\(\Leftrightarrow\left\{{}\begin{matrix}x+1\ge0\\x-1\ge0\\5x+1\ge0\\x+1+4\left(x-1\right)+4\sqrt{x^2-1}\le5x+1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\\sqrt{x^2-1}\le1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\x^2-1\le1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\-\sqrt{2}\le x\le\sqrt{2}\end{matrix}\right.\)
\(\Rightarrow1\le x\le\sqrt{2}\)