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

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:

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