Find values of $x$ so that the matrix is invertible

You just have to calculate it determinant:

$$ \det(A) = 4x^2 -4x^2 $$

Since it is always $0$ it is never invertibile.

Note that $\forall x, R_3=2R_1\implies Rank(A)<3\implies \det(A)=0$.

$\det(A)=\begin{vmatrix} x & 0 & x \\ x & 2 & 1 \\ 2x & 0 & 2x \\ \end{vmatrix}$ $=\begin{vmatrix} x & 0 & x \\ x & 2 & 1 \\ 0 & 0 & 0 \\ \end{vmatrix}=0$

The first step equals second step by row operations.