If `5x - 3 (4 -x) lt 4x - 4 lt 4x + 2x//3,` then the value of x is

Options

A. `-7`
B. 3
C. 4
D. 1

Answer

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

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

1. Distribute the -3 in the left side:
\[ 5x - 12 + 3x < 4x - 4 \]
This simplifies to:
\[ 8x - 12 < 4x - 4 \]

2. Rearranging the inequality, we subtract \( 4x \) from both sides:
\[ 8x - 4x - 12 < -4 \]
Which simplifies to:
\[ 4x - 12 < -4 \]

3. Now, add 12 to both sides:
\[ 4x < 8 \]

4. Finally, divide by 4:
\[ x < 2 \]

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

1. Subtract \( 4x \) from both sides:
\[ -4 < \frac{2x}{3} \]

2. To eliminate the fraction, multiply both sides by 3:
\[ -12 < 2x \]

3. Now, divide by 2:
\[ -6 < x \]
or equivalently:
\[ x > -6 \]

### Step 3: Combine the results from both inequalities

From the first inequality, we have:
\[ x < 2 \]
From the second inequality, we have:
\[ x > -6 \]

Thus, combining these results, we get:
\[ -6 < x < 2 \]

### Conclusion

The values of \( x \) that satisfy both inequalities are in the range \( (-6, 2) \).