📝 Practice Worksheets
🛠️ Related Tool
🔑 Key Concepts
- An inequality in two variables represents an entire region of the coordinate plane.
- An inequality involving only \(y\) has a horizontal boundary line.
- An inequality involving only \(x\) has a vertical boundary line.
- An inequality in the form \(y\lessgtr mx+b\) has a diagonal boundary line.
- Use a solid boundary line for \(\le\) or \(\ge\).
- Use a dashed boundary line for \(<\) or \(>\).
- Shade above the boundary for \(y>\) or \(y\ge\).
- Shade below the boundary for \(y<\) or \(y\le\).
- A test point can confirm which side of the boundary should be shaded.
🧠 Math Vocabulary
- Linear inequality: An inequality comparing two linear expressions in one or more variables.
- Boundary line: The line that separates the solution region from the region containing points that do not satisfy the inequality.
- Solution region: The shaded half-plane containing every ordered pair that makes the inequality true.
- Half-plane: Either of the two regions formed when a line divides the coordinate plane.
- Solid line: A boundary line showing that points on the line are included in the solution.
- Dashed line: A boundary line showing that points on the line are not included in the solution.
- Slope: The rate of vertical change compared with horizontal change along a line.
- \(y\)-intercept: The point where a line crosses the \(y\)-axis.
- Test point: A point substituted into an inequality to determine which side of the boundary should be shaded.
💡 Main Idea
To graph a linear inequality in two variables, first graph the related boundary equation. Use a solid line when equality is included and a dashed line when it is not. Then shade the half-plane containing all ordered pairs that make the inequality true. Inequalities written as \(y>\) or \(y\ge\) are shaded above the boundary, while inequalities written as \(y<\) or \(y\le\) are shaded below it. A test point can be substituted into the original inequality to verify the correct solution region.
📚 What You Should Already Know
Students should know how to plot ordered pairs, identify the four quadrants, interpret slope and \(y\)-intercept, and graph linear equations in slope-intercept form. They should also understand the inequality symbols \(<\), \(>\), \(\le\), and \(\ge\) and know how to substitute coordinates into an algebraic statement.
🚀 What Comes Next
Students can extend these skills to graphing systems of linear inequalities. The overlap between two or more shaded regions represents the ordered pairs that satisfy every inequality in the system. These ideas also support optimization, constraint modeling, and linear programming.
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