If `(-5)/(2)lt-2m+1lt(-7)/(5)`, what is the greatest possible integer value of the expression `10m-5`?

Options

A. `6`
B. `7`
C. `10`
D. `12`

Explanation

To solve the inequality \(-\frac{5}{2} < -2m + 1 < -\frac{7}{5}\) and find the greatest possible integer value of the expression \(10m - 5\), we can follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ -\frac{5}{2} < -2m + 1 < -\frac{7}{5} \]

### Step 2: Break it into two parts
We can break this into two separate inequalities:
1. \(-\frac{5}{2} < -2m + 1\)
2. \(-2m + 1 < -\frac{7}{5}\)

### Step 3: Solve the first part of the inequality
For the first inequality:
\[ -\frac{5}{2} < -2m + 1 \]
Subtract 1 from both sides:
\[ -\frac{5}{2} - 1 < -2m \]
Convert 1 to a fraction:
\[ -\frac{5}{2} - \frac{2}{2} < -2m \]
Combine the fractions:
\[ -\frac{7}{2} < -2m \]
Now, divide by -2 (remember to flip the inequality sign):
\[ \frac{7}{4} > m \]
or
\[ m < \frac{7}{4} \]

### Step 4: Solve the second part of the inequality
Now for the second inequality:
\[ -2m + 1 < -\frac{7}{5} \]
Subtract 1 from both sides:
\[ -2m < -\frac{7}{5} - 1 \]
Convert 1 to a fraction:
\[ -2m < -\frac{7}{5} - \frac{5}{5} \]
Combine the fractions:
\[ -2m < -\frac{12}{5} \]
Now, divide by -2 (remember to flip the inequality sign):
\[ m > \frac{12}{10} = \frac{6}{5} \]

### Step 5: Combine the results
Now we have:
\[ \frac{6}{5} < m < \frac{7}{4} \]

### Step 6: Express \(10m - 5\)
We need to find the expression \(10m - 5\). Substituting the bounds for \(m\):
1. For \(m = \frac{6}{5}\):
\[ 10m - 5 = 10 \cdot \frac{6}{5} - 5 = 12 - 5 = 7 \]
2. For \(m = \frac{7}{4}\):
\[ 10m - 5 = 10 \cdot \frac{7}{4} - 5 = 17.5 - 5 = 12.5 \]

### Step 7: Determine the range of \(10m - 5\)
Thus, we have:
\[ 7 < 10m - 5 < 12.5 \]

### Step 8: Find the greatest integer value
The greatest possible integer value of \(10m - 5\) is:
\[ \boxed{12} \]