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

Options

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

Answer

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

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

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

2. Subtract 2 from both sides:
\[ x < 1 - 2 \]
\[ x < -1 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 2(x - 4) \leq 4x - 1 \]

**Expanding and rearranging the inequality:**
1. Distribute \(2\):
\[ 2x - 8 \leq 4x - 1 \]

2. Subtract \(2x\) from both sides:
\[ -8 \leq 4x - 2x - 1 \]
\[ -8 \leq 2x - 1 \]

3. Add 1 to both sides:
\[ -8 + 1 \leq 2x \]
\[ -7 \leq 2x \]

4. Divide both sides by 2:
\[ -\frac{7}{2} \leq x \]
or
\[ x \geq -3.5 \]

### Step 3: Combine the results
From Step 1, we have:
\[ x < -1 \]

From Step 2, we have:
\[ x \geq -3.5 \]

Now, we combine these two results:
\[ -3.5 \leq x < -1 \]

### Step 4: Determine the possible values of x
Now we need to check which of the given options fall within the range \([-3.5, -1)\):
- (a) 0: Not in the range
- (b) 2: Not in the range
- (c) 4: Not in the range
- (d) -2: Not in the range

None of the options provided fall within the range. However, if we consider the closest option that is less than -1, we see that:
- The only value that is less than -1 is option (d) -2.

### Final Answer
The value of \(x\) that satisfies both inequalities is:
**(d) -2**