📚 Practice With Worksheets
Practice graphing linear inequalities, writing inequalities from graphs, and extending these ideas to systems of inequalities.
▶️ Learn With Videos
Review how to draw the boundary line, choose the correct shading, and graph individual or multiple linear inequalities.
🛠️ How to Use This Tool
In the exploration mode, select an inequality symbol and adjust the slope and y-intercept sliders. Watch how the boundary line and shaded solution region change.
In the inequality-entry mode, select one of the four inequality buttons and enter only the expression that belongs to the right of the symbol, such as 2x + 3 or -x - 4.
Use the test-point feature to select a point on the coordinate plane and determine whether its coordinates make the inequality true.
🧠 Things to Notice
The boundary line is the graph of the related equation formed by replacing the inequality symbol with an equals sign.
Strict inequalities use a dashed boundary because points on the line are not solutions. Inequalities that include equality use a solid boundary because points on the line are included.
For inequalities written with y by itself, greater-than symbols shade above the boundary and less-than symbols shade below the boundary.
📘 Vocabulary
Linear inequality: A mathematical statement that compares two linear expressions using <, >, ≤, or ≥.
Boundary line: The line that separates the two half-planes on the graph of a linear inequality.
Dashed boundary: A boundary used with < or > to show that points on the line are not included in the solution set.
Solid boundary: A boundary used with ≤ or ≥ to show that points on the line are included in the solution set.
Solution region: The shaded half-plane containing all ordered pairs that make the inequality true.
Test point: A point substituted into an inequality to determine which side of the boundary contains the solutions.
Half-plane: One of the two regions formed when a line divides the coordinate plane.
Solution: An ordered pair whose x- and y-values make the inequality true.
📐 Graphing Linear Inequalities
Begin by graphing the related boundary equation. The slope determines the direction and steepness of the line, while the y-intercept identifies where the line crosses the y-axis.
\(y = mx + b\)
Use a dashed line for a strict inequality and a solid line when equality is included.
\(< \text{ or } >:\ \text{dashed boundary} \qquad \leq \text{ or } \geq:\ \text{solid boundary}\)
\(y > mx+b:\ \text{shade above} \qquad y < mx+b:\ \text{shade below}\)
When the correct side is not obvious, substitute a test point that is not on the boundary. Shade the side where the test point makes the inequality true.
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