absolute-value inequality

Where |x| = 5 asks for points exactly 5 from zero, an absolute-value inequality asks for points within or beyond some distance. |x| < 5 means “closer than 5 steps to zero”; |x| > 5 means “farther than 5 steps.” One traps a value between bounds; the other pushes it out to the extremes.

The two patterns translate into the two kinds of compound inequality. A “less than” form, |expression| < k (with k positive), becomes the and statement −k < expression < k — a single squeezed interval. A “greater than” form, |expression| > k, becomes the or statement expression < −k or expression > k — two pieces with a gap between.

As always, mind the boundaries. If k is negative, |expression| < k has no solution (distance can never be negative), while |expression| > k is true for every value. A clean way to remember the split: less is and, greater is or.