Can you explain why $x>\frac 23$ is not a solution to this inequality $|2x + 2| + |x - 1| > 3$
why can't $x>\frac 23$ be a part of the solution?
Thanks for your help!

Suppose $x>2/3$, then if $x \ge 1 $, then $2x+2 + x -1 = 3x + 1 \ge 3 + 1 > 3$.
If $2/3 < x < 1$, then $2x+2 - x +1 = x + 3 > 3$.
This is, however, not a complete solution (probably why it is not a solution!?).
Consider $-1 \le x \le 2/3$, then $2x+2 - x +1 = x + 3 >3 $. This is only true if $x>0$.
If $x<-1$, then $-2x-2 - x +1 = -3x - 1 > 3.$ This holds if $x < -4/3$.
Thus the complete solution is $x \in (-\infty,-4/3) \cup (0,\infty)$.
