How will I write this as a set builder; inequality
3
x
2
−
4
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4
>
0
3x
2
−4x−4>0 in set-builder

To write the inequality \( 3x^2 - 4x - 4 > 0 \) in set-builder notation, you first need to determine the values of \( x \) that satisfy the inequality. Here's how you can express it:

1. **Solve the inequality**: First, solve the quadratic inequality \( 3x^2 - 4x - 4 > 0 \).
2. **Find the roots**: You would solve the quadratic equation \( 3x^2 - 4x - 4 = 0 \) using the quadratic formula:

   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 3 \), \( b = -4 \), and \( c = -4 \).

   Substituting the values, we get:

   \[
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-4)}}{2(3)} = \frac{4 \pm \sqrt{16 + 48}}{6} = \frac{4 \pm \sqrt{64}}{6} = \frac{4 \pm 8}{6}
   \]

   This gives the roots:
   \[
   x_1 = \frac{12}{6} = 2 \quad \text{and} \quad x_2 = \frac{-4}{6} = -\frac{2}{3}
   \]

3. **Determine the intervals**: The inequality \( 3x^2 - 4x - 4 > 0 \) will hold true in the intervals where the quadratic expression is positive. After analyzing the sign changes around the roots \( x = -\frac{2}{3} \) and \( x = 2 \), we determine that the inequality is true for \( x < -\frac{2}{3} \) and \( x > 2 \).

4. **Express in set-builder notation**:
   \[
   \{x \in \mathbb{R} \mid x < -\frac{2}{3} \text{ or } x > 2\}
   \]

This is the set-builder notation for the given inequality.

Would you like more details on any of these steps or explanations?

### Related Questions:
1. How do you find the roots of a quadratic equation using the quadratic formula?
2. What does set-builder notation represent in mathematics?
3. How can you determine where a quadratic function is positive or negative?
4. How do you graph a quadratic inequality on a number line?
5. What are the common methods for solving quadratic inequalities?

### Tip:
When working with inequalities, always pay attention to whether the inequality is strict (e.g., \( > \) or \( < \)) or inclusive (e.g., \( \geq \) or \( \leq \)), as this will affect the set notation and the intervals.

Learn how to express the quadratic inequality 3x^2 - 4x - 4 > 0 in set-builder notation. This problem involves solving a quadratic inequality using the quadratic formula and converting the solution into mathematical set notation. Suitable for high school students studying quadratic inequalities.

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