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3B.

a.

\(\dfrac{6}{7-x}+\dfrac{-4}{x}+\dfrac{6}{x-7}+\dfrac{2}{x-2}+\dfrac{4}{x}+\dfrac{-5}{x+2}\)

\(=\dfrac{6}{7-x}-\dfrac{6}{7-x}+\dfrac{-4+4}{x}+\dfrac{2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{-5\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\)

\(=0+0+\dfrac{2x+4-5x+10}{\left(x-2\right)\left(x+2\right)}\)

\(=\dfrac{-3x+14}{\left(x-2\right)\left(x+2\right)}\)

b.

\(\dfrac{x-1}{x^2-4}-\dfrac{1}{x+2}+\dfrac{x+2}{x}+\dfrac{1}{x+2}+\dfrac{x+2}{-x}-\dfrac{3}{x-2}\)

\(=\dfrac{x-1}{\left(x-2\right)\left(x+2\right)}-\dfrac{3\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\left(\dfrac{1}{x+2}-\dfrac{1}{x+2}\right)+\left(\dfrac{x+2}{x}-\dfrac{x+2}{x}\right)\)

\(=\dfrac{x-1-3x-6}{\left(x-2\right)\left(x+2\right)}+0+0\)

\(=\dfrac{-2x-7}{\left(x-2\right)\left(x+2\right)}\)

4A.

\(\dfrac{3x+1}{\left(x-1\right)^2}-\dfrac{1}{x+1}+\dfrac{x+3}{1-x^2}\)

\(=\dfrac{\left(3x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)^2}-\dfrac{\left(x-1\right)^2}{\left(x+1\right)\left(x-1\right)^2}-\dfrac{\left(x+3\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)^2}\)

\(=\dfrac{3x^2+4x+1-\left(x^2-2x+1\right)-\left(x^2+2x-3\right)}{\left(x+1\right)\left(x-1\right)^2}\)

\(=\dfrac{x^2+4x+3}{\left(x+1\right)\left(x-1\right)^2}=\dfrac{\left(x+1\right)\left(x+3\right)}{\left(x+1\right)\left(x-1\right)^2}\)

\(=\dfrac{x+3}{\left(x-1\right)^2}\)

b.

\(\dfrac{2x}{x^2-4}-\left(\dfrac{2}{x+5}+\dfrac{3-x}{x+1}\right)+\left(\dfrac{2}{x+5}-\left(\dfrac{4}{x^2-4}+\dfrac{x-3}{x+1}\right)\right)\)

\(=\dfrac{2x}{x^2-4}+\left(\dfrac{2}{x+5}-\dfrac{2}{x+5}\right)+\left(\dfrac{x-3}{x+1}-\dfrac{x-3}{x+1}\right)-\dfrac{4}{x^2-4}\)

\(=\dfrac{2x-4}{x^2-4}=\dfrac{2\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}\)

\(=\dfrac{2}{x+2}\)


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