HomeGraphing Linear Inequalities Foldable: Boundary Lines and Test-Point Shading Free original lessons
Graphing Linear Inequalities Foldable: Boundary Lines and Test-Point Shading
Build a four-flap foldable to graph two-variable linear inequalities: choose dashed or solid boundary lines, then shade a half-plane using a test point.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
Graphing a two-variable linear inequality starts with the related equation. That equation gives a boundary line. The inequality symbol decides whether the boundary is dashed or solid. Then a test point decides which half-plane to shade. This lesson uses a planned four-flap foldable: each flap is labeled by an inequality symbol type and contains a mini coordinate grid plus a test-point box. You will practice the full routine on four self-contained tasks and check each graph with simple arithmetic.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Boundary Line: Dashed or Solid
Write the inequality in slope-intercept form when possible, then graph the related equation y = mx + b. The line is the boundary between two half-planes. If the inequality is strict, using < or >, draw a dashed line because points on the line do not satisfy the inequality. If the inequality is non-strict, using ≤ or ≥, draw a solid line because points on the line do satisfy it.
For example, y > 2x - 1 has a dashed boundary through (0, -1) with slope 2. The inequality y ≤ -x + 3 has a solid boundary through (0, 3) with slope -1. Always label the boundary line with its equation so the graph is self-checking.
- Rewrite the inequality so y is isolated when convenient.
- Graph the related equation as a line using slope and intercept or two intercepts.
- Choose dashed for < or > and solid for ≤ or ≥.
- Label the line with its equation.
Test Point and Half-Plane Shading
Pick a test point that is not on the boundary line. The origin (0,0) is convenient unless the line passes through it. Substitute the coordinates into the original inequality. If the result is true, shade the half-plane containing the test point. If false, shade the opposite half-plane.
For y > 2x - 1, test (0,0): 0 > -1 is true, so shade the side containing the origin. For y ≤ -x + 3, test (0,0): 0 ≤ 3 is true, so shade the origin side. Record the test point and the true or false result in the test-point box on the foldable flap.
- Choose a test point not on the boundary.
- Substitute into the original inequality.
- If true, shade the side containing the test point; if false, shade the other side.
- Write the test point and result in the flap box.
Foldable Flap Routine
The planned foldable has four flaps labeled by symbol type: <, >, ≤, ≥. Each flap includes a mini coordinate grid for the boundary and a test-point box for the shading decision. A worked example flap models the full process, and blank practice flaps let you repeat it.
On each flap, first draw the boundary with the correct line style. Then write one test point and its true or false result. Finally shade the correct half-plane lightly so the boundary remains visible. This keeps the symbol, line style, and shading decision connected in one place.
- Label each flap with one inequality symbol type.
- Draw the boundary on the mini grid using dashed or solid.
- Write a test point and substitute it.
- Shade the half-plane indicated by the test result.
Worked example
Graph the inequality y < -2x + 4. State the boundary line style, a test point, and the shaded half-plane.
Worked answer
Boundary: dashed line through (0,4) and (2,0) with slope -2. Test (0,0): 0 < 4 is true, so shade the half-plane containing the origin, below the line.
Four practice exercises
Exercise 1
Graph y ≥ 3x - 2. Identify the boundary line style, a test point, and the shaded half-plane.
Show answer and check
Solid boundary through (0,-2) and (1,1). Test (0,0): 0 ≥ -2 is true, so shade the half-plane containing the origin, above the line.
Why: The symbol ≥ includes the boundary, so the line is solid. The test point (0,0) is not on the line and makes the inequality true, so the origin side is shaded.
Check: Substitute (0,0) into y ≥ 3x - 2: 0 ≥ -2 is true. Substitute a point on the other side, such as (0,-3): -3 ≥ -2 is false. The shading matches.
Exercise 2
Graph y < -x + 1. Identify the boundary line style, a test point, and the shaded half-plane.
Show answer and check
Dashed boundary through (0,1) and (1,0). Test (0,0): 0 < 1 is true, so shade the half-plane containing the origin, below the line.
Why: The symbol < excludes the boundary, so the line is dashed. The origin is not on the line and satisfies the inequality, so the origin side is shaded.
Check: Substitute (0,0): 0 < 1 is true. Substitute (0,2): 2 < 1 is false. The shaded side contains (0,0) and not (0,2).
Exercise 3
Graph 2x + y ≤ 4. Identify the boundary line style, a test point, and the shaded half-plane.
Show answer and check
Solid boundary through (0,4) and (2,0). Test (0,0): 0 ≤ 4 is true, so shade the half-plane containing the origin, below the line.
Why: The symbol ≤ includes the boundary, so the line is solid. Rewriting as y ≤ -2x + 4 confirms the slope is -2. The origin satisfies the inequality, so the origin side is shaded.
Check: Substitute (0,0): 0 ≤ 4 is true. Substitute (3,0): 6 ≤ 4 is false. The shading matches the test results.
Exercise 4
Graph y > 0.5x + 2. Identify the boundary line style, a test point, and the shaded half-plane.
Show answer and check
Dashed boundary through (0,2) and (2,3). Test (0,0): 0 > 2 is false, so shade the half-plane not containing the origin, above the line.
Why: The symbol > excludes the boundary, so the line is dashed. The origin makes the inequality false, so the opposite side is shaded.
Check: Substitute (0,0): 0 > 2 is false. Substitute (0,3): 3 > 2 is true. The shaded side contains (0,3) and not (0,0).
Common mistakes
- Using a solid line for < or >. Strict inequalities exclude the boundary, so use a dashed line for < and >.
- Shading before testing a point. Choose a test point not on the line, substitute, and shade only after the result is known.
- Testing a point that lies on the boundary. Pick a point clearly off the line, such as (0,0) when it is not on the boundary.
- Forgetting to reverse the inequality when multiplying or dividing by a negative number. If you solve for y and divide by a negative coefficient, reverse the symbol before choosing the line style.
Self-check rubric
- Boundary line is drawn through the correct intercepts with the correct slope.
- Line style matches the symbol: dashed for < or >, solid for ≤ or ≥.
- A test point not on the boundary is clearly stated.
- Substitution into the original inequality is shown and evaluated correctly.
- Shading is on the half-plane indicated by the test result.
- Boundary line remains visible after shading.
- Valid alternative conventions are accepted: any correct test point, any correct pair of intercepts, and reasonable handwriting or line thickness.
Questions and answers
Why does a dashed line mean the boundary is not included?
A strict inequality such as y > x + 1 is false when y equals x + 1, so points on the line do not satisfy it. The dashed line shows those points are excluded.
Can I always use (0,0) as a test point?
Only if the boundary line does not pass through the origin. If it does, choose another point such as (1,0) or (0,1).
How do I check my shading?
Pick one point in the shaded region and one in the unshaded region. Substitute both into the inequality. The shaded point should make it true and the unshaded point should make it false.
Does the foldable require exact drawing?
No. The convention is that the boundary line passes through the correct intercepts and has the correct slope, and the shading is on the correct side. Handwriting and line thickness may vary.