3 Solve for \( x \) if \( \left|\begin{array}{cc}x-3 & -x \\ 3 & 2\end{array}\right|=4 \)

To solve for \( x \), we first need to compute the determinant of the matrix: \[ \left|\begin{array}{cc}x-3 & -x \\ 3 & 2\end{array}\right| = (x-3)(2) - (-x)(3) = 2(x-3) + 3x. \] This simplifies to: \[ 2x - 6 + 3x = 5x - 6. \] Next, we set the determinant equal to 4: \[ 5x - 6 = 4. \] Adding 6 to both sides gives: \[ 5x = 10. \] Now, dividing both sides by 5 gives: \[ x = 2. \] Thus, the solution is \( x = 2 \).