If $a\leq b$ and $-a\leq b$, then $|a|\leq b$. I have arrived at the two separate conclusions:


*

*$a\leq b$ 

*$-a\leq b$
Can I conclude that $|a|\leq b$? I am missing something as it is not by definition of the absolute value.

You could prove it by proving the contrapositive, as follows.  
Assume $|a|>b$.  
Now $|a|=a$ or $|a|=-a$.  
Therefore $a>b$ or $-a>b$.  
