Solve `-5lt=(5-3x)/2lt=8.`

Explanation

To solve the inequality \(-5 \leq \frac{5 - 3x}{2} \leq 8\), we will break it down into steps.

### Step 1: Rewrite the inequality
We start with the inequality:
\[ -5 \leq \frac{5 - 3x}{2} \leq 8 \]

### Step 2: Eliminate the fraction
To eliminate the fraction, we can multiply all parts of the inequality by 2 (since 2 is positive, the direction of the inequalities will remain the same):
\[ -5 \times 2 \leq 5 - 3x \leq 8 \times 2 \]
This simplifies to:
\[ -10 \leq 5 - 3x \leq 16 \]

### Step 3: Split the compound inequality
Now we can split this compound inequality into two separate inequalities:
1. \(-10 \leq 5 - 3x\)
2. \(5 - 3x \leq 16\)

### Step 4: Solve the first inequality
Let's solve the first inequality:
\[ -10 \leq 5 - 3x \]
Subtract 5 from both sides:
\[ -10 - 5 \leq -3x \]
\[ -15 \leq -3x \]
Now, divide by -3. Remember to reverse the inequality sign:
\[ 5 \geq x \quad \text{or} \quad x \leq 5 \]

### Step 5: Solve the second inequality
Now, let's solve the second inequality:
\[ 5 - 3x \leq 16 \]
Subtract 5 from both sides:
\[ -3x \leq 16 - 5 \]
\[ -3x \leq 11 \]
Now, divide by -3, reversing the inequality sign:
\[ x \geq -\frac{11}{3} \]

### Step 6: Combine the results
Now we combine the results from both inequalities:
\[ -\frac{11}{3} \leq x \leq 5 \]

### Final Answer
The solution set is:
\[ x \in \left[-\frac{11}{3}, 5\right] \]