Solve `4-x^(2) lt 0` by algebric method and graphical method.

Answer

To solve the inequality \( 4 - x^2 < 0 \) using both algebraic and graphical methods, we will follow these steps:

### Algebraic Method

1. **Rearranging the Inequality**:
Start with the given inequality:
\[ 4 - x^2 < 0 \]
Rearranging gives:
\[ -x^2 < -4 \]

2. **Multiplying by -1**:
When we multiply both sides of the inequality by -1, we must reverse the inequality sign:
\[ x^2 > 4 \]

3. **Taking Square Roots**:
To solve for \( x \), take the square root of both sides:
\[ x > 2 \quad \text{or} \quad x < -2 \]

4. **Final Solution**:
Therefore, the solution set for the inequality \( 4 - x^2 < 0 \) is:
\[ x < -2 \quad \text{or} \quad x > 2 \]

### Graphical Method

1. **Understanding the Inequality**:
The inequality \( 4 - x^2 < 0 \) can be interpreted as finding the regions where the quadratic function \( y = 4 - x^2 \) is below the x-axis.

2. **Finding the Roots**:
Set the equation \( 4 - x^2 = 0 \) to find the points where the graph intersects the x-axis:
\[ 4 = x^2 \implies x = \pm 2 \]

3. **Sketching the Graph**:
The graph of \( y = 4 - x^2 \) is a downward-opening parabola with its vertex at (0, 4) and intersects the x-axis at \( x = -2 \) and \( x = 2 \).

4. **Identifying the Regions**:
The parabola is above the x-axis (i.e., \( 4 - x^2 > 0 \)) between the roots \( -2 \) and \( 2 \) and below the x-axis (i.e., \( 4 - x^2 < 0 \)) outside of these roots.

5. **Final Graphical Solution**:
Thus, the solution to the inequality \( 4 - x^2 < 0 \) is:
\[ x < -2 \quad \text{or} \quad x > 2 \]

### Conclusion
The solution to the inequality \( 4 - x^2 < 0 \) is:
\[ x \in (-\infty, -2) \cup (2, \infty) \]