If `2 + 2x lt 3 + 5x and 3 (x-2) lt 5 -x,` then x can take which of the following values ?

Options

A. 1
B. 3
C. `-1`
D. `-2`

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:
\[ 2 + 2x < 3 + 5x \]

**Subtract \(2x\) from both sides:**
\[ 2 < 3 + 5x - 2x \]
\[ 2 < 3 + 3x \]

**Subtract 3 from both sides:**
\[ 2 - 3 < 3x \]
\[ -1 < 3x \]

**Divide both sides by 3:**
\[ -\frac{1}{3} < x \]
or
\[ x > -\frac{1}{3} \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 3(x - 2) < 5 - x \]

**Distribute 3 on the left side:**
\[ 3x - 6 < 5 - x \]

**Add \(x\) to both sides:**
\[ 3x + x - 6 < 5 \]
\[ 4x - 6 < 5 \]

**Add 6 to both sides:**
\[ 4x < 5 + 6 \]
\[ 4x < 11 \]

**Divide both sides by 4:**
\[ x < \frac{11}{4} \]
or
\[ x < 2.75 \]

### Step 3: Combine the results
From the two inequalities, we have:
1. \( x > -\frac{1}{3} \)
2. \( x < 2.75 \)

### Conclusion
Combining these results, we find:
\[ -\frac{1}{3} < x < 2.75 \]

Thus, the values that \(x\) can take are any values greater than \(-\frac{1}{3}\) and less than \(2.75\).