A two-variable inequality has infinitely many ordered-pair solutions. Its graph consists of a boundary line and one half-plane. Boundary style records whether equality is allowed; shading records which side makes the original inequality true.
Boundary, half-plane, and solution point
- Replace the inequality with equality to obtain the boundary line.
- A point in the shaded half-plane satisfies the original inequality.
- A point on a solid boundary is included; a point on a dashed boundary is excluded.
| Operator | Boundary style | Are boundary points solutions? |
|---|---|---|
| \(<\) or \(>\) | dashed | no |
| \(\le\) or \(\ge\) | solid | yes |
A reliable graphing process
Boundary → style → test → shade
- Find the boundary
Replace the inequality sign with equality and graph that line exactly.
- Choose line style
Use dashed for strict operators and solid when equality is included.
- Choose a test point
Use \((0,0)\) when it is not on the boundary; otherwise choose another simple point.
- Test the original inequality
Substitute the point into the inequality, not merely the boundary equation.
- Shade
If the statement is true, shade the side containing the test point; otherwise shade the opposite side.
Inclusive boundary example
Graph an inclusive inequality
Graph \(x+2y\le6\).
- Boundary
\(x+2y=6\), or \(y=-\frac12x+3\).
- Style
The operator \(\le\) includes equality, so the boundary is solid.
- Test the origin
\(0+2(0)\le6\) is true.
- Shade
Shade the side containing \((0,0)\), which is below the line.
Strict boundary example
Graph a strict inequality
Graph \(2x-y>4\).
- Boundary
\(2x-y=4\), or \(y=2x-4\).
- Style
The operator \(>\) is strict, so the boundary is dashed.
- Test the origin
\(2(0)-0>4\) is false.
- Shade
Shade the side opposite the origin; algebraically, \(y<2x-4\), below the line.
Vertical and horizontal boundaries
An inequality involving only \(x\) has a vertical boundary. Smaller \(x\)-values lie left; larger values lie right. An inequality involving only \(y\) has a horizontal boundary. Smaller \(y\)-values lie below; larger values lie above.
Special boundary directions
A vertical inclusive example and a horizontal strict example.
Work backward from a graph
Graph-to-inequality reasoning
- Find the boundary equation
Use intercepts, points, or slope.
- Read inclusion
Solid means equality is possible; dashed means it is not.
- Read the shaded side
Choose a visible point in the shading and test which operator makes it true.
- Verify
Check that an unshaded point fails and that the line style matches the operator.
Check your understanding
A graph has a dashed boundary \(y=-2x+1\) and shading above. Which inequality matches?
- \(y>-2x+1\)
- \(y\ge-2x+1\)
- \(y<-2x+1\)
- \(y\le-2x+1\)
Show answer and explanation
Answer: \(y>-2x+1\)
Above means greater y-values, and dashed means equality is excluded.
What to remember
- The boundary comes from replacing the operator with equality.
- Strict operators use dashed lines; inclusive operators use solid lines.
- Test a point in the original inequality to determine shading.
- Vertical boundaries split left/right; horizontal boundaries split below/above.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.