To solve the inequality [tex]\( x^2 - x - 12 < 0 \)[/tex], follow these steps:
1. Solve the corresponding quadratic equation:
First, find the roots of the quadratic equation [tex]\( x^2 - x - 12 = 0 \)[/tex].
Factor the quadratic expression:
[tex]\[
x^2 - x - 12 = (x - 4)(x + 3) = 0
\][/tex]
By setting each factor equal to zero, we get the solutions:
[tex]\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\][/tex]
[tex]\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\][/tex]
So, the critical points (roots) are [tex]\( x = -3 \)[/tex] and [tex]\( x = 4 \)[/tex].
2. Determine the intervals:
The roots divide the number line into three intervals:
- [tex]\( (-\infty, -3) \)[/tex]
- [tex]\( (-3, 4) \)[/tex]
- [tex]\( (4, \infty) \)[/tex]
3. Test a point in each interval:
Choose a test point from each interval to see if the inequality [tex]\( x^2 - x - 12 < 0 \)[/tex] is satisfied.
- For the interval [tex]\( (-\infty, -3) \)[/tex]:
Test point [tex]\( x = -4 \)[/tex]
[tex]\[
(-4)^2 - (-4) - 12 = 16 + 4 - 12 = 8 \quad (\text{not less than 0})
\][/tex]
- For the interval [tex]\( (-3, 4) \)[/tex]:
Test point [tex]\( x = \frac{-3 + 4}{2} = \frac{1}{2} \)[/tex]
[tex]\[
\left(\frac{1}{2}\right)^2 - \frac{1}{2} - 12 = \frac{1}{4} - \frac{1}{2} - 12 = -\frac{47}{4} \quad (\text{less than 0})
\][/tex]
- For the interval [tex]\( (4, \infty) \)[/tex]:
Test point [tex]\( x = 5 \)[/tex]
[tex]\[
5^2 - 5 - 12 = 25 - 5 - 12 = 8 \quad (\text{not less than 0})
\][/tex]
4. Combine the intervals where the inequality holds true:
The inequality [tex]\( x^2 - x - 12 < 0 \)[/tex] is satisfied in the interval [tex]\( (-3, 4) \)[/tex].
Therefore, the solution to the inequality [tex]\( x^2 - x - 12 < 0 \)[/tex] is:
[tex]\[
x \in (-3, 4)
\][/tex]
