A linear inequality in two variables describes a whole region of points, not just one line. Graphing it helps you see every ordered pair that makes the inequality true. This is useful in algebra, economics, science, and any situation where there are limits or constraints.
The graph combines a boundary line with shading to show the solution set clearly.
To graph a linear inequality, first replace the inequality symbol with an equals sign and graph the boundary line. Use a solid line when points on the line are included, as with ≤ or ≥, and use a dashed line when they are not included, as with < or >. Then choose which side to shade by using the inequality direction or by testing a point such as (0, 0).
Every point in the shaded region is a solution to the original inequality.
Key Facts
- Slope-intercept form: y = mx + b, where m is slope and b is the y-intercept.
- For y ≥ mx + b or y ≤ mx + b, use a solid boundary line because equality is included.
- For y > mx + b or y < mx + b, use a dashed boundary line because equality is not included.
- If the inequality is y > mx + b or y ≥ mx + b, shade above the boundary line.
- If the inequality is y < mx + b or y ≤ mx + b, shade below the boundary line.
- A point (x, y) is a solution if substituting its coordinates makes the inequality true.
Vocabulary
- Linear inequality
- A statement comparing two linear expressions using <, >, ≤, or ≥, usually with infinitely many solution points.
- Boundary line
- The line found by replacing the inequality symbol with an equals sign.
- Solution region
- The shaded part of the coordinate plane containing all points that satisfy the inequality.
- Dashed line
- A boundary line used when the inequality does not include points on the line.
- Test point
- A point substituted into the inequality to decide which side of the boundary line should be shaded.
Common Mistakes to Avoid
- Using a solid line for < or > is wrong because points on the boundary line are not solutions.
- Using a dashed line for ≤ or ≥ is wrong because equality means the boundary line itself is included.
- Shading the wrong side of the line gives a region of points that do not satisfy the inequality. Test a point like (0, 0) when you are unsure.
- Forgetting to reverse the inequality when solving by dividing by a negative number can change the solution region. For example, if -y < 3 becomes y > -3, the symbol must flip.
Practice Questions
- 1 Graph y ≥ 2x - 1. State whether the boundary line is solid or dashed, and identify which side should be shaded.
- 2 Graph y < -1/2x + 3. State the y-intercept, the slope, whether the boundary line is solid or dashed, and which side should be shaded.
- 3 A student graphs y ≤ x + 2 using a dashed line and shades below the line. Explain what part is correct and what part must be fixed.