If `2x + 5 gt 2 + 3 x and 2 x gt 3 = 4x - 5,` then x can take which of the following values ?

Options

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

Answer

To solve the inequalities and find the possible values for \( x \), we will break down the problem step by step.

### Step 1: Solve the first inequality
The first inequality given is:
\[ 2x + 5 > 2 + 3x \]

**Rearranging the inequality:**
Subtract \( 2x \) from both sides:
\[ 5 > 2 + x \]

Now, subtract 2 from both sides:
\[ 3 > x \]
or
\[ x < 3 \]

### Step 2: Solve the second inequality
The second inequality is:
\[ 2x > 3 \]

**Rearranging the inequality:**
Divide both sides by 2:
\[ x > \frac{3}{2} \]
or
\[ x > 1.5 \]

### Step 3: Solve the equation
The equation given is:
\[ 2x = 4x - 5 \]

**Rearranging the equation:**
Subtract \( 4x \) from both sides:
\[ 2x - 4x = -5 \]
This simplifies to:
\[ -2x = -5 \]

Now, divide both sides by -2:
\[ x = \frac{5}{2} \]
or
\[ x = 2.5 \]

### Step 4: Check if \( x = 2.5 \) satisfies the inequalities
Now we need to check if \( x = 2.5 \) satisfies both inequalities:
1. From Step 1: \( x < 3 \)
- \( 2.5 < 3 \) is true.

2. From Step 2: \( x > 1.5 \)
- \( 2.5 > 1.5 \) is also true.

### Conclusion
Since \( x = 2.5 \) satisfies both inequalities, the value of \( x \) can be:
\[ x = 2.5 \]

### Final Answer
The possible value of \( x \) is \( 2.5 \).