If `|x+2|lt=9`, then

Options

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

Answer

To solve the inequality \( |x + 2| \leq 9 \), we will follow the steps outlined in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Modulus Inequality**:
The expression \( |x + 2| \leq 9 \) means that the distance of \( x + 2 \) from 0 is at most 9. This can be rewritten using the definition of absolute value.

2. **Applying the Definition**:
The inequality \( |x - a| \leq r \) can be expressed as:
\[ a - r \leq x \leq a + r \]
In our case, \( a = -2 \) and \( r = 9 \).

3. **Setting Up the Inequality**:
Substitute \( a \) and \( r \) into the inequality:
\[ -2 - 9 \leq x \leq -2 + 9 \]

4. **Calculating the Bounds**:
Now, calculate the left and right bounds:
- Left bound:
\[ -2 - 9 = -11 \]
- Right bound:
\[ -2 + 9 = 7 \]

5. **Writing the Final Interval**:
Therefore, we can express the solution as:
\[ -11 \leq x \leq 7 \]
In interval notation, this is written as:
\[ x \in [-11, 7] \]

### Conclusion:
The solution to the inequality \( |x + 2| \leq 9 \) is:
\[ x \in [-11, 7] \]