a, ĐK: \(x\ge4;x\le-4\)
\(\sqrt{x^2-4-12}\le x-4\)
\(\Leftrightarrow\sqrt{x^2-16}\le x-4\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-4\ge0\\x^2-16\le\left(x-4\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge4\\x^2-16\le x^2-8x+16\end{matrix}\right.\)
\(\Leftrightarrow x=4\left(tm\right)\)
b, ĐK: \(x\ge8;x\le0\)
\(\sqrt{x^2-8x}\ge2\left(x+1\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}2\left(x+1\right)\ge0\\x^2-8x\ge4\left(x^2+2x+1\right)\end{matrix}\right.\\2\left(x+1\right)< 0\end{matrix}\right.\)
\(\Leftrightarrow x\le\dfrac{-8+2\sqrt{13}}{3}\)
c, \(\left(x-2\right)\sqrt{x^2+4}< x^2-4\)
\(\Leftrightarrow\left(x-2\right)\left(x+2-\sqrt{x^2+4}\right)>0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-2>0\\x+2-\sqrt{x^2+4}< 0\end{matrix}\right.\left(I\right)\text{v}\left\{{}\begin{matrix}x-2< 0\\x+2-\sqrt{x^2+4}>0\end{matrix}\right.\left(II\right)\)
\(\left(I\right)\Leftrightarrow\left\{{}\begin{matrix}x>2\\x+2< \sqrt{x^2+4}\end{matrix}\right.\Leftrightarrow...\)
\(\left(II\right)\Leftrightarrow\left\{{}\begin{matrix}x-2< 0\\x+2-\sqrt{x^2+4}>0\end{matrix}\right.\Leftrightarrow...\)
