### Instruction: Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.### Response: I want to find the values of $x$ that make both $2x$ and $3x$ between 1 and 2.
I can write this as a system of inequalities: $1 < 2x < 2$ and $1 < 3x < 2$.
I can try to solve for $x$ in each inequality separately and then find the intersection of the solutions.
For the first inequality, I can divide everything by 2 to get $0.5 < x < 1$.
For the second inequality, I can divide everything by 3 to get $0.333... < x < 0.666...$
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The intersection of these two intervals is the part where they overlap, which is $(0.5, 0.666...)$.
This means that any $x$ in this interval will make both $2x$ and $3x$ in the interval $(1,2)$.
I can write the answer using interval notation as $(0.5, 0.666...)$ or using inequalities as $0.5 < x < 0.666...$
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