Graphing Inequalities Matching Activity: Pair Statements to Shaded Half-Planes

Algebra and Function Practice

This matching activity helps you connect a linear inequality in two variables to its graphed half-plane region. You will compare inequality symbols, boundary lines, and shading to decide which graph belongs with which statement. The focus is on matching, not on solving inequalities algebraically or drawing graphs from scratch. Each graph shows a boundary line and a shaded region; your job is to read both features carefully. A dashed boundary means the line is not included, while a solid boundary means it is included. The shading side is determined by testing a point or by reasoning about the inequality form. Use the worked example, mistake list, and practice tasks to build a reliable matching routine.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

Reading the boundary line and shading

Every linear inequality in two variables has a boundary line obtained by replacing the inequality symbol with an equals sign. The graph of the inequality is one side of that line. If the inequality is strict, such as y > 2x + 1 or y < 2x + 1, the boundary is dashed because points on the line do not satisfy the inequality. If the inequality is non-strict, such as y ≥ 2x + 1 or y ≤ 2x + 1, the boundary is solid because points on the line do satisfy it.

To decide which side to shade, test a point that is not on the boundary. The origin (0,0) is convenient when it is not on the line. Substitute the coordinates into the inequality. If the result is true, shade the side containing that point; if false, shade the other side. For inequalities written as y > mx + b or y ≥ mx + b, the solution is above the line. For y < mx + b or y ≤ mx + b, the solution is below the line. For vertical lines such as x > 3, shade to the right; for x < 3, shade to the left.

1. Identify the boundary line from the inequality by replacing the symbol with an equals sign.
2. Decide whether the boundary is dashed or solid from the strict or non-strict symbol.
3. Choose a test point not on the boundary, often the origin.
4. Substitute the test point into the inequality and check whether the result is true.
5. Shade the side containing the test point if true; otherwise shade the opposite side.
6. Compare your boundary and shading with the pre-drawn graph to make the match.

A reliable matching routine

Start with the boundary line, not the shading. Two inequalities may share the same boundary but have opposite shading, so the line alone is not enough. First match the slope and intercept or the vertical-line position. Then use the symbol to decide dashed versus solid. Finally, use a test point or the above/below rule to confirm the shaded side.

When several cards look similar, write a quick test in the margin. For example, for y ≤ -x + 4, test (0,0): 0 ≤ 4 is true, so the origin side is shaded and the boundary is solid. For y > -x + 4, test (0,0): 0 > 4 is false, so the opposite side is shaded and the boundary is dashed. This two-step check prevents matching by shape alone.

1. Sort inequality cards by boundary line features: slope, intercept, or vertical position.
2. Sort graph cards by boundary type: dashed or solid.
3. Use a test point to confirm the shaded side for each candidate pair.
4. Record the pair only after both boundary and shading agree.
5. For any remaining cards, recheck the inequality symbol and the test-point arithmetic.

Using a convention for hand-drawn matching work

When you sketch or annotate graphs by hand, use a clear convention so your reasoning can be checked. Draw the boundary line first. Mark a dashed line with short gaps and a solid line with a continuous stroke. Shade the chosen side with light parallel strokes or a single arrow pointing into the region. Label one test point with its coordinates and write the substituted inequality beside it. Letter styles and x-height vary, so write labels clearly enough that another person can read the coordinates and symbols.

A valid alternative is to write above or below the line instead of shading, as long as the meaning is unambiguous. Another valid alternative is to list three sample points from the shaded region and one from the unshaded region. The convention is not a single universal answer; it is a consistent way to show boundary type, shaded side, and test-point evidence.

1. Draw the boundary line and mark it dashed or solid according to the symbol.
2. Choose and label a test point not on the boundary.
3. Write the substitution and state whether it is true or false.
4. Indicate the shaded side with strokes, an arrow, or a written above/below note.
5. Check that another reader can identify the boundary type and shaded side from your notation.

Worked example

An inequality card reads y ≤ -2x + 3. A pre-drawn graph shows a solid line with slope -2 and y-intercept 3, and the region below the line is shaded. Does the graph match the inequality? Explain using a test point.

Four practice exercises

Common mistakes

• Matching only by the boundary line and ignoring the shading. Always test a point or use the above/below rule to confirm the shaded side before recording a pair.
• Using a solid boundary for a strict inequality such as y > 2x - 1. Strict symbols > and < require a dashed boundary; non-strict symbols ≥ and ≤ require a solid boundary.
• Testing a point that lies on the boundary line. Choose a point clearly inside one region, such as the origin when it is not on the line, so the test distinguishes the two sides.
• Reversing the shading rule for negative slopes. Do not rely on slope direction alone; substitute a test point into the actual inequality to decide the shaded side.
• Confusing x > 3 with y > 3 when matching vertical and horizontal graphs. Check which variable is restricted: x > 3 is a vertical boundary with shading to the right, while y > 3 is a horizontal boundary with shading above.

Self-check rubric

• Boundary line equation is identified correctly from the inequality.
• Boundary is marked dashed for strict symbols or solid for non-strict symbols.
• A test point not on the boundary is chosen and substituted correctly.
• The true or false result of the test is stated and used to justify the shaded side.
• The final match is supported by both boundary features and shading evidence.
• Hand-drawn notation clearly shows boundary type and shaded side, even if letter styles or x-height vary.
• Valid alternative notations, such as above/below labels or sample-point lists, are accepted when unambiguous.

Questions and answers