Solution set for inequality`(1)/(x-2)<0` is-

Options

A. `(2,infty)`
B. `phi`
C. (0,2)
D. (-infty,2)`

Answer

To solve the inequality \(\frac{1}{x - 2} < 0\), we need to determine when the expression is negative. Here is a step-by-step solution:

### Step 1: Understand the inequality
The inequality \(\frac{1}{x - 2} < 0\) means that the fraction is negative. A fraction is negative when its numerator and denominator have opposite signs.

### Step 2: Analyze the numerator
The numerator of the fraction is \(1\), which is always positive. Therefore, for the fraction to be negative, the denominator must be negative.

### Step 3: Set the denominator to be less than zero
We need to find when the denominator \(x - 2\) is less than zero:
\[ x - 2 < 0 \]

### Step 4: Solve for \(x\)
To solve the inequality \(x - 2 < 0\), we add \(2\) to both sides:
\[ x < 2 \]

### Step 5: Determine the solution set
The solution set for the inequality \(x < 2\) can be expressed in interval notation. Since \(x\) can take any value less than \(2\), the solution set is:
\[ (-\infty, 2) \]

### Conclusion
Thus, the solution set for the inequality \(\frac{1}{x - 2} < 0\) is \((- \infty, 2)\).

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