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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!

Find the solution set of the inequalities: x^2 - 4x - 12 < 0, x^2 - 4x - 12 ≥ 0, x^2 - 4x - 12 ≤ 0.

Learn how to solve quadratic inequalities such as x^2 - 4x - 12 < 0, x^2 - 4x - 12 ≥ 0, and x^2 - 4x - 12 ≤ 0 using factoring and the quadratic formula. This example walks through solving each inequality step by step, suitable for students in Grades 9-11.

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