Giải thích các bước giải:

1.Ta có: $x=2\to A=\dfrac{4\cdot 2+1}{2-1}=9$

2.Ta có:

$B=\dfrac{3x+1}{x^2-1}-\dfrac{2x}{x-1}+\dfrac{3x}{x+1}$

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

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

$\to B=\dfrac{x^2-2x+1}{\left(x+1\right)\left(x-1\right)}$

$\to B=\dfrac{(x-1)^2}{\left(x+1\right)\left(x-1\right)}$

$\to B=\dfrac{x-1}{x+1}$

3.Để $|AB|=4x$

$\to |\dfrac{4x+1}{x-1}\cdot \dfrac{x-1}{x+1}|=4x$

$\to |\dfrac{4x+1}{x+1}|=4x$

Vì $ |\dfrac{4x+1}{x+1}|\ge 0\to 4x\ge 0\to x\ge 0$

$\to \dfrac{4x+1}{x+1}>0$

$\to \dfrac{4x+1}{x+1}=4x$

$\to 4x+1=4x^2+4x$

$\to 4x^2=1$

$\to x^2=\dfrac14$

$\to x=\pm\dfrac12$