AI Tutor
Imagine a drone programmed to stay within 5 miles of home base, or an alarm system that triggers whenever an intruder gets more than 10 yards away from a safe zone.
Both situations depend on distance boundaries. Under its geometric definition, the absolute value expression ∣X∣|X| denotes the distance on a number line from XX to 00.
Because distance is inherently non-negative, ∣X∣≥0|X| \ge 0 for all real expressions XX. Mathematician Karl Weierstrass introduced the vertical bar notation ∣X∣|X| to standardize this non-negative distance.
What happens when we place an entire algebraic expression inside those bars?

The "Less Than" Pattern: Bounded Between

When an inequality says ∣ax+b∣<c|ax + b| < c (with c>0c > 0), it means the entire expression ax+bax + b is trapped within a distance of cc units from zero.
Because it cannot stray farther than cc in either direction, this rewrites as a single continuous compound inequality: −c<ax+b<c-c < ax + b < c.
Let's walk through a complete example: solve and graph ∣2x−5∣≤9|2x - 5| \le 9.
Unpack the absolute value into a three-part chain: −9≤2x−5≤9-9 \le 2x - 5 \le 9. Add 55 to all three sections to get −4≤2x≤14-4 \le 2x \le 14.
Next, divide every part by 22, giving −2≤x≤7-2 \le x \le 7.
On the number line, place closed circles at −2-2 and 77 to show those endpoints are included, then shade the solid segment connecting them.