If `5x + 4 (1-x ) gt 3x - 4 gt 5x //3 - x //3,` then the x can take which of the following values ?

Options

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

Answer

To solve the inequality \( 5x + 4(1 - x) > 3x - 4 > \frac{5x}{3} - \frac{x}{3} \), we will break it down into two parts and solve each part step by step.

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

1. Distribute \( 4 \) in the left side:
\[ 5x + 4 - 4x > 3x - 4 \]
2. Combine like terms:
\[ (5x - 4x) + 4 > 3x - 4 \]
Simplifying gives:
\[ x + 4 > 3x - 4 \]
3. Subtract \( x \) from both sides:
\[ 4 > 2x - 4 \]
4. Add \( 4 \) to both sides:
\[ 8 > 2x \]
5. Divide by \( 2 \):
\[ 4 > x \quad \text{or} \quad x < 4 \]

### Step 2: Solve the second inequality \( 3x - 4 > \frac{5x}{3} - \frac{x}{3} \)

1. Simplify the right side:
\[ 3x - 4 > \frac{5x - x}{3} \]
This simplifies to:
\[ 3x - 4 > \frac{4x}{3} \]
2. Multiply through by \( 3 \) to eliminate the fraction (note that multiplying by a positive number does not change the inequality):
\[ 9x - 12 > 4x \]
3. Subtract \( 4x \) from both sides:
\[ 9x - 4x - 12 > 0 \]
This simplifies to:
\[ 5x - 12 > 0 \]
4. Add \( 12 \) to both sides:
\[ 5x > 12 \]
5. Divide by \( 5 \):
\[ x > \frac{12}{5} \quad \text{or} \quad x > 2.4 \]

### Step 3: Combine the results

From the two inequalities, we have:
1. \( x < 4 \)
2. \( x > 2.4 \)

Thus, the solution for \( x \) is:
\[ 2.4 < x < 4 \]

### Conclusion

The values that \( x \) can take are those that are greater than \( 2.4 \) and less than \( 4 \).