\(\frac{1}{x-1}+\frac{1}{x-2}=\frac{1}{x+2}+\frac{1}{x+1}\)
\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x+1}+\frac{1}{x-2}-\frac{1}{x+2}=0\)
\(\Leftrightarrow\frac{2}{\left(x-1\right)\left(x+1\right)}+\frac{4}{\left(x-2\right)\left(x+2\right)}=0\)
\(\Leftrightarrow\frac{2\left(x^2-4\right)}{\left(x^2-1\right)\left(x^2-4\right)}+\frac{4\left(x^2-1\right)}{\left(x^2-1\right)\left(x^2-4\right)}=0\)
\(\Rightarrow2x^2-8+4x^2-4=0\\ \Leftrightarrow6x^2-12=0\)
\(\Leftrightarrow6x^2=12\\ \Leftrightarrow x^2=2\\ \Leftrightarrow x=\pm\sqrt{2}\)
\(\frac{1}{x-1}+\frac{1}{x-2}=\frac{1}{x+2}+\frac{1}{x+1}\)(đkxđ: \(x\ne1;2;-1;-2\))
\(\Leftrightarrow\frac{\left(x-2\right)+\left(x-1\right)}{\left(x-1\right)\left(x-2\right)}=\frac{\left(x+1\right)+\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}\)
\(\Leftrightarrow\frac{2x-3}{x^2-x-2x+2}=\frac{2x+3}{x^2+x+2x+2}\)
\(\Leftrightarrow\frac{2x-3}{x^2-3x+2}=\frac{2x+3}{x^2+3x+2}\)
<=> (2x - 3)(x2 + 3x + 2) = (2x + 3)(x2 - 3x + 2)
<=> 2x3 + 6x2 + 4x - 3x2 - 9x - 6 = 2x3 - 6x2 + 4x + 3x2 - 9x + 6
<=> 3x2 - 6 = -3x2 + 6
<=> 3x2 + 3x2 = 6 + 6
<=> 6x2 = 12 <=> x2 = 2
\(\Rightarrow\left[\begin{matrix}x=\sqrt{2}\\x=-\sqrt{2}\end{matrix}\right.\) (TM)
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