Find all $x$ such that neither $2+x+5x$ nor $2-x-8-3x$ is in the interval $(-2,11]$. Express your answer as an interval or as a union of intervals.

Answer 1

First, we set $2+x+5x<-2$. Simplifying the left-hand side gives $6x<-4$, so $x<-\frac{2}{3}$.

Next, we set $2-x-8-3x>11$. Simplifying the left-hand side gives $-4x<-21$, so $x>\frac{21}{4}$.

Therefore, neither inequality is satisfied if $-\frac{2}{3}<x<\frac{21}{4}$, so the values of $x$ that satisfy neither inequality are $x\leq-\frac{2}{3}$ or $x\geq\frac{21}{4}$. In interval notation, this is $\boxed{(-\infty, -\frac23] \cup [\frac{21}{4}, \infty)}$.