$$\frac{ x+2 }{ 6 } - \frac{ x+1 }{ 5 } = \frac{ x-3 }{ 4 } -1$$
$x = \frac{113}{17} = 6\frac{11}{17} \approx 6.647058824$
$$10\left(x+2\right)-12\left(x+1\right)=15\left(x-3\right)-60$$
$$10x+20-12\left(x+1\right)=15\left(x-3\right)-60$$
$$10x+20-12x-12=15\left(x-3\right)-60$$
$$-2x+20-12=15\left(x-3\right)-60$$
$$-2x+8=15\left(x-3\right)-60$$
$$-2x+8=15x-45-60$$
$$-2x+8=15x-105$$
$$-2x+8-15x=-105$$
$$-17x+8=-105$$
$$-17x=-105-8$$
$$-17x=-113$$
$$x=\frac{-113}{-17}$$
$$x=\frac{113}{17}$$
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