Solution of the inequation `{x} ({x}-1)({x}+2)ge0`
where `{.}` denots fractin part function) is :

Options

A. `x in (-2, 1)`
B. `x in I ` (I denote set of integers)
C. `x in [0,1)`
D. `x in [-2, 0)`

Answer

To solve the inequation \(\{x\} ( \{x\} - 1)( \{x\} + 2) \geq 0\), where \(\{x\}\) denotes the fractional part of \(x\), we will follow these steps:

### Step 1: Understand the Fractional Part Function
The fractional part function \(\{x\}\) is defined as:
\[ \{x\} = x - \lfloor x \rfloor \]
This means that \(\{x\}\) is always between \(0\) and \(1\) (inclusive), i.e., \(0 \leq \{x\} < 1\).

### Step 2: Analyze the Expression
We need to analyze the expression \(\{x\} ( \{x\} - 1)( \{x\} + 2)\):
- The term \(\{x\}\) is non-negative (since it is either \(0\) or positive).
- The term \(\{x\} - 1\) will be negative for \(0 \leq \{x\} < 1\) and equals \(-1\) when \(\{x\} = 1\) (which is not possible since \(\{x\} < 1\)).
- The term \(\{x\} + 2\) is always positive since it is greater than \(2\).

### Step 3: Determine the Sign of the Product
Now, we can analyze the product:
\[ \{x\} ( \{x\} - 1)( \{x\} + 2) \]
- Since \(\{x\} \geq 0\) and \(\{x\} + 2 > 0\), the sign of the product depends on \(\{x\} - 1\).
- Therefore, the product will be non-negative if \(\{x\} = 0\).

### Step 4: Identify Conditions for Non-negativity
The only case where the product \(\{x\} ( \{x\} - 1)( \{x\} + 2) \geq 0\) holds true is when \(\{x\} = 0\). This occurs when \(x\) is an integer.

### Step 5: Conclusion
Thus, the solution to the inequation is:
\[ x \in \mathbb{Z} \quad \text{(the set of all integers)} \]