1. **State the problem:** We need to find the least integer value of $x$ that satisfies the inequality $$\frac{5x - 3}{2} > 5.$$\n\n2. **Isolate the variable:** Multiply both sides of the inequality by 2 to eliminate the denominator:\n$$5x - 3 > 10.$$\n\n3. **Solve for $x$:** Add 3 to both sides:\n$$5x > 13.$$\n\n4. **Divide both sides by 5:**\n$$x > \frac{13}{5} = 2.6.$$\n\n5. **Find the least integer satisfying the inequality:** Since $x$ must be greater than 2.6, the smallest integer greater than 2.6 is 3.\n\n**Final answer:** The least integer value of $x$ that satisfies the inequality is **3**.