`-(12)/(5)lt6-9ylt-(9)/(4)`
In the inequality above, what is one possible value of `3y-2`?

Options

A. `0.75`
B. `0.76`
C. `0.80`
D. `0.81`

Answer

To solve the inequality \(-\frac{12}{5} < 6 - 9y < -\frac{9}{4}\) and find a possible value of \(3y - 2\), we can follow these steps:

### Step 1: Rewrite the inequality
The given inequality is:
\[ -\frac{12}{5} < 6 - 9y < -\frac{9}{4} \]

### Step 2: Multiply the entire inequality by -1
To eliminate the negative sign in front of the first term, we multiply the entire inequality by -1. Remember that when we multiply or divide an inequality by a negative number, we must reverse the inequality signs:
\[ \frac{12}{5} > -6 + 9y > \frac{9}{4} \]

### Step 3: Rearrange the middle term
Now, we can rearrange the middle term:
\[ \frac{12}{5} > 9y - 6 > \frac{9}{4} \]

### Step 4: Add 6 to all parts of the inequality
Next, we add 6 to all parts of the inequality:
\[ \frac{12}{5} + 6 > 9y > \frac{9}{4} + 6 \]

### Step 5: Convert 6 to a fraction
Convert 6 into a fraction with a denominator of 5:
\[ 6 = \frac{30}{5} \]
Now, adding:
\[ \frac{12}{5} + \frac{30}{5} = \frac{42}{5} \]
And for the right side:
\[ \frac{9}{4} + 6 = \frac{9}{4} + \frac{24}{4} = \frac{33}{4} \]

### Step 6: Write the new inequality
Now we have:
\[ \frac{42}{5} > 9y > \frac{33}{4} \]

### Step 7: Divide the entire inequality by 9
Next, divide the entire inequality by 9 to isolate \(y\):
\[ \frac{42}{5 \cdot 9} > y > \frac{33}{4 \cdot 9} \]
Calculating the fractions:
\[ \frac{42}{45} > y > \frac{33}{36} \]
This simplifies to:
\[ \frac{14}{15} > y > \frac{11}{12} \]

### Step 8: Find \(3y - 2\)
Now, we need to find \(3y - 2\). We can multiply the entire inequality by 3:
\[ 3 \cdot \frac{14}{15} > 3y > 3 \cdot \frac{11}{12} \]
Calculating:
\[ \frac{42}{15} > 3y > \frac{33}{12} \]
This simplifies to:
\[ 2.8 > 3y > 2.75 \]

### Step 9: Subtract 2 from all parts
Now, subtract 2 from all parts of the inequality to find \(3y - 2\):
\[ 2.8 - 2 > 3y - 2 > 2.75 - 2 \]
This gives:
\[ 0.8 > 3y - 2 > 0.75 \]

### Conclusion
Thus, \(3y - 2\) lies between 0.75 and 0.8. A possible value for \(3y - 2\) is:
\[ \boxed{0.76} \]