1. **State the problem:** Find the number of integer values of $x$ that satisfy the inequality $$3 < 2x + 1 \leq 11.$$\n\n2. **Rewrite the inequality:** We want to find all integers $x$ such that $$3 < 2x + 1 \leq 11.$$\n\n3. **Isolate $x$:** Subtract 1 from all parts of the inequality:\n$$3 - 1 < 2x + 1 - 1 \leq 11 - 1$$\nwhich simplifies to $$2 < 2x \leq 10.$$\n\n4. **Divide all parts by 2:** Since 2 is positive, the inequality signs remain the same:\n$$\frac{2}{2} < x \leq \frac{10}{2}$$\nwhich simplifies to $$1 < x \leq 5.$$\n\n5. **Find integer values of $x$:** The integers $x$ satisfying $$1 < x \leq 5$$ are $$x = 2, 3, 4, 5.$$\n\n6. **Count the integers:** There are 4 such integers.\n\n**Final answer:** There are **4** integer values of $x$ that satisfy the inequality.