$$\frac{2}{x+1}+\frac{3}{3x-1}=\frac{3}{x}$$
$x=\frac{3}{5}=0.6$
$$x\left(3x-1\right)\times 2+x\left(x+1\right)\times 3=\left(3x-1\right)\left(x+1\right)\times 3$$
$$\left(3x^{2}-x\right)\times 2+x\left(x+1\right)\times 3=\left(3x-1\right)\left(x+1\right)\times 3$$
$$6x^{2}-2x+x\left(x+1\right)\times 3=\left(3x-1\right)\left(x+1\right)\times 3$$
$$6x^{2}-2x+\left(x^{2}+x\right)\times 3=\left(3x-1\right)\left(x+1\right)\times 3$$
$$6x^{2}-2x+3x^{2}+3x=\left(3x-1\right)\left(x+1\right)\times 3$$
$$9x^{2}-2x+3x=\left(3x-1\right)\left(x+1\right)\times 3$$
$$9x^{2}+x=\left(3x-1\right)\left(x+1\right)\times 3$$
$$9x^{2}+x=\left(3x^{2}+2x-1\right)\times 3$$
$$9x^{2}+x=9x^{2}+6x-3$$
$$9x^{2}+x-9x^{2}=6x-3$$
$$x=6x-3$$
$$x-6x=-3$$
$$-5x=-3$$
$$x=\frac{-3}{-5}$$
$$x=\frac{3}{5}$$
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