If `5 - 2x lt 4 - x and 3 (2-x) gt 2 -4x,` then x can take which of the following values ?

Options

A. -5
B. `-2`
C. 3
D. `-3`

Answer

To solve the inequalities given in the question, we will break it down step by step.

### Step 1: Solve the first inequality
The first inequality is:
\[ 5 - 2x < 4 - x \]

**Rearranging the inequality:**
1. Subtract 4 from both sides:
\[ 5 - 4 - 2x < -x \]
\[ 1 - 2x < -x \]

2. Add \(2x\) to both sides:
\[ 1 < -x + 2x \]
\[ 1 < x \]

This can be rewritten as:
\[ x > 1 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 3(2 - x) > 2 - 4x \]

**Expanding and rearranging:**
1. Distribute \(3\) on the left side:
\[ 6 - 3x > 2 - 4x \]

2. Add \(4x\) to both sides:
\[ 6 - 3x + 4x > 2 \]
\[ 6 + x > 2 \]

3. Subtract \(6\) from both sides:
\[ x > 2 - 6 \]
\[ x > -4 \]

### Step 3: Combine the results
From the first inequality, we found:
\[ x > 1 \]

From the second inequality, we found:
\[ x > -4 \]

Since \(x > 1\) is a stricter condition than \(x > -4\), the final solution is:
\[ x > 1 \]

### Conclusion
Thus, the values that \(x\) can take must satisfy \(x > 1\).