\(\frac{\sqrt{51-2x-x^2}}{1-x}< 1\)
ĐKXĐ: \(\left\{{}\begin{matrix}-1-2\sqrt{13}\le x\le-1+2\sqrt{13}\\x\ne1\end{matrix}\right.\)
bpt \(\Leftrightarrow\frac{\sqrt{51-2x-x^2}}{1-x}-1< 0\)
\(\Leftrightarrow\frac{\sqrt{51-2x-x^2}-1+x}{1-x}< 0\)
TH1: \(\left\{{}\begin{matrix}\sqrt{51-2x-x^2}-1+x< 0\\1-x>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{51-2x-x^2}< 1-x\\1-x>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x< 1\\51-2x-x^2< 1-2x+x^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x< 1\\-2x^2+50< 0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x< 1\\\left[{}\begin{matrix}x< -5\\x>5\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow x< -5\)
Kết hợp ĐKXĐ: \(-1-2\sqrt{13}< x< -5\)
TH2: \(\left\{{}\begin{matrix}\sqrt{51-2x-x^2}-1+x>0\\1-x< 0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x>1\\\sqrt{51-2x-x^2}>1-x\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x>1\\\sqrt{51-2x-x^2}>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x>1\\51-2x-x^2>0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x>1\\-1-2\sqrt{13},x< -1+2\sqrt{13}\end{matrix}\right.\)\(\Leftrightarrow1< x< -1+2\sqrt{13}\)
Vậy bpt có nghiệm \(x\in\left(-1-2\sqrt{13};-5\right)\cup\left(1;-1+2\sqrt{13}\right)\)
