show me all the methods to solve this inequality 

Certainly! Solving a quadratic inequality like \( 3x^2 - 4x - 4 > 0 \) can be approached using several methods. I'll walk you through four common methods:

### 1. **Algebraic Method (Using the Quadratic Formula and Sign Analysis)**
   - **Step 1:** Solve the quadratic equation \( 3x^2 - 4x - 4 = 0 \) using the quadratic formula:
     \[
     x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
     \]
     The roots are \( x_1 = 2 \) and \( x_2 = -\frac{2}{3} \).
   - **Step 2:** Determine the sign of the quadratic expression in the intervals defined by these roots: \( (-\infty, -\frac{2}{3}) \), \( (-\frac{2}{3}, 2) \), and \( (2, \infty) \).
   - **Step 3:** Since the inequality is \( > 0 \), the solution will be in the intervals where the quadratic expression is positive:
     \[
     x \in (-\infty, -\frac{2}{3}) \cup (2, \infty)
     \]

### 2. **Graphical Method**
   - **Step 1:** Graph the quadratic function \( y = 3x^2 - 4x - 4 \).
   - **Step 2:** Identify the points where the graph intersects the x-axis (i.e., the roots \( x_1 = 2 \) and \( x_2 = -\frac{2}{3} \)).
   - **Step 3:** Observe the regions where the graph is above the x-axis (where \( y > 0 \)). The solution is the x-values corresponding to these regions:
     \[
     x \in (-\infty, -\frac{2}{3}) \cup (2, \infty)
     \]

### 3. **Test Point Method**
   - **Step 1:** Solve the quadratic equation \( 3x^2 - 4x - 4 = 0 \) to find the roots \( x_1 = 2 \) and \( x_2 = -\frac{2}{3} \).
   - **Step 2:** Select test points in each interval: \( (-\infty, -\frac{2}{3}) \), \( (-\frac{2}{3}, 2) \), and \( (2, \infty) \).
     - Test \( x = -1 \) in \( (-\infty, -\frac{2}{3}) \): \( 3(-1)^2 - 4(-1) - 4 = 3 \), positive.
     - Test \( x = 0 \) in \( (-\frac{2}{3}, 2) \): \( 3(0)^2 - 4(0) - 4 = -4 \), negative.
     - Test \( x = 3 \) in \( (2, \infty) \): \( 3(3)^2 - 4(3) - 4 = 11 \), positive.
   - **Step 3:** Conclude that the quadratic expression is positive in the intervals \( (-\infty, -\frac{2}{3}) \) and \( (2, \infty) \).

### 4. **Factoring Method (if applicable)**
   - **Step 1:** Try to factor the quadratic expression \( 3x^2 - 4x - 4 \). However, in this case, factoring is not straightforward, so we revert to using the quadratic formula.
   - **Step 2:** Solve the inequality by determining where the factored expression (if it were factorizable) would be positive or negative, leading to similar intervals as above.

### Summary
Regardless of the method chosen, the solution to \( 3x^2 - 4x - 4 > 0 \) is:

\[
x \in (-\infty, -\frac{2}{3}) \cup (2, \infty)
\]

Would you like to explore any of these methods in more detail?

Here are 5 related questions you might consider:
1. How can you verify the solution of a quadratic inequality?
2. What are the advantages and disadvantages of each method?
3. How do the roots of the quadratic equation relate to the solution of the inequality?
4. Can you solve a quadratic inequality without finding the exact roots?
5. What happens if the quadratic expression cannot be factored easily?

**Tip:** When using the test point method, ensure you pick points that are clearly within each interval to avoid errors.

Explore multiple methods to solve the quadratic inequality 3x^2 - 4x - 4 > 0, including algebraic, graphical, and test point methods. Understand the application of the quadratic formula and sign analysis to determine intervals of solutions. Suitable for high school students studying quadratic equations and inequalities.

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