The inequality given is:

$x^2 + 2x - 12 < 0$

Steps to solve the inequality:

1. Solve the corresponding equation: First, we solve the quadratic equation: $x^2 + 2x - 12 = 0$ We can factor or use the quadratic formula to solve this.

2. Find the roots using the quadratic formula: The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 2$, and $c = -12$. Substituting these values into the formula: $x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-12)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 48}}{2} = \frac{-2 \pm \sqrt{52}}{2} = \frac{-2 \pm 2\sqrt{13}}{2}$ Simplifying further: $x = -1 \pm \sqrt{13}$

   So the roots are: $x_1 = -1 + \sqrt{13}, \quad x_2 = -1 - \sqrt{13}$

3. Determine the intervals: The quadratic expression changes sign at the roots. So, the number line is divided into three intervals:

   • $(-\infty, x_2)$
   • $(x_2, x_1)$
   • $(x_1, \infty)$

4. Test the sign of the expression in each interval:

   • For $x \in (-\infty, x_2)$, the expression is positive.
   • For $x \in (x_2, x_1)$, the expression is negative.
   • For $x \in (x_1, \infty)$, the expression is positive.

5. Conclusion: The solution to $x^2 + 2x - 12 < 0$ is the interval where the quadratic expression is negative, which occurs between the two roots: $x_2 < x < x_1$ Thus, the solution is: $(-1 - \sqrt{13}, -1 + \sqrt{13})$

Would you like more details on any step or further explanation?

Related Questions:

1. How can you graphically represent this quadratic inequality?
2. What is the significance of the discriminant in solving quadratic inequalities?
3. How do you solve a quadratic inequality by factoring instead of using the quadratic formula?
4. How does the solution of a quadratic inequality differ from a quadratic equation?
5. What real-world scenarios might involve solving quadratic inequalities?

Tip:

When solving quadratic inequalities, always check the sign of the expression in each interval divided by the roots to ensure correct solution intervals.