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ĐKXĐ : \(x\ne\pm2;x\ne0;x\ne3\)
\(A=\left(\frac{4x}{2+x}+\frac{8x^2}{4-x^2}\right):\left(\frac{x-1}{x^2-2x}-\frac{2}{x}\right)\)
\(=\frac{4x\left(2-x\right)+8x^2}{\left(2-x\right)\left(2+x\right)}:\frac{x-1-2\left(x-2\right)}{x\left(x-2\right)}\)
\(=\frac{8x-4x^2+8x^2}{\left(2-x\right)\left(2+x\right)}:\frac{x-1-2x+4}{x\left(x-2\right)}\)
\(=\frac{8x+4x^2}{\left(2-x\right)\left(2+x\right)}:\frac{3-x}{x\left(x-2\right)}\)
\(=\frac{8x+4x^2}{\left(2-x\right)\left(2+x\right)}.\frac{x\left(x-2\right)}{3-x}\) \(=\frac{4x\left(2+x\right)}{\left(2-x\right)\left(2+x\right)}.\frac{x\left(2-x\right)}{x-3}\)
\(=\frac{4x^2}{x-3}\)
\(A< 0\Leftrightarrow\frac{4x^2}{x-3}< 0\Leftrightarrow x-3< 0\) ( do \(4x^2>0\) )
\(\Leftrightarrow x< 3\)
Vậy :........
ĐKXĐ: \(x\ne\pm2;x\ne0\)
\(A=\left[\frac{4x\left(x-2\right)}{x^2-4}-\frac{8x^2}{x^2-4}\right]:\left[\frac{x-1}{x\left(x-2\right)}-\frac{2\left(x-2\right)}{x\left(x-2\right)}\right]\)
\(=\frac{-4x^2-8x}{x^2-4}:\frac{-x+3}{x\left(x-2\right)}\)
\(=\frac{-4x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}.\frac{x\left(x-2\right)}{-x+3}\)
\(=\frac{4x^2}{x-3}\)
Vì \(4x^2\ge0\)với mọi x nên:
để A > 0 thì x - 3 >0 <=> x > 3
\(B=\left(\frac{2+x}{2-x}-\frac{4x^2}{x^2-4}-\frac{2-x}{x+2}\right):\frac{x^2-3x}{2x^2-x^3}\left(ĐKXĐ:x\ne2;-2;0\right)\)
a)\(B=\left(-\frac{\left(x+2\right)^2}{x^2-4}-\frac{4x^2}{x^2-4}+\frac{\left(x-2\right)^2}{x^2-4}\right):\frac{x\left(x-3\right)}{x^2\left(2-x\right)}\)
\(B=\left(\frac{-\left(x+2\right)^2-4x^2+\left(x-2\right)^2}{x^2-4}\right).\frac{-x\left(x-2\right)}{\left(x-3\right)}\)
\(B=\left(\frac{-x^2-4x-4-4x^2+x-4x+4}{\left(x-2\right)\left(x+2\right)}\right).-\frac{x\left(x-2\right)}{x-3}\)
\(B=\frac{-5x^2-7x}{\left(x+2\right)}.\frac{-x}{x-3}\)
\(B=\frac{\left(-5x^2-7x\right)-x}{\left(x+2\right)\left(x-3\right)}\)
\(B=\frac{5x^3+7x^2}{\left(x+2\right)\left(x+3\right)}\)
ĐKXĐ:\(x\ne\pm2;x\ne-3;x\ne0\)
\(P=1+\frac{x-3}{x^2+5x+6}\left(\frac{8x^2}{4x^3-8x^2}-\frac{3x}{3x^2-12}-\frac{1}{x+2}\right)\)
\(=1+\frac{x-3}{\left(x+2\right)\left(x+3\right)}\left[\frac{8x^2}{4x^2\left(x-2\right)}-\frac{3x}{3\left(x^2-4\right)}-\frac{1}{x+2}\right]\)
\(=1+\frac{x-3}{\left(x+2\right)\left(x+3\right)}\left(\frac{2}{x-2}-\frac{x}{x^2-4}-\frac{1}{x+2}\right)\)
\(=1+\frac{x-3}{\left(x+2\right)\left(x+3\right)}\left[\frac{2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{x}{\left(x-2\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+2\right)}\right]\)
\(=1+\frac{x-3}{\left(x+2\right)\left(x+3\right)}\cdot\frac{2x+4-x-x+4}{\left(x-2\right)\left(x+2\right)}\)
\(=1+\frac{8\left(x-3\right)}{\left(x+2\right)^2\left(x+3\right)\left(x-2\right)}\)
Đề sai à ??
Ta có: \(B=\left(\frac{4x}{x+2}+\frac{8x^2}{4-x^2}\right):\left(\frac{x-1}{x^2-2x}-\frac{2}{x}\right)\)
\(=\left(\frac{4x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{8x^2}{\left(x-2\right)\left(x+2\right)}\right):\left(\frac{x-1}{x\left(x-2\right)}-\frac{2\left(x-2\right)}{x\left(x-2\right)}\right)\)
\(=\frac{4x^2-8x-8x^2}{\left(x-2\right)\left(x+2\right)}:\frac{x-1-2x+4}{\left(x-2\right)}\)
\(=\frac{-4x^2-8x}{\left(x-2\right)\left(x+2\right)}\cdot\frac{x-2}{3-x}\)
\(=\frac{-4x\left(x+2\right)}{x+2}\cdot\frac{1}{3-x}\)
\(=-\frac{4x}{3-x}=\frac{4x}{x-3}\)
a) ĐKXĐ: \(x\notin\left\{2;-2;0;3\right\}\)
Để B=-1 thì \(\frac{4x}{x-3}=-1\)
\(\Leftrightarrow4x=3-x\)
\(\Leftrightarrow4x+x=3\)
\(\Leftrightarrow5x=3\)
hay \(x=\frac{3}{5}\)(nhận)
Vậy: Để B=-1 thì \(x=\frac{3}{5}\)
b) Sửa đề: Tìm x để B<0
Để B<0 thì \(\frac{4x}{x-3}< 0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}4x>0\\x-3< 0\end{matrix}\right.\\\left\{{}\begin{matrix}4x< 0\\x-3>0\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x>0\\x< 3\end{matrix}\right.\\\left\{{}\begin{matrix}x< 0\\x>3\end{matrix}\right.\end{matrix}\right.\Leftrightarrow0< x< 3\)
Kết hợp ĐKXĐ, ta được:
\(\left\{{}\begin{matrix}0< x< 3\\x\ne2\end{matrix}\right.\)
Vậy: Để B<0 thì \(\left\{{}\begin{matrix}0< x< 3\\x\ne2\end{matrix}\right.\)