1. **State the problem:** Find the number of integer values of $x$ that satisfy the inequality $$5 < 2x + 1 \leq 9.$$\n\n2. **Isolate the variable term:** Subtract 1 from all parts of the inequality to get $$5 - 1 < 2x + 1 - 1 \leq 9 - 1,$$ which simplifies to $$4 < 2x \leq 8.$$\n\n3. **Divide by 2:** Since 2 is positive, dividing by 2 preserves the inequality direction: $$\frac{4}{2} < x \leq \frac{8}{2},$$ so $$2 < x \leq 4.$$\n\n4. **Find integer values:** The integers $x$ satisfying $2 < x \leq 4$ are $3$ and $4$.\n\n5. **Count the integers:** There are 2 such integers.\n\n**Final answer:** There are 2 integer values of $x$ that satisfy the inequality.