If `2x + 2 (1-x ) gt 3x -2 gt 2x -5,` then x can take which of the following values ?

Options

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

Answer

To solve the inequality \( 2x + 2(1 - x) > 3x - 2 > 2x - 5 \), we will break it down into two parts and solve each part step by step.

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

1. Distribute the 2:
\[ 2x + 2 - 2x > 3x - 2 \]
2. Simplify the left side:
\[ 2 > 3x - 2 \]
3. Add 2 to both sides:
\[ 4 > 3x \]
4. Divide by 3:
\[ \frac{4}{3} > x \quad \text{or} \quad x < \frac{4}{3} \]

### Step 2: Solve the second part of the inequality
Now, we solve the right part of the inequality:
\[ 3x - 2 > 2x - 5 \]

1. Subtract \( 2x \) from both sides:
\[ 3x - 2x - 2 > -5 \]
2. Simplify:
\[ x - 2 > -5 \]
3. Add 2 to both sides:
\[ x > -3 \]

### Step 3: Combine the results
Now we combine the results from both parts:
1. From the first part, we have:
\[ x < \frac{4}{3} \]
2. From the second part, we have:
\[ x > -3 \]

Thus, the solution for \( x \) is:
\[ -3 < x < \frac{4}{3} \]

### Conclusion
The values that \( x \) can take are those that lie between -3 and \( \frac{4}{3} \).