📚 Practice With Worksheets
Practice graphing and interpreting systems of linear inequalities with the resources below. The 🛠️ icon identifies a companion worksheet designed specifically for use with the interactive tool above.
▶️ Learn With Videos
Review how to graph each inequality, choose the correct boundary and shading, and identify the overlapping region that solves the complete system.
🛠️ How to Use This Tool
In Explore Systems mode, adjust the controls for both inequalities or generate a new random system. Watch how each boundary and shaded half-plane change.
In Build a System mode, choose \(x\) or \(y\), select an inequality symbol, and enter the expression or boundary value for each inequality.
Use Test a Point to select an ordered pair. The tool checks both inequalities separately and determines whether the point belongs to the common solution region.
🧠 Things to Notice
Each inequality has its own shaded half-plane. The common green region contains the ordered pairs that satisfy both inequalities at the same time.
Parallel boundaries may create a strip of solutions, nested regions, or no common solution, depending on the direction of the inequality symbols.
Horizontal and vertical boundaries follow the same inclusion rules as slanted boundaries: strict inequalities use dashed lines, while inclusive inequalities use solid lines.
📘 Vocabulary
System of linear inequalities: Two or more linear inequalities considered together.
Solution to a system: An ordered pair that makes every inequality in the system true.
Solution region: The shaded part of the coordinate plane containing all solutions to an inequality.
Common solution region: The overlapping region containing the points that satisfy both inequalities.
Boundary line: The line that separates solutions from non-solutions for one inequality.
Dashed boundary: A boundary used with \(<\) or \(>\) to show that points on the line are not included.
Solid boundary: A boundary used with \(\leq\) or \(\geq\) to show that points on the line are included.
Half-plane: One of the two regions formed when a line divides the coordinate plane.
Test point: An ordered pair substituted into an inequality to determine whether it belongs to the solution region.
Feasible region: Another name for the common region containing all possible solutions to a system.
Boundary-only solution: A system whose only common solutions lie on a shared solid boundary.
No solution: A system in which the shaded regions do not share any points.
📐 Understanding Overlapping Solution Regions
Each linear inequality shades one half-plane. A point solves the complete system only when it belongs to every shaded half-plane. The overlapping region therefore represents all possible ordered pairs that make both inequalities true.
\(\text{Solution to the system}=\text{points that satisfy Inequality 1 and Inequality 2}\)
The boundary style determines whether points on the line are included. A dashed boundary is used with \(<\) or \(>\), so points on that line are not solutions. A solid boundary is used with \(\leq\) or \(\geq\), so points on the line may be included.
\(<\text{ or }>:\text{ dashed and not included}\qquad \leq\text{ or }\geq:\text{ solid and included}\)
Some systems produce a large overlapping region, a strip between parallel lines, or a wedge-shaped region. Other systems may share only a solid boundary line. When no point satisfies both inequalities, the system has no solution.
A test point belongs to the common solution region only when substitution produces a true statement for both inequalities. If even one inequality is false, the point is not a solution to the system.
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