MTC = x2 - 4
\(\dfrac{1-x}{1+2}+\dfrac{x}{x-2}+\dfrac{6-7x}{x^2-4}\\ =\dfrac{\left(1-x\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}+\dfrac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{6-7x}{x^2-4}\\ =\dfrac{x-2-x^2+2x+x^2+2x+6-7x}{x^2-4}\\ =\dfrac{-2x+4}{x^2-4}=\dfrac{2\left(-x+2\right)}{x^2-4}\)
\(\dfrac{1-x}{x+2}+\dfrac{x}{x-2}+\dfrac{6-7x}{x^2-4}\\ =\dfrac{1-x}{x+2}+\dfrac{x}{x-2}+\dfrac{6-7x}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{1-x\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}+\dfrac{x\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{6-7x}{\left(x-2\right)\left(x+2\right)}\)
\(=\dfrac{1+4x+6-7x}{\left(x-2\right)\left(x+2\right)}\\ =\dfrac{7-3x}{\left(x-2\right)\left(x+2\right)}.\)
\(=\dfrac{1-x}{x+2}+\dfrac{x}{x-2}+\dfrac{6-7x}{\left(x+2\right)\left(x-2\right)}=\dfrac{\left(1-x\right)\left(x-2\right)+x\left(x+2\right)+6-7x}{\left(x-2\right)\left(x+2\right)}=\dfrac{x-x^2-2+2x+x^2+2x+6-7x}{\left(x-2\right)\left(x+2\right)}=\dfrac{-2x+4}{\left(x-2\right)\left(x+2\right)}=\dfrac{-2}{x+2}\)
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