Why is x> 2where u get that and why not the inverse of that

The reason we conclude \( x > 2 \) as one of the intervals where the inequality \( 3x^2 - 4x - 4 > 0 \) holds true comes from analyzing the behavior of the quadratic function around its roots.

### Detailed Explanation:

1. **Roots of the Quadratic**:
   - We found the roots of the quadratic equation \( 3x^2 - 4x - 4 = 0 \) to be \( x = -\frac{2}{3} \) and \( x = 2 \).
   - These roots divide the real number line into three intervals: \( (-\infty, -\frac{2}{3}) \), \( (-\frac{2}{3}, 2) \), and \( (2, \infty) \).

2. **Behavior of the Quadratic**:
   - The quadratic function \( 3x^2 - 4x - 4 \) is a parabola that opens upwards because the coefficient of \( x^2 \) (which is 3) is positive.
   - This means the parabola will be positive (above the x-axis) outside the interval between the roots, and negative (below the x-axis) within the interval between the roots.

3. **Sign Analysis**:
   - **Interval 1: \( x < -\frac{2}{3} \)**: 
     - To the left of the smaller root \( x = -\frac{2}{3} \), the quadratic expression is positive.
   - **Interval 2: \( -\frac{2}{3} < x < 2 \)**:
     - Between the roots \( -\frac{2}{3} \) and \( 2 \), the quadratic expression is negative.
   - **Interval 3: \( x > 2 \)**:
     - To the right of the larger root \( x = 2 \), the quadratic expression is again positive.

Therefore, the inequality \( 3x^2 - 4x - 4 > 0 \) is satisfied in the intervals where the quadratic expression is positive, which are:

- \( x < -\frac{2}{3} \)
- \( x > 2 \)

### Why Not the Inverse?

The intervals where \( -\frac{2}{3} < x < 2 \) would correspond to the quadratic expression being negative (i.e., \( 3x^2 - 4x - 4 < 0 \)). Since we're solving the inequality \( 3x^2 - 4x - 4 > 0 \), we are interested only in the regions where the expression is positive, hence the intervals \( x < -\frac{2}{3} \) and \( x > 2 \).

### Visualization:
If you were to graph the quadratic function \( y = 3x^2 - 4x - 4 \), you'd see a parabola crossing the x-axis at \( x = -\frac{2}{3} \) and \( x = 2 \), with the parabola being above the x-axis (positive) outside of these roots, and below the x-axis (negative) between these roots.

Does this clarify your question? Feel free to ask for further explanations!

### Related Questions:
1. How can you graph a quadratic function to determine its positive and negative regions?
2. Why does the sign of the quadratic coefficient (positive or negative) affect the direction of the parabola?
3. What are the steps to solve a quadratic inequality graphically?
4. How does completing the square help in understanding the vertex and direction of a parabola?
5. What is the difference between solving \( f(x) > 0 \) and \( f(x) < 0 \) for a quadratic function?

### Tip:
When solving inequalities, a sign analysis on the intervals created by the roots can help you quickly determine where the function is positive or negative.

Explore why the inequality x > 2 is true in the context of the quadratic equation 3x^2 - 4x - 4 > 0. Learn how to analyze quadratic inequalities, intervals, and graphical representations. Suitable for high school students.

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