### Instruction: Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.### Response: I need to find the values of $x$ that satisfy two inequalities: $1 < 2x < 2$ and $1 < 3x < 2$.
I can divide both sides of the first inequality by 2 to get $0.5 < x < 1$.
Similarly, I can divide both sides of the second inequality by 3 to get $0.333 \ldots < x < 0.666 \ldots$.
To find the values of $x$ that satisfy both inequalities, I need to find the intersection of these two intervals.
The intersection is the part of the number line where both intervals overlap.
I can see that the intersection is $(0.5, 0.666 \ldots)$, since $x$ has to be larger than 0.5 to satisfy the first inequality, and smaller than 0.666 \ldots to satisfy the second inequality.
This is the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.