` -12 lt 4 -(3x)/(-5) le 2`

Explanation

To solve the inequality \( -12 < 4 - \frac{3x}{-5} \leq 2 \), we will break it down step by step.

### Step 1: Rewrite the Inequality
The given inequality is:
\[ -12 < 4 - \frac{3x}{-5} \leq 2 \]
Since \(-\frac{3x}{-5}\) is the same as \(\frac{3x}{5}\), we can rewrite the inequality as:
\[ -12 < 4 + \frac{3x}{5} \leq 2 \]

### Step 2: Split the Compound Inequality
We can split the compound inequality into two parts:
1. \( -12 < 4 + \frac{3x}{5} \)
2. \( 4 + \frac{3x}{5} \leq 2 \)

### Step 3: Solve the First Inequality
Starting with the first part:
\[ -12 < 4 + \frac{3x}{5} \]
Subtract 4 from both sides:
\[ -12 - 4 < \frac{3x}{5} \]
This simplifies to:
\[ -16 < \frac{3x}{5} \]
Now, multiply both sides by 5 to eliminate the fraction:
\[ -80 < 3x \]
Finally, divide by 3:
\[ -\frac{80}{3} < x \]

### Step 4: Solve the Second Inequality
Now, let's solve the second part:
\[ 4 + \frac{3x}{5} \leq 2 \]
Subtract 4 from both sides:
\[ \frac{3x}{5} \leq 2 - 4 \]
This simplifies to:
\[ \frac{3x}{5} \leq -2 \]
Multiply both sides by 5:
\[ 3x \leq -10 \]
Finally, divide by 3:
\[ x \leq -\frac{10}{3} \]

### Step 5: Combine the Results
Now we have two inequalities:
1. \( x > -\frac{80}{3} \)
2. \( x \leq -\frac{10}{3} \)

Combining these gives:
\[ -\frac{80}{3} < x \leq -\frac{10}{3} \]

### Final Answer
Thus, the solution set is:
\[ x \in \left(-\frac{80}{3}, -\frac{10}{3}\right] \]

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