The square root of $x$ is greater than 2 and less than 4.
This means $2<\sqrt{x}<4$.
Squaring both sides, we have $4<x<16$.
The integers satisfying this condition are 5, 6, 7, ..., 15.
Therefore, there are $\boxed{11}$ integer values of $x$ that satisfy this condition.
The answer is: 11
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
The square root of $x$ is greater than 2 and less than 4. How many integer values of $x$ satisfy this condition?

### Answer:
