Question

1 (x-3).(x-1)<1

2 (x+2).(x+3)>0

Answer 1

1/ \(\left(x-3\right)\left(x-1\right)< 1\)

\(\Leftrightarrow x^2-x-3x+3< 1\)

\(\Leftrightarrow x^2-4x+2< 0\)

\(\Leftrightarrow\left(x-2-\sqrt{2}\right)\left(x-2+\sqrt{2}\right)< 0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-2-\sqrt{2}< 0\\x-2+\sqrt{2}>0\end{matrix}\right.\\\left\{{}\begin{matrix}x-2-\sqrt{2}>0\\x-2+\sqrt{2}< 0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow2-\sqrt{2}< x< 2+\sqrt{2}\)

2/ \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+2>0\\x+3>0\end{matrix}\right.\\\left\{{}\begin{matrix}x+2< 0\\x+3< 0\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x>-2\\x< -3\end{matrix}\right.\)