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.

Yes, it is! Since $\lvert a\rvert$ is one of the numbers $a$ or $-a$, and since both of these numbers are smaller than or equal to $b$, $\lvert a\rvert\leqslant b$.
