The solution set of the inequation `|(1)/(x)-2| lt 4`, is

Options

A. `(-oo, -1//2)`
B. `(1//6, oo)`
C. `(-1//2, 1//6)`
D. `(-oo, -1//2) cup (1//6, oo)`

Explanation

To solve the inequality \( \left| \frac{1}{x} - 2 \right| < 4 \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Remove the absolute value
The inequality \( \left| A \right| < B \) can be rewritten as:
\[ -B < A < B \]
For our case, we have:
\[ -4 < \frac{1}{x} - 2 < 4 \]

### Step 2: Split into two inequalities
This gives us two separate inequalities to solve:
1. \( \frac{1}{x} - 2 < 4 \)
2. \( \frac{1}{x} - 2 > -4 \)

### Step 3: Solve the first inequality
Starting with the first inequality:
\[ \frac{1}{x} - 2 < 4 \]
Add 2 to both sides:
\[ \frac{1}{x} < 6 \]
Taking the reciprocal (and remembering to reverse the inequality since \( x \) must be positive):
\[ x > \frac{1}{6} \]

### Step 4: Solve the second inequality
Now, solve the second inequality:
\[ \frac{1}{x} - 2 > -4 \]
Add 2 to both sides:
\[ \frac{1}{x} > -2 \]
Taking the reciprocal (again reversing the inequality since \( x \) must be positive):
\[ x < -\frac{1}{2} \]

### Step 5: Combine the results
Now we have two conditions:
1. \( x > \frac{1}{6} \)
2. \( x < -\frac{1}{2} \)

Since \( x \) cannot be both greater than \( \frac{1}{6} \) and less than \( -\frac{1}{2} \) at the same time, we conclude that the solution set is:
\[ x \in (-\infty, -\frac{1}{2}) \cup (\frac{1}{6}, \infty) \]

### Final Answer
The solution set of the inequation \( \left| \frac{1}{x} - 2 \right| < 4 \) is:
\[ (-\infty, -\frac{1}{2}) \cup (\frac{1}{6}, \infty) \]