$$\frac{x}{2}-\frac{1}{4}(x-\frac{1}{3})=\frac{1}{6}(x+1)+\frac{1}{12}$$
$x=2$
$$6x-3\left(x-\frac{1}{3}\right)=2\left(x+1\right)+1$$
$$6x-3x-3\left(-\frac{1}{3}\right)=2\left(x+1\right)+1$$
$$6x-3x+1=2\left(x+1\right)+1$$
$$3x+1=2\left(x+1\right)+1$$
$$3x+1=2x+2+1$$
$$3x+1=2x+3$$
$$3x+1-2x=3$$
$$x+1=3$$
$$x=3-1$$
$$x=2$$
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