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

Options

A. `(-oo, -5)cup(5, oo)`
B. `(-3, 3)`
C. `(-oo, -5) cup (-3, 3) cup (5, oo)`
D. none of these

Answer

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

### Step 1: Set up the inequality
We start with the inequality:
\[ \frac{1}{|x| - 3} < \frac{1}{2} \]

### Step 2: Rewrite the inequality
To eliminate the fractions, we can cross-multiply (keeping in mind the sign of the expressions):
\[ 2 < |x| - 3 \]
This simplifies to:
\[ |x| - 3 > 2 \]

### Step 3: Solve for |x|
Adding 3 to both sides gives:
\[ |x| > 5 \]

### Step 4: Break it into cases
Now we will consider two cases based on the definition of absolute value.

#### Case 1: \( x \geq 0 \)
In this case, \( |x| = x \), so the inequality becomes:
\[ x > 5 \]

#### Case 2: \( x < 0 \)
Here, \( |x| = -x \), so the inequality becomes:
\[ -x > 5 \quad \Rightarrow \quad x < -5 \]

### Step 5: Combine the results
From both cases, we have:
1. From Case 1: \( x > 5 \)
2. From Case 2: \( x < -5 \)

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

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