If 3x + 5 (4-3x) > 2 - 4x < 3x - x/3, then the value of x is.

Options

A. 3
B. 0
C. 2
D. 4

Answer

To solve the inequality \( 3x + 5(4 - 3x) > 2 - 4x < 3x - \frac{x}{3} \), we will break it down into two parts and solve each inequality separately.

### Step 1: Solve the first inequality \( 3x + 5(4 - 3x) > 2 - 4x \)

1. Distribute \( 5 \) in the left-hand side:
\[ 3x + 20 - 15x > 2 - 4x \]

2. Combine like terms:
\[ -12x + 20 > 2 - 4x \]

3. Add \( 4x \) to both sides:
\[ -12x + 4x + 20 > 2 \]
\[ -8x + 20 > 2 \]

4. Subtract \( 20 \) from both sides:
\[ -8x > 2 - 20 \]
\[ -8x > -18 \]

5. Divide by \( -8 \) (remember to flip the inequality sign):
\[ x < \frac{18}{8} \]
\[ x < \frac{9}{4} \]

### Step 2: Solve the second inequality \( 2 - 4x < 3x - \frac{x}{3} \)

1. First, simplify the right-hand side:
\[ 3x - \frac{x}{3} = \frac{9x}{3} - \frac{x}{3} = \frac{8x}{3} \]

2. Now, rewrite the inequality:
\[ 2 - 4x < \frac{8x}{3} \]

3. Multiply through by \( 3 \) to eliminate the fraction:
\[ 3(2) - 3(4x) < 8x \]
\[ 6 - 12x < 8x \]

4. Add \( 12x \) to both sides:
\[ 6 < 20x \]

5. Divide by \( 20 \):
\[ \frac{6}{20} < x \]
\[ \frac{3}{10} < x \]

### Step 3: Combine the results

Now we have two inequalities:
1. \( x < \frac{9}{4} \)
2. \( \frac{3}{10} < x \)

This means:
\[ \frac{3}{10} < x < \frac{9}{4} \]

### Step 4: Check the options

We need to check which of the given options (0, 2, 3, 4) fall within this range.

- **Option 0**: \( 0 \) is not greater than \( \frac{3}{10} \).
- **Option 2**: \( 2 \) is greater than \( \frac{3}{10} \) and less than \( \frac{9}{4} \) (which is \( 2.25 \)).
- **Option 3**: \( 3 \) is greater than \( \frac{9}{4} \).
- **Option 4**: \( 4 \) is greater than \( \frac{9}{4} \).

Thus, the only value that satisfies both inequalities is \( 2 \).

### Final Answer:
The value of \( x \) is \( 2 \).

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