If x is an integer satisfying `x^(2)-6x+5 le 0 " and " x^(2)-2x gt 0`, then the number of possible values of x, is

Options

A. 3
B. 4
C. 2
D. infinite

Answer

To solve the inequalities \( x^2 - 6x + 5 \leq 0 \) and \( x^2 - 2x > 0 \) for integer values of \( x \), we will follow these steps:

### Step 1: Solve the first inequality \( x^2 - 6x + 5 \leq 0 \)

1. **Factor the quadratic expression**:
\[ x^2 - 6x + 5 = (x - 1)(x - 5) \]
Thus, we rewrite the inequality:
\[ (x - 1)(x - 5) \leq 0 \]

2. **Find the critical points**:
The critical points are \( x = 1 \) and \( x = 5 \).

3. **Test intervals**:
We will test the intervals determined by the critical points: \( (-\infty, 1) \), \( (1, 5) \), and \( (5, \infty) \).

- For \( x < 1 \) (e.g., \( x = 0 \)):
\((0 - 1)(0 - 5) = 5 > 0\)
- For \( 1 < x < 5 \) (e.g., \( x = 3 \)):
\((3 - 1)(3 - 5) = -4 < 0\)
- For \( x > 5 \) (e.g., \( x = 6 \)):
\((6 - 1)(6 - 5) = 5 > 0\)

4. **Conclusion for the first inequality**:
The solution to \( (x - 1)(x - 5) \leq 0 \) is:
\[ 1 \leq x \leq 5 \]

### Step 2: Solve the second inequality \( x^2 - 2x > 0 \)

1. **Factor the quadratic expression**:
\[ x^2 - 2x = x(x - 2) \]
Thus, we rewrite the inequality:
\[ x(x - 2) > 0 \]

2. **Find the critical points**:
The critical points are \( x = 0 \) and \( x = 2 \).

3. **Test intervals**:
We will test the intervals determined by the critical points: \( (-\infty, 0) \), \( (0, 2) \), and \( (2, \infty) \).

- For \( x < 0 \) (e.g., \( x = -1 \)):
\((-1)(-1 - 2) = 3 > 0\)
- For \( 0 < x < 2 \) (e.g., \( x = 1 \)):
\((1)(1 - 2) = -1 < 0\)
- For \( x > 2 \) (e.g., \( x = 3 \)):
\((3)(3 - 2) = 3 > 0\)

4. **Conclusion for the second inequality**:
The solution to \( x(x - 2) > 0 \) is:
\[ x < 0 \quad \text{or} \quad x > 2 \]

### Step 3: Find the intersection of the two solutions

1. **First inequality**: \( 1 \leq x \leq 5 \)
2. **Second inequality**: \( x < 0 \) or \( x > 2 \)

The intersection of these two sets is:
- From \( 1 \leq x \leq 5 \) and \( x > 2 \), we find:
\[ 2 < x \leq 5 \]

### Step 4: Identify integer solutions

The integer solutions in the interval \( 2 < x \leq 5 \) are:
- \( x = 3, 4, 5 \)

### Conclusion

The number of possible integer values of \( x \) is:
\[ \text{Possible values: } 3, 4, 5 \quad \Rightarrow \quad \text{Total: } 3 \]

Thus, the answer is **3**.

---