Find the solution set of `|(1)/(x)-2|gt4`.

Answer

To solve the inequality \( \left| \frac{1}{x} - 2 \right| > 4 \), we will break it down into steps.

### Step 1: Remove the Absolute Value
The expression \( \left| A \right| > B \) can be rewritten as two separate inequalities:
1. \( A > B \)
2. \( A < -B \)

In our case, \( A = \frac{1}{x} - 2 \) and \( B = 4 \). Therefore, we have:
1. \( \frac{1}{x} - 2 > 4 \)
2. \( \frac{1}{x} - 2 < -4 \)

### Step 2: Solve the First Inequality
Starting with the first inequality:
\[ \frac{1}{x} - 2 > 4 \]
Add 2 to both sides:
\[ \frac{1}{x} > 6 \]
Now, take the reciprocal (remembering that the inequality sign flips when we multiply or divide by a negative number):
\[ x < \frac{1}{6} \]

### Step 3: Solve the Second Inequality
Now, let's solve the second inequality:
\[ \frac{1}{x} - 2 < -4 \]
Add 2 to both sides:
\[ \frac{1}{x} < -2 \]
Again, take the reciprocal:
\[ x > -\frac{1}{2} \]

### Step 4: Combine the Results
From the first inequality, we found \( x < \frac{1}{6} \). From the second inequality, we found \( x > -\frac{1}{2} \). Therefore, we can combine these results:
\[ -\frac{1}{2} < x < \frac{1}{6} \]

### Step 5: Consider the Excluded Value
Since \( x \) cannot be 0 (as it would make the original expression undefined), we need to ensure that our solution set does not include 0.

### Final Solution Set
Thus, the solution set is:
\[ x \in \left(-\frac{1}{2}, 0\right) \cup \left(0, \frac{1}{6}\right) \]