\(A=\dfrac{x^2-x+1}{x^2+x+1}\)
\(\Leftrightarrow Ax^2+Ax+A-x^2+x-1=0\)
\(\Leftrightarrow x^2\left(A-1\right)+x\left(A+1\right)+\left(A-1\right)=0\)
\(\Delta=\left(A+1\right)^2-4\left(A-1\right)^2\ge0\)
\(\Leftrightarrow\left(A+1\right)^2-\left(2A-2\right)^2\ge0\)
\(\Leftrightarrow\left(A+1+2A-2\right)\left(A+1-2A+2\right)\ge0\)
\(\Leftrightarrow\left(3A-1\right)\left(3-A\right)\ge0\)
\(\Leftrightarrow\dfrac{1}{3}\le A\le3\)
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