To find the highest possible rank of matrix A in an inconsistent system of equations, we need to understand the concept of rank and its relationship with the given system.

Rank of a Matrix:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.

Inconsistent System of Equations:
An inconsistent system of equations is a system in which there are no solutions. In other words, the system of equations is contradictory and cannot be satisfied simultaneously.

Given that AX = b is an inconsistent system of equations, it means that there are no solutions to the system. This implies that the vector b is not in the column space of matrix A.

Now, let's analyze the possible values for the rank of matrix A in this scenario.

Rank of A in an Inconsistent System:
When the system AX = b is inconsistent, it means that the equation b is not in the column space of matrix A. Therefore, the rank of A cannot be equal to the number of columns of A.

To understand this, consider that the column space of A represents all possible linear combinations of the columns of A. Since b is not in the column space, it cannot be expressed as a linear combination of the columns of A.

If the rank of A were equal to the number of columns of A, it would imply that the column space of A spans the entire vector space, and any vector b can be expressed as a linear combination of the columns of A. However, this contradicts the fact that the system is inconsistent.

Therefore, the highest possible rank of matrix A in an inconsistent system of equations is less than the number of columns of A.

Conclusion:
The highest possible rank of matrix A in an inconsistent system of equations is less than the number of columns of A. In this case, A is a 3 x 4 matrix, so the highest possible rank of A is 2 (option B).