If `2x + 3 (5-2 x) gt 2 - 3x lt 2x - x//3,` then x can take which of the following values ?

Options

A. 0
B. `-2`
C. `-4`
D. 2

Answer

To solve the inequality \( 2x + 3(5 - 2x) > 2 - 3x < 2x - \frac{x}{3} \), we will break it down into two separate inequalities and solve them step by step.

### Step 1: Solve the first inequality
We start with the first part of the inequality:
\[ 2x + 3(5 - 2x) > 2 - 3x \]

**Expand and simplify:**
1. Distribute \(3\) in \(3(5 - 2x)\):
\[ 2x + 15 - 6x > 2 - 3x \]
2. Combine like terms:
\[ -4x + 15 > 2 - 3x \]

**Rearrange the inequality:**
3. Add \(3x\) to both sides:
\[ -4x + 3x + 15 > 2 \]
\[ -x + 15 > 2 \]
4. Subtract \(15\) from both sides:
\[ -x > 2 - 15 \]
\[ -x > -13 \]
5. Multiply by \(-1\) (remember to flip the inequality sign):
\[ x < 13 \]

### Step 2: Solve the second inequality
Now we solve the second part of the inequality:
\[ 2 - 3x < 2x - \frac{x}{3} \]

**Rearranging the terms:**
1. Combine like terms on the right side:
\[ 2 - 3x < \frac{6x - x}{3} \]
\[ 2 - 3x < \frac{5x}{3} \]

**Eliminate the fraction:**
2. Multiply the entire inequality by \(3\) (to eliminate the fraction):
\[ 3(2) - 3(3x) < 5x \]
\[ 6 - 9x < 5x \]

**Rearranging the terms:**
3. Add \(9x\) to both sides:
\[ 6 < 5x + 9x \]
\[ 6 < 14x \]
4. Divide by \(14\):
\[ \frac{6}{14} < x \]
\[ \frac{3}{7} < x \]

### Step 3: Combine the results
From our two inequalities, we have:
1. \( x < 13 \)
2. \( x > \frac{3}{7} \)

Thus, the combined result is:
\[ \frac{3}{7} < x < 13 \]

### Conclusion
The values of \(x\) can take any value in the range \( \left( \frac{3}{7}, 13 \right) \).