`x^2-4lt0`

Options

A. `-2lexle2`
B. `-2ltxlt2`
C. `(-4,4)`
D. none of these

Answer

To solve the inequality \( x^2 - 4 < 0 \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:
\[ x^2 - 4 < 0 \]

### Step 2: Factor the Expression
We can factor the left-hand side using the difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]
So, we rewrite the inequality as:
\[ (x - 2)(x + 2) < 0 \]

### Step 3: Identify Critical Points
Next, we find the critical points by setting each factor to zero:
1. \( x - 2 = 0 \) gives \( x = 2 \)
2. \( x + 2 = 0 \) gives \( x = -2 \)

The critical points are \( x = -2 \) and \( x = 2 \).

### Step 4: Test Intervals
We will test the intervals determined by the critical points:
1. Interval 1: \( (-\infty, -2) \)
2. Interval 2: \( (-2, 2) \)
3. Interval 3: \( (2, \infty) \)

**Testing Interval 1: \( (-\infty, -2) \)**
Choose \( x = -3 \):
\[ (-3 - 2)(-3 + 2) = (-5)(-1) = 5 > 0 \]

**Testing Interval 2: \( (-2, 2) \)**
Choose \( x = 0 \):
\[ (0 - 2)(0 + 2) = (-2)(2) = -4 < 0 \]

**Testing Interval 3: \( (2, \infty) \)**
Choose \( x = 3 \):
\[ (3 - 2)(3 + 2) = (1)(5) = 5 > 0 \]

### Step 5: Determine the Solution Set
From our tests, we find that the inequality \( (x - 2)(x + 2) < 0 \) holds true in the interval \( (-2, 2) \).

### Step 6: Write the Final Answer
Thus, the solution to the inequality \( x^2 - 4 < 0 \) is:
\[ \boxed{(-2, 2)} \]