If `5 -3 x lt 4 - x and 5 (2-x ) gt 2 - 2x,` then x can take which of the following values ? (a) 0 (b) −1 (c) 1 (d) 3

Options

A. 0
B. `-1`
C. 1
D. 3

Answer

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

### Step 1: Solve the first inequality
The first inequality is:
\[ 5 - 3x < 4 - x \]

**Rearranging the inequality:**
1. Move \( -x \) to the left side and \( 5 \) to the right side:
\[ 5 - 4 < -x + 3x \]
2. Simplifying both sides:
\[ 1 < 2x \]
3. Dividing both sides by 2:
\[ \frac{1}{2} < x \]
or
\[ x > \frac{1}{2} \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 5(2 - x) > 2 - 2x \]

**Expanding the left side:**
1. Distributing \( 5 \):
\[ 10 - 5x > 2 - 2x \]

**Rearranging the inequality:**
2. Move \( -2 \) to the left side and \( -5x \) to the right side:
\[ 10 - 2 > -2x + 5x \]
3. Simplifying both sides:
\[ 8 > 3x \]
4. Dividing both sides by 3:
\[ \frac{8}{3} > x \]
or
\[ x < \frac{8}{3} \]

### Step 3: Combine the results
From the two inequalities, we have:
1. \( x > \frac{1}{2} \)
2. \( x < \frac{8}{3} \)

This means:
\[ \frac{1}{2} < x < \frac{8}{3} \]

### Step 4: Determine the valid options
Now we check which of the given options satisfy this combined inequality:
- (a) 0: Not valid since \( 0 < \frac{1}{2} \)
- (b) -1: Not valid since \( -1 < \frac{1}{2} \)
- (c) 1: Valid since \( \frac{1}{2} < 1 < \frac{8}{3} \)
- (d) 3: Not valid since \( 3 > \frac{8}{3} \)

### Conclusion
The only value that satisfies both inequalities is:
**(c) 1**

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