Don't understand adding a system of compound inequalities

Question

I'm reading a proof of the Division Theorem and one line that comes up is Since 0 ≤ r1 < b and 0 ≤ r2 < b , we have −b < r1 − r2 < b. I do not understand how they combined these two inequalities. Where does the -b come from, and why does the sign ≤ turn into <? Help would be greatly appreciated. Thanks!

Answer 1

$r_1 -r_2 < r_1 < b \Rightarrow r_1-r_2 < b$, and $r_2 - r_1 < r_2 < b\Rightarrow r_2-r_1 < b \Rightarrow r_1-r_2 > -b \Rightarrow -b < r_1-r_2 < b$

Answer 2

I'll argue geometrically. I'm sure we could play around combining inequalities, but it's less fun (to me at least).

The two inequalities $0 \leq r_1 < b$ and $0 \leq r_2 < b$ let us place $r_1$ and $r_2$ on a numberline; they both live in the interval $[0, b)$.

Since they both live in this interval, which is $b$ units long, the distance between $r_1$ and $r_2$ is at most $b$. In fact, since neither can be exactly $b$, the distance between the two must be strictly less than $b$. One way to write this is by saying $|r_1 - r_2| < b$, which can in turn be written as the compound inequality $-b < r_1 - r_2 < b$.