If `|x^2-7|le 9 ` then find the values of x

Explanation

To solve the inequality \( |x^2 - 7| \leq 9 \), we will follow these steps:

### Step 1: Remove the absolute value
The expression \( |x^2 - 7| \leq 9 \) can be rewritten as two separate inequalities:
\[ -9 \leq x^2 - 7 \leq 9 \]

### Step 2: Split into two inequalities
This gives us two inequalities to solve:
1. \( x^2 - 7 \geq -9 \)
2. \( x^2 - 7 \leq 9 \)

### Step 3: Solve the first inequality
For the first inequality:
\[ x^2 - 7 \geq -9 \]
Adding 7 to both sides:
\[ x^2 \geq -2 \]
Since \( x^2 \) is always non-negative, this inequality is always true for all real numbers \( x \).

### Step 4: Solve the second inequality
For the second inequality:
\[ x^2 - 7 \leq 9 \]
Adding 7 to both sides:
\[ x^2 \leq 16 \]

### Step 5: Find the values of \( x \)
Taking the square root of both sides:
\[ -\sqrt{16} \leq x \leq \sqrt{16} \]
This simplifies to:
\[ -4 \leq x \leq 4 \]

### Conclusion
Thus, the solution set for \( x \) is:
\[ x \in [-4, 4] \]