An inequality compares two mathematical expressions using signs like $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), or $\geq$ (greater than or equal to). Graphing an inequality visually represents its infinite solution set on a number line or a coordinate plane. [1, 2, 3]
1. Key Rules for Graphing
Whether on a line or a plane, three core rules dictate the visual representation:
- Boundary Type: Use a solid line/dot for inclusive inequalities ($\leq$, $\geq$) because the boundary values are included in the solution. Use a dashed/dotted line or open circle for strict inequalities ($<$, $>$).
- Shading: Shade the region or number line corresponding to values that make the inequality true.
- Test Points: For complex two-variable inequalities, plug a coordinate (like $(0,0)$) into the equation. If the statement is true, shade the side containing the test point. [6, 7]
2. Graphing Inequalities on a Number Line (1 Variable)
This applies to simple inequalities (e.g., involving just $x$) on a 1-dimensional line.
3. Graphing Inequalities on a Coordinate Plane (2 Variables)
This applies to inequalities with both $x$ and $y$ variables (e.g., $y < 2x + 1$).
- Step 1: Replace the inequality sign with an equals sign to find the boundary line (e.g., $y = 2x + 1$).
- Step 2: Draw the line (solid or dashed depending on the sign).
- Step 3: Shade the area.
4. Systems of Inequalities
When given multiple inequalities, their joint solution is found by graphing them together and identifying the overlapping shaded regions. This overlapping zone is called the "feasible region," where any coordinate placed inside will satisfy all inequalities in the system. [2, 11, 12]
For comprehensive tutorials and visual examples of graphing, check out the Khan Academy Graphing Inequalities or the Math LibreTexts Graphing Inequalities Guide. [13]
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