Multi-Step Inequalities and the Reversal Rule
I can solve multi-step inequalities and reverse the sign when I divide by a negative.
The problem this sheet solves
Students learn 'minus sign means flip'. That gets −2x > 8 right and then quietly breaks 4x > −12, x − 4 > 12 and 4x + 3 > −5, none of which reverse. Because the genuine flips still come out right, the error is nearly invisible in a mark total. Half the items on this sheet contain a visible negative and do NOT reverse, which is the only way to break the false cue. The anatomy row shows a matched pair — same relation, same right side, opposite coefficient — so students can see that only the coefficient could have made the difference, and the reason is taught before the rule: 3 −5.
The key rule
Solve a multi-step inequality exactly as you would an equation, with one exception. If you multiply or divide BOTH sides by a negative number, reverse the direction of the sign. Adding or subtracting a negative never reverses anything.
3 is less than 5, but −3 is greater than −5. A negative turns the line around.
Watch out
The mistake: Seeing any minus sign and reversing, so 4x > −12 becomes x < −3.
The fix: Nothing was divided by a negative there. Dividing by positive 4 gives x > −3. The cue is the operation, not the sign you can see.
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.