If `2x - 6 le x -3 and 1 + 3x lt 4x + 4,` then x can take which of the following values ?

Options

A. - 4
B. 4
C. 6
D. 2

Answer

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

### Step 1: Solve the first inequality
The first inequality is:
\[ 2x - 6 \leq x - 3 \]

**Rearranging the inequality:**
1. Subtract \(x\) from both sides:
\[ 2x - x - 6 \leq -3 \]
\[ x - 6 \leq -3 \]

2. Add 6 to both sides:
\[ x \leq -3 + 6 \]
\[ x \leq 3 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 1 + 3x < 4x + 4 \]

**Rearranging the inequality:**
1. Subtract \(3x\) from both sides:
\[ 1 < 4x - 3x + 4 \]
\[ 1 < x + 4 \]

2. Subtract 4 from both sides:
\[ 1 - 4 < x \]
\[ -3 < x \]
\[ x > -3 \]

### Step 3: Combine the results
From the two inequalities, we have:
1. \( x \leq 3 \)
2. \( x > -3 \)

Combining these results, we find:
\[ -3 < x \leq 3 \]

### Conclusion
The values that \(x\) can take are in the interval:
\[ (-3, 3] \]

### Possible values for \(x\)
Given the options, \(x\) can take the value of \(2\) since it lies within the interval \((-3, 3]\).

### Final Answer
Thus, the correct answer is:
**Option 4: \(x = 2\)**

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