Triangles $ABC$ and $DEF$ are similar. The length of side $AB$ is $x+2$, the length of side $BC$ is $x+5$, and the length of side $AC$ is $12$. The length of side $DE$ is $x$, the length of side $EF$ is $x+2$, and the length of side $DF$ is $8$. What is the value of $x$?