### Instruction: The square root of $2x$ is greater than 3 and less than 4. How many integer values of $x$ satisfy this condition?### Response: I need to find the integer values of $x$ that make the square root of $2x$ between 3 and 4.
I can start by squaring both sides of the inequality to get rid of the square root.
That gives me $2x > 9$ and $2x < 16$.
Dividing everything by 2, I get $x > 4.5$ and $x < 8$.
Now I can see that the possible integer values of $x$ are 5, 6, and 7, since they are the only ones that are greater than 4.5 and less than 8.
Therefore, there are 3 integer values of $x$ that satisfy the condition.