How to solve this multiple-absolute-value equation using regions-in a number line method? How to solve this multiple-absolute-value equation using three-region number line?

I can solve it with combination of giving each absolute value a negative sign and leaving it as it is. There are four combinations. The method using regions in number line only requires three combinations instead of four. But it fails. So please help me to solve it using region-in-a number line method. (The answer should be {-6, -2/3})

Case 1: Let $x<-2$ therefore $$|2x+4|-|3-x|=-2x-4-(3-x)=-1$$which yields to valid answer $x=-6$
Case 2: Let $-2\le x\le3$ therefore $$|2x+4|-|3-x|=2x+4-(3-x)=-1$$which yields to valid answer $x=-\dfrac{2}{3}$
Case 3: Let $x>3$ therefore $$|2x+4|-|3-x|=2x+4+(3-x)=-1$$which yields to invalid answer $x=-8$
