1. **State the problem:** Solve the quadratic equation $x^2 - 7x + 11 = 0$ and then find the integer values of $x$ that satisfy the inequality $x^2 - 7x + 11 < 0$. 2. **Recall the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-7$, and $c=11$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-7)^2 - 4 \times 1 \times 11 = 49 - 44 = 5$$ 4. **Find the roots:** $$x = \frac{-(-7) \pm \sqrt{5}}{2 \times 1} = \frac{7 \pm \sqrt{5}}{2}$$ 5. **Approximate the roots:** $$\sqrt{5} \approx 2.236$$ So, $$x_1 = \frac{7 - 2.236}{2} = \frac{4.764}{2} = 2.382$$ $$x_2 = \frac{7 + 2.236}{2} = \frac{9.236}{2} = 4.618$$ 6. **Analyze the inequality $x^2 - 7x + 11 < 0$:** Since the parabola opens upwards ($a=1 > 0$), the quadratic is less than zero between the roots. 7. **Determine integer values satisfying the inequality:** The inequality holds for $2.382 < x < 4.618$. The integers in this interval are $3$ and $4$. **Final answer:** The integer values of $x$ that satisfy $x^2 - 7x + 11 < 0$ are $3$ and $4$.