How do I complete a proof for an inequality by using the triangle inequality theorem? Prove that the inequality will hold for every real number, $x$
$$\left| 2+x \right| \le \left| 2x+1 \right| +\left| 1-x \right| $$
Proof: This proof is by case analysis. 
Case 1: 
1) Let $a=2x+1$ and let $b=1-x$ and $a,b\in\mathbb{R}$
2) Suppose $a+b \ge 0$, then $\left| a+b \right| =a+b$
3) Assume that $a\le \left| a \right|$ and $b\le \left| b \right| $, by adding these two inequalities, we get $a+b\le \left| a \right| +\left| b \right|$
4) Therefore we get that $\left| a+b \right| =a+b\le \left| a \right| +\left| b \right|$
Case 2:
1) Let $a=2x+1$ and let $b=1-x$ and $a,b\in\mathbb{R}$
2) Suppose $a+b < 0$, then $\left| a+b \right| = -a-b$
3) Assume that $-a\le \left| -a \right|$ and $-b\le \left| -b \right| $, by adding these two inequalities, we get $-a-b\le \left| a \right| +\left| b \right|$
4) Therefore we get that $\left| -a-b \right| =a+b\le \left| a \right| +\left| b \right|$
At this point, I have no idea what to do to complete my proof or to even show that my original claim is correct. I would like to be guided in the right direction. I am new to proof writing, so this is quite difficult for me to wrap my head around.

By the triangle inequality $|a| + |b| \ge |a + b|$
Let $a = 2x + 1$.  Let $b = 1 - x.$
Then $|2x + 1| + |1 - x| \ge |2x + 1 + 1 - x| = |x + 2|$
