Sure, let's work through the problem step-by-step.
### Step 1: Solve the Equation [tex]\( x^2 - 7x + 11 = 0 \)[/tex]
To solve the quadratic equation [tex]\( x^2 - 7x + 11 = 0 \)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the equation [tex]\( x^2 - 7x + 11 = 0 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -7 \)[/tex]
- [tex]\( c = 11 \)[/tex]
Plug these values into the quadratic formula:
[tex]\[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot 11}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{49 - 44}}{2} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{5}}{2} \][/tex]
Thus, the roots are:
[tex]\[ x_1 = \frac{7 - \sqrt{5}}{2} \][/tex]
[tex]\[ x_2 = \frac{7 + \sqrt{5}}{2} \][/tex]
### Step 2: Identify the Intervals
The quadratic [tex]\( x^2 - 7x + 11 \)[/tex] is a parabola that opens upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive). The roots [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the points where the parabola intersects the x-axis.
The inequality [tex]\( x^2 - 7x + 11 < 0 \)[/tex] is satisfied where the parabola is below the x-axis. This happens between the roots [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex].
### Step 3: Find the Interval Where the Inequality is True
From the roots:
[tex]\[ x_1 = \frac{7 - \sqrt{5}}{2} \][/tex]
[tex]\[ x_2 = \frac{7 + \sqrt{5}}{2} \][/tex]
The inequality [tex]\( x^2 - 7x + 11 < 0 \)[/tex] holds for:
[tex]\[ \frac{7 - \sqrt{5}}{2} < x < \frac{7 + \sqrt{5}}{2} \][/tex]
### Step 4: Determine the Integer Values in the Interval
Now we need to find the integer values of [tex]\( x \)[/tex] that satisfy:
[tex]\[ \frac{7 - \sqrt{5}}{2} < x < \frac{7 + \sqrt{5}}{2} \][/tex]
First, calculate the approximate values of [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
- [tex]\( x_1 \approx \frac{7 - \sqrt{5}}{2} \approx \frac{7 - 2.236}{2} \approx \frac{4.764}{2} \approx 2.382 \)[/tex]
- [tex]\( x_2 \approx \frac{7 + \sqrt{5}}{2} \approx \frac{7 + 2.236}{2} \approx \frac{9.236}{2} \approx 4.618 \)[/tex]
The integer values between [tex]\( 2.382 \)[/tex] and [tex]\( 4.618 \)[/tex] are:
[tex]\[ x = 3, \ 4 \][/tex]
Thus, the integer values that satisfy the inequality [tex]\( x^2 - 7x + 11 < 0 \)[/tex] are [tex]\( \boxed{3 \text{ and } 4} \)[/tex].
