Number of integers satisfying the inequality
`(1/3)^(|x+2|/(2-|x|)) > 9` is

Answer

To solve the inequality \((1/3)^{(|x+2|/(2-|x|))} > 9\), we can follow these steps:

### Step 1: Rewrite the Inequality
We start by rewriting the inequality in terms of base 3:
\[ (1/3)^{(|x+2|/(2-|x|))} > 9 \]
This can be rewritten as:
\[ 3^{-1 \cdot (|x+2|/(2-|x|))} > 3^2 \]
Since both sides have the same base, we can drop the base and reverse the inequality:
\[ -\frac{|x+2|}{2-|x|} > 2 \]

### Step 2: Multiply by -1
Multiplying both sides by -1 (which reverses the inequality):
\[ \frac{|x+2|}{2-|x|} < -2 \]
Since the left side is always non-negative (as it is an absolute value divided by a positive quantity), this inequality has no solutions. Thus, we need to analyze the cases for \(x\) to see if we can find any valid solutions.

### Step 3: Break into Cases
We need to consider different cases based on the value of \(x\).

#### Case 1: \(x < -2\)
In this case:
- \(|x+2| = -x - 2\)
- \(|x| = -x\)

Substituting these into the inequality:
\[ \frac{-x - 2}{2 - (-x)} < -2 \]
This simplifies to:
\[ \frac{-x - 2}{2 + x} < -2 \]
Cross-multiplying (noting that \(2 + x > 0\) for \(x < -2\)):
\[ -x - 2 > -2(2 + x) \]
This leads to:
\[ -x - 2 > -4 - 2x \]
Rearranging gives:
\[ x < -2 \]
This does not provide any new information, so we check for solutions in this range.

#### Case 2: \(-2 \leq x < 0\)
Here:
- \(|x+2| = x + 2\)
- \(|x| = -x\)

Substituting gives:
\[ \frac{x + 2}{2 - (-x)} < -2 \]
This simplifies to:
\[ \frac{x + 2}{2 + x} < -2 \]
Cross-multiplying:
\[ x + 2 > -2(2 + x) \]
This leads to:
\[ x + 2 > -4 - 2x \]
Rearranging gives:
\[ 3x > -6 \implies x > -2 \]
So, in this case, we have \(-2 < x < 0\).

#### Case 3: \(x \geq 0\)
Here:
- \(|x+2| = x + 2\)
- \(|x| = x\)

Substituting gives:
\[ \frac{x + 2}{2 - x} < -2 \]
Cross-multiplying:
\[ x + 2 > -2(2 - x) \]
This leads to:
\[ x + 2 > -4 + 2x \]
Rearranging gives:
\[ 2 > x \]
Thus, we have \(0 \leq x < 2\).

### Step 4: Combine the Cases
From Case 2 and Case 3, we have:
1. From Case 2: \(-2 < x < 0\)
2. From Case 3: \(0 \leq x < 2\)

Combining these gives:
\[ -2 < x < 2 \]

### Step 5: Find Integer Solutions
The integers satisfying \(-2 < x < 2\) are:
- \( -1, 0, 1 \)

### Final Count
Thus, the number of integers satisfying the inequality is:
\[ \text{Number of integers} = 3 \]