Solve : ` (1)/(x-3)lt 0`

Explanation

To solve the inequality \( \frac{1}{x-3} < 0 \), we will follow these steps:

### Step 1: Understand the inequality
The inequality states that the expression \( \frac{1}{x-3} \) is less than zero. This means we need to find the values of \( x \) for which this expression is negative.

### Step 2: Identify when the expression is negative
The expression \( \frac{1}{x-3} \) will be negative when the denominator \( x-3 \) is negative. This is because a positive numerator (1) divided by a negative denominator results in a negative value.

### Step 3: Set up the inequality for the denominator
To find when \( x-3 < 0 \):
\[ x - 3 < 0 \]

### Step 4: Solve the inequality
Now, we solve for \( x \):
\[ x < 3 \]

### Step 5: Write the solution in interval notation
The solution \( x < 3 \) means that \( x \) can take any value less than 3. In interval notation, this is expressed as:
\[ (-\infty, 3) \]

### Final Answer
Thus, the solution to the inequality \( \frac{1}{x-3} < 0 \) is:
\[ x \in (-\infty, 3) \]

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