Find all $z$ such that neither $2+z-5z$ nor $2-z+8z$ is in the interval $(-7,13]$. Express your answer as an interval or as a union of intervals.
1 answer
Simplifying $2+z-5z$ gives $2(1-4z)$, while simplifying $2-z+8z$ gives $2(1+7z)$. We want to find all $z$ such that $2(1-4z)\leq-7$ or $2(1-4z)>13$, and such that $2(1+7z)\leq-7$ or $2(1+7z)>13$. Simplifying these inequalities gives $-4z\leq-9$ or $-4z>11$ and $7z\leq-9$ or $7z>11$. The first inequality is satisfied when $3/2\leq z$, and the second is satisfied when $z<-11/4$. Taking the intersection of these two possibilities, we have $3/2\leq z<-11/4$, or we can express this as $\boxed{\left(\frac32, -\frac{11}{4}\right)}$.