If `3x + 2 ge x-1 and 2x - 4 le 2 - (x)/(3),` then x can take which of the following values ?

Options

A. `-2`
B. 3
C. 5
D. `-1`

Answer

To solve the inequalities given in the question, we will break them down step by step.

### Step 1: Solve the first inequality
The first inequality is:
\[ 3x + 2 \geq x - 1 \]

**Rearranging the inequality:**
1. Subtract \(x\) from both sides:
\[ 3x - x + 2 \geq -1 \]
This simplifies to:
\[ 2x + 2 \geq -1 \]

2. Now, subtract 2 from both sides:
\[ 2x \geq -1 - 2 \]
This simplifies to:
\[ 2x \geq -3 \]

3. Finally, divide both sides by 2:
\[ x \geq -\frac{3}{2} \]
or
\[ x \geq -1.5 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 2x - 4 \leq 2 - \frac{x}{3} \]

**Rearranging the inequality:**
1. First, let's eliminate the fraction by multiplying the entire inequality by 3 (to avoid dealing with fractions):
\[ 3(2x - 4) \leq 3(2) - x \]
This simplifies to:
\[ 6x - 12 \leq 6 - x \]

2. Now, add \(x\) to both sides:
\[ 6x + x - 12 \leq 6 \]
This simplifies to:
\[ 7x - 12 \leq 6 \]

3. Next, add 12 to both sides:
\[ 7x \leq 6 + 12 \]
This simplifies to:
\[ 7x \leq 18 \]

4. Finally, divide both sides by 7:
\[ x \leq \frac{18}{7} \]
or approximately:
\[ x \leq 2.57 \]

### Step 3: Combine the results
Now we have two inequalities:
1. \( x \geq -1.5 \)
2. \( x \leq \frac{18}{7} \) (approximately \( 2.57 \))

This means:
\[ -1.5 \leq x \leq 2.57 \]

### Step 4: Check the options
Now we need to check which of the given options fall within the range \([-1.5, 2.57]\):
- Option A: -2 (not in range)
- Option B: 3 (not in range)
- Option C: 5 (not in range)
- Option D: -1 (in range)

Thus, the correct answer is **Option D: -1**.