The solution set of `-12 lt 4 -(3x)/(-5) le 2 , x in R ` is

Options

A. `(10/3,80/3]`
B. `[-80/3,10/3)`
C. `(-80/3,-10/3]`
D. `(80/3,-10/3)`

Answer

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

### Step 1: Rewrite the inequality
We start with the inequality:
\[ -12 < 4 - \frac{3x}{-5} \leq 2 \]

### Step 2: Simplify the inequality
We can rewrite the inequality as two separate inequalities:
1. \(-12 < 4 - \frac{3x}{-5}\)
2. \(4 - \frac{3x}{-5} \leq 2\)

### Step 3: Solve the first inequality
For the first inequality:
\[ -12 < 4 - \frac{3x}{-5} \]
Subtract 4 from both sides:
\[ -12 - 4 < -\frac{3x}{-5} \]
This simplifies to:
\[ -16 < -\frac{3x}{-5} \]
Multiplying both sides by -5 (remember to flip the inequality sign):
\[ 80 > 3x \]
or
\[ 3x < 80 \]
Dividing by 3:
\[ x < \frac{80}{3} \]

### Step 4: Solve the second inequality
Now for the second inequality:
\[ 4 - \frac{3x}{-5} \leq 2 \]
Subtract 4 from both sides:
\[ -\frac{3x}{-5} \leq 2 - 4 \]
This simplifies to:
\[ \frac{3x}{5} \leq -2 \]
Multiplying both sides by 5:
\[ 3x \leq -10 \]
Dividing by 3:
\[ x \leq -\frac{10}{3} \]

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

Since \(x\) must satisfy both conditions, we combine them:
\[ -\frac{10}{3} \leq x < \frac{80}{3} \]

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