If `3x - 3 lt 3 + x //2 and x -2 le 6 + 2x,` then x can take which of the following values ? (a) 6 (b) 2 (c) 10 (d) -10

Options

A. 6
B. 2
C. 10
D. `-10`

Answer

To solve the inequalities given in the question, we will break it down into two parts and solve each inequality step by step.

### Step 1: Solve the first inequality
The first inequality is:
\[ 3x - 3 < 3 + \frac{x}{2} \]

**Step 1.1:** Rearrange the inequality.
- Move \(\frac{x}{2}\) to the left side and \(-3\) to the right side:
\[ 3x - \frac{x}{2} < 3 + 3 \]

**Step 1.2:** Simplify the right side.
- The right side simplifies to \(6\):
\[ 3x - \frac{x}{2} < 6 \]

**Step 1.3:** Combine the terms on the left side.
- To combine \(3x\) and \(-\frac{x}{2}\), convert \(3x\) to a fraction:
\[ 3x = \frac{6x}{2} \]
- Now combine:
\[ \frac{6x}{2} - \frac{x}{2} = \frac{5x}{2} \]
- So, we have:
\[ \frac{5x}{2} < 6 \]

**Step 1.4:** Multiply both sides by \(2\) to eliminate the fraction:
\[ 5x < 12 \]

**Step 1.5:** Divide by \(5\):
\[ x < \frac{12}{5} \]
- This simplifies to:
\[ x < 2.4 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ x - 2 \leq 6 + 2x \]

**Step 2.1:** Rearrange the inequality.
- Move \(2x\) to the left side and \(-2\) to the right side:
\[ x - 2 - 2x \leq 6 \]

**Step 2.2:** Combine the terms:
\[ -x - 2 \leq 6 \]

**Step 2.3:** Add \(2\) to both sides:
\[ -x \leq 8 \]

**Step 2.4:** Multiply by \(-1\) (remember to reverse the inequality):
\[ x \geq -8 \]

### Step 3: Combine the results
From the first inequality, we have:
\[ x < 2.4 \]

From the second inequality, we have:
\[ x \geq -8 \]

### Final Result
Combining these results, we have:
\[ -8 \leq x < 2.4 \]

### Step 4: Determine which values are valid
Now we check the options given:
- (a) 6: Not valid (6 is not less than 2.4)
- (b) 2: Valid (2 is between -8 and 2.4)
- (c) 10: Not valid (10 is not less than 2.4)
- (d) -10: Not valid (-10 is less than -8)

Thus, the only value that \(x\) can take from the options provided is:
**(b) 2**