Writing Systems of Inequalities from Shaded Graphs: Reverse Reasoning Practice

Algebra and Function Practice

Most graphing lessons start with inequalities and ask you to shade. This reverse task starts with a shaded region and asks you to write the system that produces it. You will identify each boundary line from its slope and intercept, decide whether the line is solid or dashed, and use a test point to choose the correct inequality symbol. The key habit is verification: after writing both inequalities, test a point inside the shaded region and a point outside it to confirm your system matches the graph.

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

Reading boundary lines from a graph

Each boundary is a straight line. Find two clear points on the line, compute slope as rise over run, and write the equation in slope-intercept form y = mx + b. If the line is vertical, write x = c instead. Record whether the line is solid or dashed before choosing the inequality symbol.

A solid boundary means points on the line are included, so use ≤ or ≥. A dashed boundary means points on the line are excluded, so use < or >. This decision is separate from shading direction and must be checked independently.

1. Locate two points where the boundary crosses grid intersections.
2. Compute slope m = (change in y) / (change in x).
3. Identify the y-intercept b and write y = mx + b.
4. Note solid or dashed to select ≤, ≥, <, or >.
5. Choose a test point not on the line and substitute to decide the symbol direction.

Choosing the inequality symbol with test points

After writing the boundary equation, pick a test point that is clearly inside the shaded region. Substitute its coordinates into the equation form. If the resulting statement is true for the intended symbol, keep that symbol; if false, reverse it. Repeat with a point outside the region to confirm the reverse is false.

For example, suppose a boundary is y = 2x + 1 and the shaded region lies below a solid line. Test (0,0): 0 ≤ 2(0)+1 is true, so y ≤ 2x + 1 is consistent. Test a point above the line such as (0,3): 3 ≤ 1 is false, confirming the outside point is excluded.

1. Select an inside test point with integer coordinates when possible.
2. Substitute into the boundary equation using the candidate symbol.
3. Keep the symbol if the statement is true; reverse it if false.
4. Test an outside point to confirm it makes the inequality false.
5. Write the final pair and verify both inequalities together at one inside point.

Combining two inequalities into a system

A system of inequalities is the pair of statements joined by 'and' because the shaded region is the overlap of two half-planes. Write both inequalities with the same variable isolated when possible, such as y ≤ 2x + 1 and y > -x + 4. If one boundary is vertical, write x ≥ c or x ≤ c.

Check that the overlap of the two half-planes matches the shaded region. A common error is reversing one symbol while keeping the other correct, which produces a region on the wrong side of one boundary. Testing one inside point against both inequalities catches this quickly.

1. Write each boundary equation separately.
2. Decide solid or dashed for each line.
3. Use test points to fix each symbol.
4. Join the two inequalities as a system.
5. Verify one inside point satisfies both and one outside point fails at least one.

Checking boundary type and shading direction

Boundary type and shading direction are independent decisions. A solid line can shade above or below; a dashed line can also shade above or below. Do not assume that solid means 'above' or dashed means 'below'. Read each feature from the graph.

When a line is vertical, shading to the right means x ≥ c or x > c depending on solid or dashed. Shading to the left means x ≤ c or x < c. For horizontal lines, shading above means y ≥ c or y > c, and shading below means y ≤ c or y < c.

1. Mark each boundary as solid or dashed.
2. Mark the shaded side of each boundary.
3. Match solid to ≤ or ≥ and dashed to < or >.
4. Match the shaded side using a test point.
5. Recheck both decisions after writing the system.

Worked example

A graph shows two solid boundary lines. Line A passes through (0,1) and (2,5). Line B passes through (0,4) and (2,0). The shaded region is below Line A and below Line B. Write the system of inequalities.

Four practice exercises

Common mistakes

• Using the wrong symbol because the student confuses solid with 'above' and dashed with 'below'. Treat boundary type and shading direction as two separate checks. First decide ≤ or ≥ versus < or > from solid or dashed. Then use a test point to decide which side is shaded.
• Testing a point that lies exactly on the boundary line. Choose a test point clearly inside or outside the shaded region, not on the line. A point on the line makes the equation equal and does not distinguish the inequality direction.
• Writing both inequalities with the same symbol without checking each boundary separately. Test each boundary with its own inside point. One boundary may shade above while the other shades below, so the symbols can differ.
• Forgetting to reverse the inequality when the boundary equation is rearranged. Keep each inequality in the form y ≤ mx + b or y ≥ mx + b when possible. If you multiply or divide by a negative number, reverse the symbol and recheck with a test point.

Self-check rubric

• Correctly identifies both boundary equations in slope-intercept or vertical form.
• Correctly labels each boundary as solid or dashed and selects ≤/≥ or </> accordingly.
• Uses at least one test point per boundary to justify the shading direction.
• Writes the final system as a pair of inequalities joined by 'and'.
• Verifies one inside point satisfies both inequalities and one outside point fails at least one.
• Accepts equivalent forms such as 2x - y ≥ -1 for y ≤ 2x + 1, provided the region is unchanged.

Questions and answers