If `5- xgt 1 + 5x and 5 -3 x ge 2 - 4x ,` then x can take which of the following values ?

Options

A. 1
B. 2
C. `-2`
D. `-4`

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:
\[ 5 - x > 1 + 5x \]

**Rearranging the inequality:**
1. Move \(x\) and \(5x\) to one side:
\[ 5 - 1 > 5x + x \]
\[ 4 > 6x \]

2. Divide both sides by 6:
\[ \frac{4}{6} > x \]
\[ \frac{2}{3} > x \]
or
\[ x < \frac{2}{3} \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 5 - 3x \geq 2 - 4x \]

**Rearranging the inequality:**
1. Move \(3x\) and \(2\) to one side:
\[ 5 - 2 \geq -4x + 3x \]
\[ 3 \geq -x \]

2. Multiply both sides by -1 (remember to flip the inequality sign):
\[ -3 \leq x \]
or
\[ x \geq -3 \]

### Step 3: Combine the results
Now we have two inequalities:
1. \( x < \frac{2}{3} \)
2. \( x \geq -3 \)

This means \( x \) can take values in the range:
\[ -3 \leq x < \frac{2}{3} \]

### Step 4: Identify the possible values from the options
Given the options are \( 1, 2, -2, -4 \):
- \( 1 \) is not in the range.
- \( 2 \) is not in the range.
- \( -2 \) is in the range.
- \( -4 \) is not in the range.

Thus, the only value that satisfies both inequalities is:
\[ \boxed{-2} \]