Systems of Linear Inequalities
I can graph a system of linear inequalities and identify the region that satisfies all of them.
The problem this sheet solves
The error is not graphing, it is the word AND. A student picks a point sitting in exactly one of the two shaded regions and calls it a solution. It looks like carelessness and it is actually a wrong definition of the word solution. Almost none of this sheet re-teaches graphing. It spends its space on checking a point in EVERY inequality, and the overlap region is drawn as its own hatched object rather than left as an accident of two transparent washes — so it survives a black-and-white classroom printer, which is where most of these are printed. The grading guide sends a student who misses the group back to compound inequalities, not to more graphing.
The key rule
Graph each inequality as you already do, boundary and edge and side, on the same axes. The solution of the system is only where the regions OVERLAP, because a solution must satisfy all of them at once.
Graph each one, then keep only what they share. Every inequality, every time.
Watch out
The mistake: Choosing (4, 1) because it lies in the shaded region for y ≤ x + 2.
The fix: Check it in BOTH: 1 ≤ 6 is true, but 4 + 1 < 4 is false. A point in one region only is not a solution of the system.
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.