To solve the inequality [tex]\(\frac{1}{2} x^2 - x - 1 > x + 1\)[/tex], let's follow these steps:
1. Rewrite the inequality:
[tex]\[
\frac{1}{2} x^2 - x - 1 > x + 1
\][/tex]
2. Move all terms to one side of the inequality:
[tex]\[
\frac{1}{2} x^2 - x - 1 - (x + 1) > 0
\][/tex]
Let's simplify the left-hand side:
3. Combine like terms:
[tex]\[
\frac{1}{2} x^2 - x - 1 - x - 1 > 0
\][/tex]
[tex]\[
\frac{1}{2} x^2 - 2x - 2 > 0
\][/tex]
4. Multiply through by 2 to clear the fraction:
[tex]\[
x^2 - 4x - 4 > 0
\][/tex]
5. Solve the quadratic inequality:
We start by solving the corresponding equation:
[tex]\[
x^2 - 4x - 4 = 0
\][/tex]
6. Find the roots of the quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = -4\)[/tex].
[tex]\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{16 + 16}}{2}
\][/tex]
[tex]\[
x = \frac{4 \pm \sqrt{32}}{2}
\][/tex]
[tex]\[
x = \frac{4 \pm 4\sqrt{2}}{2}
\][/tex]
[tex]\[
x = 2 \pm 2\sqrt{2}
\][/tex]
Therefore, the roots are:
[tex]\[
x = 2 + 2\sqrt{2} \quad \text{and} \quad x = 2 - 2\sqrt{2}
\][/tex]
7. Determine the intervals defined by these roots:
[tex]\[
x = 2 + 2\sqrt{2} \quad \text{and} \quad x = 2 - 2\sqrt{2}
\][/tex]
8. Check the sign of the quadratic expression in each interval:
The intervals are:
[tex]\[
(-\infty, 2 - 2\sqrt{2}), \quad (2 - 2\sqrt{2}, 2 + 2\sqrt{2}), \quad (2 + 2\sqrt{2}, \infty)
\][/tex]
By testing points in these intervals, we can determine where the expression [tex]\(x^2 - 4x - 4 > 0\)[/tex].
9. Intervals that satisfy the inequality:
Through testing, it can be determined that the inequality [tex]\(x^2 - 4x - 4 > 0\)[/tex] is satisfied in the intervals
[tex]\[
(-\infty, 2 - 2\sqrt{2}) \quad \text{and} \quad (2 + 2\sqrt{2}, \infty)
\][/tex]
10. Express the solution set in interval notation:
The solution set is:
[tex]\[
(-\infty, 2 - 2\sqrt{2}) \cup (2 + 2\sqrt{2}, \infty)
\][/tex]
Using approximations for the roots,
[tex]\[
2 - 2\sqrt{2} \approx -0.82842712474619 \quad \text{and} \quad 2 + 2\sqrt{2} \approx 4.82842712474619
\][/tex]
This can be written as:
[tex]\[
(-\infty, -0.82842712474619) \cup (4.82842712474619, \infty)
\][/tex]
Hence, the solution to the inequality [tex]\(\frac{1}{2} x^2 - x - 1 > x + 1\)[/tex] in interval notation is:
[tex]\[
(-\infty, -0.82842712474619) \cup (4.82842712474619, \infty)
\][/tex]
