If `-3lt 4/3 h +1/6lt 1,` then what is one possible value of `12 h- 4` ?

Explanation

To solve the inequality \(-3 < \frac{4}{3}h + \frac{1}{6} < 1\) and find a possible value of \(12h - 4\), we will follow these steps:

### Step 1: Multiply the entire inequality by 9
Since we want to eliminate the fractions, we can multiply the entire inequality by 9 (a positive integer, which does not change the inequality signs).

\[ -3 \times 9 < \left(\frac{4}{3}h + \frac{1}{6}\right) \times 9 < 1 \times 9 \]

This simplifies to:

\[ -27 < 12h + \frac{3}{2} < 9 \]

### Step 2: Subtract \(\frac{3}{2}\) from all parts of the inequality
Next, we need to isolate \(12h\). We can do this by subtracting \(\frac{3}{2}\) from each part of the inequality.

\[ -27 - \frac{3}{2} < 12h < 9 - \frac{3}{2} \]

Calculating the left side:

\[ -27 = -\frac{54}{2} \quad \text{so} \quad -\frac{54}{2} - \frac{3}{2} = -\frac{57}{2} \]

Calculating the right side:

\[ 9 = \frac{18}{2} \quad \text{so} \quad \frac{18}{2} - \frac{3}{2} = \frac{15}{2} \]

Thus, we have:

\[ -\frac{57}{2} < 12h < \frac{15}{2} \]

### Step 3: Write the inequality for \(12h - 4\)
Now, we want to find the range of \(12h - 4\). We can express this as:

\[ 12h - 4 = 12h + (-4) \]

To find the new bounds, we subtract 4 from each part of the inequality:

\[ -\frac{57}{2} - 4 < 12h - 4 < \frac{15}{2} - 4 \]

Calculating the left side:

\[ -4 = -\frac{8}{2} \quad \text{so} \quad -\frac{57}{2} - \frac{8}{2} = -\frac{65}{2} \]

Calculating the right side:

\[ -4 = -\frac{8}{2} \quad \text{so} \quad \frac{15}{2} - \frac{8}{2} = \frac{7}{2} \]

Thus, we have:

\[ -\frac{65}{2} < 12h - 4 < \frac{7}{2} \]

### Step 4: Choose a possible value
From the inequality \(-\frac{65}{2} < 12h - 4 < \frac{7}{2}\), we can choose any value in this range. For instance, we can choose \(2\) as a possible value since:

\[ -\frac{65}{2} \approx -32.5 \quad \text{and} \quad \frac{7}{2} = 3.5 \]

Thus, \(2\) is a valid choice.

### Final Answer
One possible value of \(12h - 4\) is \(2\).