The image presents three inequalities to solve involving the quadratic equation $x^2 - 4x - 12$:

1. $x^2 - 4x - 12 < 0$
2. $x^2 - 4x - 12 \geq 0$
3. $x^2 - 4x - 12 \leq 0$

To solve these inequalities, we first solve the corresponding equation:

$x^2 - 4x - 12 = 0$

We use the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)}$ $x = \frac{4 \pm \sqrt{16 + 48}}{2} = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2}$

Thus, the solutions to the equation are:

$x = \frac{4 + 8}{2} = 6 \quad \text{and} \quad x = \frac{4 - 8}{2} = -2$

So, the quadratic equation factors as:

$(x - 6)(x + 2) = 0$

Now, we use these solutions to solve each inequality.

1. $x^2 - 4x - 12 < 0$

We need to find where the quadratic expression is less than zero. The quadratic has roots at $x = -2$ and $x = 6$, so it changes sign at these points. To determine where it's negative, we check the intervals defined by the roots:

• For $x < -2$, both factors $(x + 2)$ and $(x - 6)$ are negative, so the product is positive.
• For $-2 < x < 6$, $(x + 2)$ is positive and $(x - 6)$ is negative, so the product is negative.
• For $x > 6$, both factors are positive, so the product is positive.

Thus, $x^2 - 4x - 12 < 0$ on the interval $(-2, 6)$.

2. $x^2 - 4x - 12 \geq 0$

This inequality holds where the quadratic expression is zero or positive. From the analysis above, we see that the expression is zero at $x = -2$ and $x = 6$, and it is positive for $x < -2$ and $x > 6$.

Thus, the solution is $(-\infty, -2] \cup [6, \infty)$.

3. $x^2 - 4x - 12 \leq 0$

This inequality holds where the quadratic is less than or equal to zero. Combining the results from the previous two parts, we have:

$x^2 - 4x - 12 \leq 0 \quad \text{on} \quad [-2, 6]$

Summary of solutions:

1. $x^2 - 4x - 12 < 0$ on $(-2, 6)$
2. $x^2 - 4x - 12 \geq 0$ on $(-\infty, -2] \cup [6, \infty)$
3. $x^2 - 4x - 12 \leq 0$ on $[-2, 6]$

Let me know if you would like more details or have questions!

Here are some related questions:

1. How does the sign of a quadratic change across its roots?
2. What are the general methods for solving quadratic inequalities?
3. How can you verify the solution to a quadratic inequality using a graph?
4. What role do the discriminant and roots play in determining the sign of a quadratic?
5. How does factoring help simplify solving quadratic inequalities?

Tip: When solving inequalities, always consider the intervals around the roots where the quadratic expression changes sign!