}\\
   x + 2 \ge 3 – x + 2\sqrt {\left( {3 – x} \right)\left( {5 – 2x} \right)}  + 5 – 2x
   \end{array} \right.
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x < \frac{1}{2}\\
   \left\{ \begin{array}{l}
   x > \frac{1}{2}\\
   2x – 3 \ge \sqrt {\left( {3 – x} \right)\left( {5 – 2x} \right)} 
   \end{array} \right.
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x < \frac{1}{2}\\
   \left\{ \begin{array}{l}
   x > \frac{1}{2}\\
   x \ge \frac{3}{2}\\
   4{x^2} – 12x + 9 \ge 2{x^2} – 11x + 15
   \end{array} \right.
   \end{array} \right.\\
    \Leftrightarrow \left[ \begin{array}{l}
   x < \frac{1}{2}\\
   \left\{ \begin{array}{l}
   x \ge \frac{3}{2}\\
   2{x^2} – x – 6 \ge 0
   \end{array} \right.
   \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
   x < \frac{1}{2}\\
   \left\{ \begin{array}{l}
   x \ge \frac{3}{2}\\
   \left[ \begin{array}{l}
   x \ge 2\\
   x \le  – \frac{3}{2}
   \end{array} \right.
   \end{array} \right.
   \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
   x < \frac{1}{2}\\
   x \ge 2
   \end{array} \right.
   \end{array}\)

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