HomePlot Linear Inequality Regions: Boundary Lines and Test-Point Checks Free original lessons
Plot Linear Inequality Regions: Boundary Lines and Test-Point Checks
Learn to graph two-variable linear inequalities by drawing the correct boundary line and shading the region using a test point. Includes worked example, practice, and rubric.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
Graphing a linear inequality in two variables means showing all points that make the inequality true. Start by graphing the boundary line, choosing solid or dashed based on the inequality symbol, then use a test point to decide which side to shade. This lesson gives a clear convention for boundary style and shading, a worked example, common mistakes, and four self-contained practice tasks with reasoning and check methods.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Boundary Line: Solid or Dashed
Write the inequality in slope-intercept form y = mx + b to identify the boundary line. If the inequality is y > mx + b or y < mx + b, draw a dashed line because points on the line do not satisfy the inequality. If it is y ≥ mx + b or y ≤ mx + b, draw a solid line because points on the line do satisfy it.
For inequalities not solved for y, such as Ax + By ≤ C, you can still graph the line Ax + By = C by finding intercepts. The same rule applies: use a solid line for ≤ or ≥ and a dashed line for < or >.
- Rewrite the inequality as an equation to get the boundary line.
- Find two points on the line, such as intercepts or a point and the slope.
- Draw the line dashed for strict inequalities (<, >) or solid for inclusive inequalities (≤, ≥).
- Label the line with its equation.
Test Point and Shading
Choose a test point that is not on the boundary line. The origin (0,0) is convenient when it is not on the line. Substitute the coordinates into the original inequality. If the result is true, shade the side containing the test point. If false, shade the opposite side.
Use a consistent shading convention: shade the region that makes the inequality true. You may use light pencil shading or a pattern, but the boundary style must match the inequality symbol. Always record the test point and the true/false result near the graph.
- Pick a test point not on the boundary line.
- Substitute its coordinates into the original inequality.
- If the inequality is true, shade the side containing the test point.
- If false, shade the other side.
- Write the test point and result on the worksheet.
Reading and Checking Your Graph
After shading, check a point inside the shaded region and a point outside. The inside point should make the inequality true; the outside point should make it false. This confirms both the boundary style and the shaded side.
For accuracy, use simple integer coordinates when possible. If the boundary line passes through your test point, choose a different test point. Keep units consistent and label axes clearly.
- Choose one point clearly inside the shaded region and one clearly outside.
- Substitute both into the original inequality.
- Confirm inside is true and outside is false.
- Adjust shading or boundary style if the check fails.
Worked example
Graph the inequality y < 2x + 1. Draw the boundary line, choose a test point, and shade the correct region.
Worked answer
Boundary line: y = 2x + 1. Since the inequality is <, draw a dashed line. Test point (0,0): 0 < 2(0) + 1 gives 0 < 1, which is true. Shade the side containing (0,0), which is below the dashed line. Check: point (0,0) inside is true; point (0,2) outside gives 2 < 1 false.
Four practice exercises
Exercise 1
Graph y ≥ -x + 2. State the boundary line style, choose a test point, and describe the shaded region.
Show answer and check
Boundary line y = -x + 2 is solid because the inequality is ≥. Test point (0,0): 0 ≥ -0 + 2 gives 0 ≥ 2, false. Shade the side not containing (0,0), which is above the solid line.
Why: The symbol ≥ means points on the line are included, so the line is solid. The test point (0,0) is not on the line. Substituting gives a false statement, so shade the opposite side from (0,0).
Check: Check (0,3): 3 ≥ -0 + 2 is 3 ≥ 2 true, so it should be inside. Check (0,0): 0 ≥ 2 false, so it should be outside. Confirm shading matches.
Exercise 2
Graph 2x + y < 4. Find the boundary line by intercepts, state the line style, and use a test point to decide shading.
Show answer and check
Boundary line 2x + y = 4 passes through (2,0) and (0,4). Draw a dashed line because the inequality is <. Test point (0,0): 2(0) + 0 < 4 gives 0 < 4, true. Shade the side containing (0,0), which is below the dashed line.
Why: The strict inequality < requires a dashed boundary. The intercepts give two easy points. The origin is not on the line, and substitution is true, so shade toward the origin.
Check: Check (0,0) inside: 0 < 4 true. Check (3,0) outside: 6 + 0 < 4 is 6 < 4 false. Confirm the shaded side contains (0,0) and not (3,0).
Exercise 3
Graph y > 3. State the boundary line, line style, test point, and shaded region.
Show answer and check
Boundary line y = 3 is a horizontal line. Draw it dashed because the inequality is >. Test point (0,0): 0 > 3 is false. Shade the side not containing (0,0), which is above the dashed line.
Why: The inequality y > 3 includes all points with y-coordinate greater than 3. The line y = 3 is not included, so it is dashed. The origin has y = 0, which is not greater than 3, so shade above.
Check: Check (0,4): 4 > 3 true, so inside. Check (0,2): 2 > 3 false, so outside. Confirm shading is above the dashed line.
Exercise 4
Graph x ≤ -1. State the boundary line, line style, test point, and shaded region.
Show answer and check
Boundary line x = -1 is a vertical line. Draw it solid because the inequality is ≤. Test point (0,0): 0 ≤ -1 is false. Shade the side not containing (0,0), which is to the left of the solid line.
Why: The inequality x ≤ -1 includes all points with x-coordinate less than or equal to -1. The line x = -1 is included, so it is solid. The origin has x = 0, which is not ≤ -1, so shade left.
Check: Check (-2,0): -2 ≤ -1 true, so inside. Check (0,0): 0 ≤ -1 false, so outside. Confirm shading is to the left of the solid line.
Common mistakes
- Using a solid line for < or >. Use a dashed line for strict inequalities because points on the line are not included.
- Shading the side containing the test point without checking whether the inequality is true or false. Substitute the test point into the inequality. Shade the side containing the test point only if the result is true; otherwise shade the opposite side.
- Choosing a test point that lies on the boundary line. Pick a point not on the line, such as (0,0) when it is not on the boundary, or (1,1) if the origin is on the line.
- Forgetting to reverse the inequality symbol when solving for y by multiplying or dividing by a negative number. When isolating y, reverse the inequality symbol if you multiply or divide both sides by a negative number.
Self-check rubric
- Boundary line is correctly identified from the inequality.
- Line style matches the inequality symbol: dashed for < or >, solid for ≤ or ≥.
- A test point not on the boundary line is clearly stated.
- Substitution into the original inequality is shown and evaluated correctly.
- Shaded region matches the true/false result of the test point.
- A check point inside the shaded region makes the inequality true.
- A check point outside the shaded region makes the inequality false.
- Graph is labeled with the inequality and axes are clear.
Questions and answers
How do I know which side to shade?
Use a test point not on the boundary line. If substituting the test point makes the inequality true, shade the side containing that point. If false, shade the other side.
When is the boundary line dashed?
Use a dashed line for strict inequalities < and > because points on the line do not satisfy the inequality. Use a solid line for ≤ and ≥.
Can I always use (0,0) as a test point?
Only if (0,0) is not on the boundary line. If the line passes through the origin, choose another point such as (1,0) or (0,1).
What if the inequality is not in y = mx + b form?
You can graph the boundary line by finding intercepts or by solving for y. The same rules for solid/dashed and test-point shading apply.