Find all $x$ such that neither $2x-5$ nor $-20x+13x-7$ is in the interval $(8,12)$.

Answer 1

First, $2x-5 \in (8,12)$ means that $8 < 2x-5 < 12$, so $13 < 2x < 17$, and $6.5 < x < 8.5$. Since we cannot have $x$ in the interval $(6.5, 8.5)$, it follows that the numbers which satisfy $2x-5 \in (8,12)$ are those that are equal to or less than $6.5$ or equal to or greater than $8.5$.

Now, $-20x+13x-7 \in (8,12)$ means that $8 < -20x+13x-7 < 12$. Rearranging this gives $15 < -7x < 19$, and dividing each term by $-7$ gives $-15/7 > x > -19/7$. Hence, any $x$ satisfying $-15/7 > x > -19/7$ is acceptable.

Graphing both inequalities, we have that $2x-5$ has all points to the left of $6.5$ shaded, while $-20x+13x-7$ has all points in between $-15/7$ and $-19/7$ shaded. To find the union of these shaded regions, which correspond to the numbers that satisfy neither $2x-5$ nor $-20x+13x-7$ in the interval $(8,12)$, subtract $x$ instead of adding. The answer is any $x$ that is $\boxed{\leq -19/7 \text{ or }\geq 8.5}$.