Absolute Value Inequalities
I can solve absolute value inequalities and graph the one-piece or two-piece answer.
The problem this sheet solves
'Less than means and, greater than means or' gives students nothing to rebuild from in June. And it leaves them helpless on the two sign cases, which are opposites: |x| −2 is satisfied by every real number. This sheet teaches the distance instead — within the radius, or beyond it — so the connector is a description of the picture rather than a rule handed down. The error analysis item shows the trap that looks most reasonable on paper: a greater-than split like a less-than, which produces a tidy interval that is the exact answer to the opposite question.
The key rule
Read an absolute value as a distance from a centre. Less-than asks which numbers are WITHIN that distance, so the answer is one bounded piece and the parts are joined by AND. Greater-than asks which are BEYOND it, so the answer is two rays joined by OR.
Within is one piece. Beyond is two. Picture the distance, do not memorise a rhyme.
Watch out
The mistake: Splitting |x − 2| > 5 into −5 < x − 2 < 5, the way a less-than is split.
The fix: Greater-than asks for numbers FAR from the centre, so the answer is two rays: x < −3 or x > 7. A one-piece answer to a greater-than is backwards.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.