1. \(\left|2x-7\right|< x^2+2x+2\)
+) Xét \(x\ge\frac{7}{2}\):
Bpt \(\Leftrightarrow2x-7< x^2+2x+2\)
\(\Leftrightarrow x^2+9>0\) ( luôn đúng )
+) Xét \(x< \frac{7}{2}\):
Bpt \(\Leftrightarrow7-2x< x^2+2x+2\)
\(\Leftrightarrow x^2+4x-5>0\)
\(\Leftrightarrow\left(x+5\right)\left(x-1\right)>0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+5>0\\x-1>0\end{matrix}\right.\\\left\{{}\begin{matrix}x+5< 0\\x-1< 0\end{matrix}\right.\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x>1\\x< -5\end{matrix}\right.\)
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2. \(a+b+c=0\Leftrightarrow a+c=-b\)
Ta có: \(P=\Sigma\frac{1}{b^2+\left(c-a\right)\left(c+a\right)}=\Sigma\frac{1}{b^2-b\left(c-a\right)}\)
\(=\Sigma\frac{1}{b\left(b-c+a\right)}=\Sigma\frac{1}{b\left(a+b+c-2c\right)}=\Sigma\frac{1}{-2bc}\)
\(=\frac{1}{2ab}+\frac{1}{2bc}+\frac{1}{2ca}=\frac{a+b+c}{2abc}=0\)
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