If `x^(2)le4`,`xin`___________

Answer

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

### Step 1: Rewrite the inequality
We start with the inequality:
\[ x^2 \leq 4 \]

### Step 2: Move all terms to one side
We can rewrite the inequality as:
\[ x^2 - 4 \leq 0 \]

### Step 3: Factor the expression
Next, we factor the left-hand side:
\[ (x - 2)(x + 2) \leq 0 \]

### Step 4: Find the critical points
The critical points occur when each factor is equal to zero:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x + 2 = 0 \) → \( x = -2 \)

So, the critical points are \( x = -2 \) and \( x = 2 \).

### Step 5: Test intervals
We will test the intervals defined by the critical points:
1. \( (-\infty, -2) \)
2. \( (-2, 2) \)
3. \( (2, \infty) \)

- **Interval 1: \( (-\infty, -2) \)**
Choose \( x = -3 \):
\[ (-3 - 2)(-3 + 2) = (-5)(-1) = 5 \quad (\text{not } \leq 0) \]

- **Interval 2: \( (-2, 2) \)**
Choose \( x = 0 \):
\[ (0 - 2)(0 + 2) = (-2)(2) = -4 \quad (\text{is } \leq 0) \]

- **Interval 3: \( (2, \infty) \)**
Choose \( x = 3 \):
\[ (3 - 2)(3 + 2) = (1)(5) = 5 \quad (\text{not } \leq 0) \]

### Step 6: Include the critical points
Since the inequality is \( \leq 0 \), we include the critical points where the expression equals zero:
- At \( x = -2 \):
\[ (-2 - 2)(-2 + 2) = (0)(-4) = 0 \quad (\text{is } = 0) \]
- At \( x = 2 \):
\[ (2 - 2)(2 + 2) = (0)(4) = 0 \quad (\text{is } = 0) \]

### Final Solution
Thus, the solution to the inequality \( x^2 \leq 4 \) is:
\[ x \in [-2, 2] \]