If \begin{vmatrix} x+1 & x \ x & x+2 \end{vmatrix} = \begin{vmatrix} 4 & 1 \ 1 & 3 \end{vmatrix}, then write the value of x.

Compute the determinants as shown below:
\begin{vmatrix} x+1 & x \ x & x+2 \end{vmatrix} = \begin{vmatrix} 4 & 1 \ 1 & 3 \end{vmatrix}
(x+1)(x+2) - (x)(x) = 4 \times 3 - 1 \times 1
\implies x^2 + 2x + x + 2 - (x^2) = 12 - 1
\implies x^2 + 3x + 2 - x^2 = 11
\implies 3x + 2 = 11
\implies 3x = 9
\implies x = 3