Find all $x$ so that neither $-2(6+2x)\le -16-x+8-5x$ nor $-3x\ge 18-7x+25$.

1 answer

To deal with the inequality $-2(6+2x)\le -16-x+8-5x$, we simplify the left side and the right side separately. On the left side, we distribute the $-2$ to get $-12-4x$. On the right side, we combine like terms and get $-8-6x$. Thus our inequality is \begin{align*}
-12-4x\le-8-6x.
\end{align*}We add $4x$ to both sides and add $8$ to both sides to obtain \begin{align*}
12&\le 2x.
\end{align*}Dividing both sides by $2$ and reversing the inequality gives $x\ge 6$.

To deal with the inequality $-3x\ge 18-7x+25$, we simplify the left side and the right side separately. On the right side, we combine like terms and get $43-7x$. Thus our inequality is \begin{align*}
-3x\ge 43-7x.
\end{align*}Adding $3x$ to both sides and adding $7x$ to both sides to obtain \begin{align*}
43&\le 10x.
\end{align*}Dividing both sides by $10$ gives $x\ge \frac{43}{10}=\frac{86}{20}=\boxed{\frac{43}{10}}$.

Finally, we notice that the problem statement asks for values of $x$ that satisfy neither of the inequalities. To do this, we look for values of $x$ that satisfy both $x\ge 6$ and $x\ge \frac{43}{10}$. When $x\ge \frac{43}{10}$, it is guarantted that $x\ge 6$, so the only solutions to both inequalities are $x\ge \frac{43}{10}$. Therefore, the final solution is $\boxed{x>\frac{43}{10}}$.