Split the compound inequality into two separate inequalities.

Convert the mixed number to an improper fraction: $5\frac{1}{2} = \frac{11}{2}$.

Multiply the entire inequality by 2 to eliminate fractions:

Subtract $2x$ from both sides:

Add 4 to both sides:

Divide by 3:

Convert mixed numbers to improper fractions: $5\frac{1}{2} = \frac{11}{2}$ and $1\frac{1}{2} = \frac{3}{2}$.

Multiply the entire inequality by 2 to eliminate fractions:

Subtract $2x$ from both sides:

Subtract 3 from both sides:

Divide by 4:

From Step 2, we have $x \le 5$.
From Step 3, we have $x > 2$.
Combining these two conditions gives $2 < x \le 5$.

The integral values of $x$ that are greater than 2 and less than or equal to 5 are 3, 4, and 5.

The integral values are $\boxed{3, 4, 5}$.