Q.The solution of $|x| \le 5$ is to be shown on the number line as a shaded segment with circles at both ends. Find the two endpoints and state, with reason, whether each circle is open or closed.

[!TLDR] $|x|\le 5$ is the slack version of Rule A with $a=5$, giving a bounded interval with both ends included. [!ANSWER] Endpoints $-5$ and $5$; both circles are closed (filled), since th …

$|x| \le 5$ is the slack version of Rule A ($|x|<a \iff -a<x<a$), so $|x|\le 5 \iff -5 \le x \le 5$. The two endpoints are $-5$ and $5$. Since the inequality is $\le$ (slack) at both ends simultaneously (a modulus inequality of this type always produces matching circle types at both ends, unlike a general compound inequality), both circles are closed (filled): substituting $x=-5$ gives $|-5|=5 …

Apply the slack version of the modulus rule $|x|\le a \iff -a\le x\le a$ directly with $a=5$, then c …

Students sometimes assume a modulus inequality always gives an open interval; here, because the original inequality is slack ($\le$, not $<$), both end …