If `|x+2| le 9`, then

Options

A. ` x in (-7,11)`
B. `x in [-11,7]`
C. `x in (-infty, -7) cup (11,infty)`
D. `x in (-infty, -7)cup [11,infty)`

Explanation

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

### Step 1: Understand the Absolute Value Inequality
The absolute value inequality \( |A| \leq B \) implies that \( -B \leq A \leq B \). In our case, \( A = x + 2 \) and \( B = 9 \).

### Step 2: Set Up the Two Inequalities
From the absolute value inequality, we can set up the following two inequalities:
\[ -9 \leq x + 2 \leq 9 \]

### Step 3: Solve the Left Inequality
Starting with the left part of the inequality:
\[ -9 \leq x + 2 \]
Subtract 2 from both sides:
\[ -9 - 2 \leq x \]
\[ -11 \leq x \quad \text{or} \quad x \geq -11 \]

### Step 4: Solve the Right Inequality
Now, we solve the right part of the inequality:
\[ x + 2 \leq 9 \]
Subtract 2 from both sides:
\[ x \leq 9 - 2 \]
\[ x \leq 7 \]

### Step 5: Combine the Results
Now we have two inequalities:
1. \( x \geq -11 \)
2. \( x \leq 7 \)

Combining these gives us:
\[ -11 \leq x \leq 7 \]

### Step 6: Write the Final Answer
In interval notation, the solution can be expressed as:
\[ x \in [-11, 7] \]