We begin by solving the first inequality, $-2(6+2w)\le -16 + 7w.$ We distribute the $-2,$ giving $-12-4w\le -16 + 7w.$ Adding $16 + 4w$ to both sides gives \[16 + 3w \le 7w,\]so $-7w\le -16 - 3w.$ Adding $3w$ to both sides gives \[16\ge 4w.\]Therefore, $w\le 4.$
Next, we solve the second inequality, $-3w\ge 18 - 5w + 3.$ Adding $5w$ to both sides gives \[2w\ge 21.\]Dividing both sides by 2 gives $w\ge \frac{21}{2}.$
The solution to the first inequality is $w\le 4,$ and the solution to the second inequality is $w\ge \frac{21}{2}.$ Therefore, we take the overlap of these solutions, which gives \[\boxed{w\in \left[ \frac{21}{2}, 4\right]}.\]
Find all $w$ so that neither $-2(6+2w)\le -16 + 7w$ nor $-3w\ge 18 - 5w + 3$.
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