Let's analyze each of the equations step-by-step to determine the nature of their solutions.
### Equation A: \( 6 + 3x = 3x - 3 \)
1. Initial equation:
[tex]\[ 6 + 3x = 3x - 3 \][/tex]
2. Subtract \(3x\) from both sides:
[tex]\[ 6 + 3x - 3x = 3x - 3 - 3x \][/tex]
3. Simplify:
[tex]\[ 6 = -3 \][/tex]
The statement \(6 = -3\) is a contradiction, indicating there is no solution for Equation A. Therefore, Equation A has no solution.
### Equation B: \( 2(4x - 1) = 8x - 2 \)
1. Distribute the 2 on the left-hand side:
[tex]\[ 2 \cdot 4x - 2 \cdot 1 = 8x - 2 \][/tex]
2. Simplify:
[tex]\[ 8x - 2 = 8x - 2 \][/tex]
This simplifies to an identity, which is \( 8x - 2 = 8x - 2 \). Since both sides of the equation are always equal regardless of the value of \( x \), Equation B has an infinite number of solutions.
### Summary of the Solutions:
- Equation A has no solution.
- Equation B has an infinite number of solutions.
### Verify the Given Statements:
1. Equation \( A \) and Equation \( B \) have an infinite number of solutions.
- This is false because Equation A has no solution.
2. Equation \( A \) has no solution and Equation \( B \) has an infinite number of solutions.
- This is true.
3. Equation \( A \) has an infinite number of solutions and Equation \( B \) has no solution.
- This is false because it is the reverse of the actual solution.
4. Equation \( A \) and Equation \( B \) have no solution.
- This is false because Equation B has an infinite number of solutions.
Based on the above analysis, the true statement is:
Equation \( A \) has no solution and Equation \( B \) has an infinite number of solutions.
Thus, the correct answer is 2.
