CCSS.Math.Content.HSA-REI.D.12
The Standard
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students turn each inequality into a boundary line, choose solid or dashed, and shade the side containing valid points. For several inequalities, they identify the region where all shaded areas overlap.
What Mastery Looks Like
- A student graphs the correct boundary, chooses a solid or dashed line, and shades the correct side. For a system, the student marks the overlap and verifies a sample point in every inequality.
Common Misconceptions
- Students may use a solid line for a strict inequality or a dashed line when equality is included. They may shade the wrong side, forget to reverse the sign when dividing by a negative, or treat overlap as either shaded region.
How to Assess It
- Exit ticket: Graph y > -2x + 3 and y ≤ x - 1 on the same axes. Shade the solution and label one point that works and one that fails.
Lesson moves
Ways to Teach It
On transparent coordinate-grid sheets, students graph one inequality per sheet, stack them, and trace the overlap.
Ask students to explain why a boundary line is dashed for y < 2x + 1 but solid for y ≤ 2x + 1.
Match cards showing an inequality, its boundary style, a shaded graph, and a point that satisfies it.
Model ticket sales with adult and student tickets, a seat limit, and a revenue target, then graph the feasible combinations.
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Printable HSA-REI.D.12 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSA-REI.D.12, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
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Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.HSA-REI.D.10
Graphing inequality half-planes requires seeing a two-variable relation as all coordinate pairs that make the statement true.
- CCSS.Math.Content.8.F.A.3
Graphing linear inequalities relies on recognizing the related linear equation as the boundary line before shading the correct half-plane.
- CCSS.Math.Content.HSA-REI.B.3
Solving one-variable inequalities supports rearranging two-variable inequalities and understanding which side of a boundary satisfies the inequality.
What This Unlocks
Mastery here sets students up for these next.
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Related Standards
- CCSS.Math.Content.HSA-REI.D
Represent and solve equations and inequalities graphically
- CCSS.Math.Content.HSA-REI.C.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
- CCSS.Math.Content.8.EE.C.8a
Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersect...
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