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) uu (1//6,oo)`

Explanation

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

### Step 1: Rewrite the Inequality
The inequality \( \left| \frac{1}{x} - 2 \right| < 4 \) can be expressed as:
\[ -4 < \frac{1}{x} - 2 < 4 \]

### Step 2: Solve the Left Inequality
First, we solve the left part of the compound inequality:
\[ \frac{1}{x} - 2 > -4 \]
Adding 2 to both sides gives:
\[ \frac{1}{x} > -2 \]
Now, we can rewrite this as:
\[ 1 > -2x \]
Dividing both sides by -2 (remember to reverse the inequality):
\[ -\frac{1}{2} < x \]
This means:
\[ x > -\frac{1}{2} \]

### Step 3: Solve the Right Inequality
Now we solve the right part of the compound inequality:
\[ \frac{1}{x} - 2 < 4 \]
Adding 2 to both sides gives:
\[ \frac{1}{x} < 6 \]
Rearranging this gives:
\[ 1 < 6x \]
Dividing both sides by 6 gives:
\[ \frac{1}{6} < x \]
This means:
\[ x > \frac{1}{6} \]

### Step 4: Combine the Results
Now we have two inequalities:
1. \( x > -\frac{1}{2} \)
2. \( x > \frac{1}{6} \)

The more restrictive condition is \( x > \frac{1}{6} \).

### Step 5: Consider the Domain
Since \( \frac{1}{x} \) is undefined at \( x = 0 \), we must also consider that \( x \) cannot be zero. Therefore, the solution set must exclude zero.

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