The solution set of the inequation `(1)/(|x|-3) lt (1)/(2)` is

Options

A. ` x in ( -oo , -3) cup ( 3 , oo) `
B. `x in (-oo,-5) cup ( -3,3) cup ( 5 , oo)`
C. `x in ( -oo , -5) cup (3,oo)`
D. `x in (-oo, -3) cup ( 5 ,oo)`

Answer

To solve the inequation \(\frac{1}{|x| - 3} < \frac{1}{2}\), we will break it down into two cases based on the definition of the absolute value.

### Step 1: Rewrite the Inequation
We start with the given inequation:
\[ \frac{1}{|x| - 3} < \frac{1}{2} \]
This can be rewritten as:
\[ \frac{1}{|x| - 3} - \frac{1}{2} < 0 \]

### Step 2: Find a Common Denominator
To combine the fractions, we find a common denominator:
\[ \frac{2 - (|x| - 3)}{2(|x| - 3)} < 0 \]
This simplifies to:
\[ \frac{5 - |x|}{2(|x| - 3)} < 0 \]

### Step 3: Analyze the Expression
The expression \(\frac{5 - |x|}{2(|x| - 3)} < 0\) will be negative when the numerator and denominator have opposite signs.

### Case 1: \(x \geq 0\)
In this case, \(|x| = x\). The inequation becomes:
\[ \frac{5 - x}{2(x - 3)} < 0 \]
This means we need to analyze when \(5 - x\) and \(2(x - 3)\) have opposite signs.

1. **Numerator \(5 - x < 0\)**:
- This occurs when \(x > 5\).

2. **Denominator \(2(x - 3) > 0\)**:
- This occurs when \(x > 3\).

Now we check the intervals:
- For \(x < 5\) and \(x > 3\), the expression is negative.
- For \(x > 5\), both numerator and denominator are negative, hence the expression is positive.

Thus, for \(x \geq 0\), the solution is:
\[ 3 < x < 5 \]

### Case 2: \(x < 0\)
In this case, \(|x| = -x\). The inequation becomes:
\[ \frac{5 + x}{2(-x - 3)} < 0 \]
Again, we analyze the signs:

1. **Numerator \(5 + x < 0\)**:
- This occurs when \(x < -5\).

2. **Denominator \(2(-x - 3) < 0\)**:
- This occurs when \(x > -3\).

Now we check the intervals:
- For \(x < -5\), the numerator is negative and the denominator is negative, hence the expression is positive.
- For \(-5 < x < -3\), the numerator is positive and the denominator is negative, hence the expression is negative.

Thus, for \(x < 0\), the solution is:
\[ -5 < x < -3 \]

### Step 4: Combine the Solutions
Combining the solutions from both cases, we have:
1. From Case 1: \(3 < x < 5\)
2. From Case 2: \(-5 < x < -3\)

Thus, the final solution set is:
\[ (-5, -3) \cup (3, 5) \]

### Final Answer
The solution set of the inequation \(\frac{1}{|x| - 3} < \frac{1}{2}\) is:
\[ (-5, -3) \cup (3, 5) \]