### Instruction: Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.### Response: To solve this problem, I need to find the values of $x$ that satisfy both $2x>1$ and $3x<2$.
I can divide both sides of each inequality by the coefficient of $x$ to get $x>1/2$ and $x<2/3$.
Now I have two inequalities that define an interval for $x$: $(1/2,2/3)$.
To check if this is correct, I can plug in some values of $x$ in this interval and see if they work.
For example, if $x=0.6$, then $2x=1.2$ and $3x=1.8$, both of which are in the interval $(1,2)$.
If $x=0.5$ or $x=2/3$, then one of the inequalities is not satisfied, so they are not in the solution set.
Therefore, the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$ is $(1/2,2/3)$.