If x is a prime number and ` - 1 le ( 2 x - 7)/( 5) le 1` then the number of values of x is

Options

A. 4
B. 3
C. 2
D. 5

Answer

To solve the inequality \(-1 \leq \frac{2x - 7}{5} \leq 1\) for prime numbers \(x\), we will break it down into two parts.

### Step 1: Solve the left part of the inequality
We start with the left part of the inequality:
\[ -1 \leq \frac{2x - 7}{5} \]
To eliminate the fraction, we multiply all terms by 5:
\[ -5 \leq 2x - 7 \]
Next, we add 7 to both sides:
\[ 2 \leq 2x \]
Now, we divide both sides by 2:
\[ 1 \leq x \]
This simplifies to:
\[ x \geq 1 \]

### Step 2: Solve the right part of the inequality
Now we solve the right part of the inequality:
\[ \frac{2x - 7}{5} \leq 1 \]
Again, we multiply all terms by 5:
\[ 2x - 7 \leq 5 \]
Next, we add 7 to both sides:
\[ 2x \leq 12 \]
Now, we divide both sides by 2:
\[ x \leq 6 \]

### Step 3: Combine the results
From the two parts, we have:
\[ 1 \leq x \leq 6 \]
This means \(x\) can take values from 1 to 6.

### Step 4: Identify the prime numbers
Now we need to find the prime numbers within this range. The prime numbers between 1 and 6 are:
- 2
- 3
- 5

### Conclusion
Thus, the prime numbers that satisfy the given inequality are 2, 3, and 5. Therefore, the number of values of \(x\) is:
\[ \text{Number of values} = 3 \]