If 4 + 3x <=6 + x and 3x + 5 > 2 + 2x. then x can take which of the following values ?

Options

A. 2
B. `-2`
C. 4
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 \( 4 + 3x \leq 6 + x \)

1. Start by isolating the variable \( x \):
\[ 4 + 3x \leq 6 + x \]
2. Subtract \( x \) from both sides:
\[ 4 + 3x - x \leq 6 \]
This simplifies to:
\[ 4 + 2x \leq 6 \]
3. Next, subtract 4 from both sides:
\[ 2x \leq 6 - 4 \]
This simplifies to:
\[ 2x \leq 2 \]
4. Now, divide both sides by 2:
\[ x \leq 1 \]

### Step 2: Solve the second inequality \( 3x + 5 > 2 + 2x \)

1. Start by isolating the variable \( x \):
\[ 3x + 5 > 2 + 2x \]
2. Subtract \( 2x \) from both sides:
\[ 3x - 2x + 5 > 2 \]
This simplifies to:
\[ x + 5 > 2 \]
3. Now, subtract 5 from both sides:
\[ x > 2 - 5 \]
This simplifies to:
\[ x > -3 \]

### Step 3: Combine the results from both inequalities

From the first inequality, we have:
\[ x \leq 1 \]
From the second inequality, we have:
\[ x > -3 \]

### Step 4: Determine the range of \( x \)

Combining both results, we find:
\[ -3 < x \leq 1 \]

### Step 5: Identify possible values for \( x \)

The values that \( x \) can take are any real numbers between -3 and 1, including 1 but not including -3. Therefore, possible values could be:
- \( -2 \)
- \( 0 \)
- \( 1 \)

### Conclusion

Thus, \( x \) can take values such as \( -2, 0, \) or \( 1 \).

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