$$\frac{ x+1 }{ 2 } - \frac{ 5x+9 }{ 28 } = \frac{ x+6 }{ 21 } +5- \frac{ x-12 }{ 3 }$$
$x=15$
$$42\left(x+1\right)-3\left(5x+9\right)=4\left(x+6\right)+420-28\left(x-12\right)$$
$$42x+42-3\left(5x+9\right)=4\left(x+6\right)+420-28\left(x-12\right)$$
$$42x+42-15x-27=4\left(x+6\right)+420-28\left(x-12\right)$$
$$27x+42-27=4\left(x+6\right)+420-28\left(x-12\right)$$
$$27x+15=4\left(x+6\right)+420-28\left(x-12\right)$$
$$27x+15=4x+24+420-28\left(x-12\right)$$
$$27x+15=4x+444-28\left(x-12\right)$$
$$27x+15=4x+444-28x+336$$
$$27x+15=-24x+444+336$$
$$27x+15=-24x+780$$
$$27x+15+24x=780$$
$$51x+15=780$$
$$51x=780-15$$
$$51x=765$$
$$x=\frac{765}{51}$$
$$x=15$$
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