If `-9/5 lt -5t +2 lt -7/4` . What is one possible value of 10t-4 ?

Answer

To solve the inequality \(-\frac{9}{5} < -5t + 2 < -\frac{7}{4}\) and find one possible value of \(10t - 4\), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the given inequality:
\[ -\frac{9}{5} < -5t + 2 < -\frac{7}{4} \]

### Step 2: Multiply by -1
When we multiply the entire inequality by -1, we must reverse the inequality signs:
\[ \frac{9}{5} > 5t - 2 > \frac{7}{4} \]

### Step 3: Rearrange the Inequality
Now, we can rearrange the middle part of the inequality:
\[ \frac{9}{5} > 5t - 2 \quad \text{and} \quad 5t - 2 > \frac{7}{4} \]

### Step 4: Solve for \(5t\)
We will solve both parts of the inequality separately.

**For the left part:**
\[ 5t - 2 < \frac{9}{5} \]
Add 2 to both sides:
\[ 5t < \frac{9}{5} + 2 \]
Convert 2 to a fraction:
\[ 2 = \frac{10}{5} \]
So,
\[ 5t < \frac{9}{5} + \frac{10}{5} = \frac{19}{5} \]
Now divide by 5:
\[ t < \frac{19}{25} \]

**For the right part:**
\[ 5t - 2 > \frac{7}{4} \]
Add 2 to both sides:
\[ 5t > \frac{7}{4} + 2 \]
Convert 2 to a fraction:
\[ 2 = \frac{8}{4} \]
So,
\[ 5t > \frac{7}{4} + \frac{8}{4} = \frac{15}{4} \]
Now divide by 5:
\[ t > \frac{15}{20} = \frac{3}{4} \]

### Step 5: Combine the Results
Now we combine the results from both parts:
\[ \frac{3}{4} < t < \frac{19}{25} \]

### Step 6: Find a Value for \(10t - 4\)
Now we need to find \(10t - 4\). We can choose a value for \(t\) within the interval \(\left(\frac{3}{4}, \frac{19}{25}\right)\).

Let's choose \(t = \frac{4}{5}\) (which is 0.8 and lies between \(\frac{3}{4}\) and \(\frac{19}{25}\)):
\[ 10t - 4 = 10 \cdot \frac{4}{5} - 4 = 8 - 4 = 4 \]

Thus, one possible value of \(10t - 4\) is:
\[ \boxed{4} \]