Find the value of $x$ that makes $\triangle ABC\sim\triangle DEF$ .

Triangle A B C and triangle D E F are shown. The length of side A B is 4 units. The length of side B C is 6 units. The length of side A C is x plus 1 units. The length of side D E is 12 units. The length of side E F is 6 left-parenthesis x minus 4 right-parenthesis units. The length of side D F is 24 units.