Solve `(1)/(|x|-3) lt 1/2`

Explanation

To solve the inequality \( \frac{1}{|x| - 3} < \frac{1}{2} \), we will follow these steps:

### Step 1: Understand the inequality
We start with the inequality:

\[ \frac{1}{|x| - 3} < \frac{1}{2} \]

### Step 2: Identify the conditions for the absolute value
The expression \( |x| - 3 \) can be positive or negative, so we need to consider two cases based on the value of \( |x| \).

1. **Case 1**: \( |x| - 3 > 0 \) which implies \( |x| > 3 \) (i.e., \( x > 3 \) or \( x < -3 \)).
2. **Case 2**: \( |x| - 3 < 0 \) which implies \( |x| < 3 \) (i.e., \( -3 < x < 3 \)).

### Step 3: Solve Case 1
For Case 1, since \( |x| - 3 > 0 \), we can multiply both sides of the inequality by \( |x| - 3 \) without changing the inequality:

\[ 1 < \frac{1}{2} (|x| - 3) \]

Multiplying both sides by 2 gives:

\[ 2 < |x| - 3 \]

Adding 3 to both sides results in:

\[ |x| > 5 \]

This implies:

\[ x > 5 \quad \text{or} \quad x < -5 \]

### Step 4: Solve Case 2
For Case 2, since \( |x| - 3 < 0 \), we multiply both sides of the inequality by \( |x| - 3 \), but we need to reverse the inequality sign:

\[ 1 > \frac{1}{2} (|x| - 3) \]

Multiplying both sides by 2 gives:

\[ 2 > |x| - 3 \]

Adding 3 to both sides results in:

\[ 5 > |x| \]

This implies:

\[ -5 < x < 5 \]

### Step 5: Combine the solutions
Now we combine the solutions from both cases:

1. From Case 1: \( x > 5 \) or \( x < -5 \)
2. From Case 2: \( -5 < x < 5 \)

The overall solution is:

\[ (-\infty, -5) \cup (-3, 3) \cup (5, \infty) \]

### Final Answer
Thus, the solution to the inequality \( \frac{1}{|x| - 3} < \frac{1}{2} \) is:

\[ (-\infty, -5) \cup (-3, 3) \cup (5, \infty) \]