Solve : `-5 le (2-3x)/(4) le 9`.

Answer

To solve the inequality \(-5 \leq \frac{2 - 3x}{4} \leq 9\), we will break it down into two parts and solve them step by step.

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ -5 \leq \frac{2 - 3x}{4} \leq 9 \]

### Step 2: Eliminate the fraction
To eliminate the fraction, multiply all parts of the inequality by 4 (note that 4 is positive, so the direction of the inequalities remains the same):
\[ -5 \times 4 \leq 2 - 3x \leq 9 \times 4 \]
This simplifies to:
\[ -20 \leq 2 - 3x \leq 36 \]

### Step 3: Split into two inequalities
Now we can split this into two separate inequalities to solve:
1. \(-20 \leq 2 - 3x\)
2. \(2 - 3x \leq 36\)

### Step 4: Solve the first inequality
Starting with the first inequality:
\[ -20 \leq 2 - 3x \]
Subtract 2 from both sides:
\[ -20 - 2 \leq -3x \]
This simplifies to:
\[ -22 \leq -3x \]
Now, divide by -3 (remember to flip the inequality sign):
\[ \frac{-22}{-3} \geq x \]
This simplifies to:
\[ \frac{22}{3} \geq x \quad \text{or} \quad x \leq \frac{22}{3} \]

### Step 5: Solve the second inequality
Now, we solve the second inequality:
\[ 2 - 3x \leq 36 \]
Subtract 2 from both sides:
\[ -3x \leq 36 - 2 \]
This simplifies to:
\[ -3x \leq 34 \]
Now, divide by -3 (again, flip the inequality sign):
\[ x \geq \frac{34}{-3} \]
This simplifies to:
\[ x \geq -\frac{34}{3} \]

### Step 6: Combine the results
Now we combine the results from both inequalities:
\[ -\frac{34}{3} \leq x \leq \frac{22}{3} \]

### Final Answer
Thus, the solution to the inequality is:
\[ -\frac{34}{3} \leq x \leq \frac{22}{3} \]