The Boundary Line
Graph linear inequalities in Grade 9 algebra by drawing a solid or dashed boundary line and shading the half-plane that satisfies the inequality, using a test point to confirm direction.
Key Concepts
Property The boundary line is a dashed line when the inequality contains the symbol < or . The boundary line is a solid line when the inequality contains the symbol $\le$ or $\ge$. Explanation Imagine the boundary line is a fence for the solution zone. A solid line ($\le, \ge$) is a sturdy fence, meaning points on the fence are included in the solution area. A dashed line ($<, $) is just for show; the points on that line are not part of the solution. Examples The graph of $y 3x 2$ has a dashed boundary line because points on the line are not solutions. The graph of $y \le x + 4$ has a solid boundary line because points on the line are included as solutions.
Common Questions
What is a boundary line in graphing a linear inequality?
The boundary line is the graph of the related linear equation. It divides the coordinate plane into two half-planes. For y > 2x + 1, the boundary is the line y = 2x + 1, drawn dashed to exclude points on the line.
When should the boundary line be solid versus dashed?
Use a solid line for inequalities with ≤ or ≥ because points on the line are included in the solution. Use a dashed line for strict inequalities < or > because points on the line are NOT solutions.
How do you determine which side of the boundary line to shade?
Pick a test point not on the line (the origin (0,0) is usually easiest) and substitute into the inequality. If the inequality holds true, shade the side containing the test point. If not, shade the opposite side.