Solve : `(2)/(|x-3|) gt1,x ne 3`

Explanation

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

### Step 1: Rewrite the Inequality
Start by rewriting the inequality:
\[ \frac{2}{|x-3|} > 1 \]

### Step 2: Multiply Both Sides by \( |x-3| \)
Since \( |x-3| > 0 \) (because \( x \neq 3 \)), we can multiply both sides by \( |x-3| \) without changing the direction of the inequality:
\[ 2 > |x-3| \]

### Step 3: Interpret the Absolute Value
The inequality \( |x-3| < 2 \) means that the distance between \( x \) and 3 is less than 2. This can be expressed as:
\[ -2 < x - 3 < 2 \]

### Step 4: Solve the Compound Inequality
Now, we will solve the compound inequality:
1. Add 3 to all parts of the inequality:
\[ -2 + 3 < x < 2 + 3 \]
This simplifies to:
\[ 1 < x < 5 \]

### Step 5: Write the Solution Set
The solution set in interval notation is:
\[ (1, 5) \]

### Step 6: Exclude the Point \( x = 3 \)
Since \( x \neq 3 \), we need to exclude this point from our solution. Therefore, the final solution set is:
\[ (1, 3) \cup (3, 5) \]

### Final Answer
Thus, the solution to the inequality \( \frac{2}{|x-3|} > 1 \) with \( x \neq 3 \) is:
\[ (1, 3) \cup (3, 5) \]

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