The solution set of the inequation... `2/(|x-4|) >1,x != 4` is ...

Options

A. `(2, 6)`
B. `(2, 4) cup (4,6)`
C. `[-1, 1]cup [3, oo)`
D. none of these

Answer

To solve the inequation \( \frac{2}{|x-4|} > 1 \) with the condition \( x \neq 4 \), we will consider two cases based on the definition of the absolute value.

### Step 1: Rewrite the Inequation
The given inequation is:
\[ \frac{2}{|x-4|} > 1 \]

### Step 2: Consider Case 1: \( x > 4 \)
In this case, \( |x-4| = x-4 \). Therefore, the inequation becomes:
\[ \frac{2}{x-4} > 1 \]

### Step 3: Solve the Inequation
To solve the inequation, we can take the reciprocal of both sides. Remember that since \( x - 4 > 0 \), the direction of the inequality will not change:
\[ 2 > x - 4 \]
Rearranging gives:
\[ x < 6 \]

### Step 4: Combine Conditions
Since we are considering \( x > 4 \) and \( x < 6 \), we have:
\[ 4 < x < 6 \]
This means:
\[ x \in (4, 6) \]

### Step 5: Consider Case 2: \( x < 4 \)
In this case, \( |x-4| = -(x-4) = 4-x \). Therefore, the inequation becomes:
\[ \frac{2}{4-x} > 1 \]

### Step 6: Solve the Inequation
Taking the reciprocal of both sides (the direction of the inequality will change since \( 4 - x > 0 \)):
\[ 2 < 4 - x \]
Rearranging gives:
\[ x < 2 \]

### Step 7: Combine Conditions
Since we are considering \( x < 4 \) and \( x < 2 \), we have:
\[ x < 2 \]
This means:
\[ x \in (-\infty, 2) \]

### Step 8: Final Solution Set
Combining the results from both cases, we have:
\[ x \in (-\infty, 2) \cup (4, 6) \]

### Conclusion
The solution set of the inequation \( \frac{2}{|x-4|} > 1 \) is:
\[ (-\infty, 2) \cup (4, 6) \]