Linear Inequalities in Two Variables: Graphing Regions

A linear inequality in two variables, such as 2x + 3y ≤ 12, is satisfied not by isolated points but by an entire half-plane: every point (x, y) lying on one side of the boundary line 2x + 3y = 12.

They matter because real limits usually involve two quantities at once — hours split between two jobs, or money split between two products — and these inequalities draw the exact region of allowed choices on a graph.

Key Concepts

• Boundary line: the line ax + by = c obtained by replacing the inequality sign with an equals sign.
• Half-plane: the set of points on one side of the boundary line; exactly one of the two half-planes satisfies the inequality.
• Test point: any point off the line, usually (0, 0), substituted to decide which side to shade.
• Solid vs dashed: ≤ and ≥ use a solid line (boundary included); < and > use a dashed line (boundary excluded).
• Solution region: every shaded point, not just the line, is a solution.

How It Works

1. Draw the boundary line ax + by = c — solid for ≤ or ≥, dashed for < or >.
2. Pick a test point not on the line; (0, 0) is the easiest when it is off the line.
3. Substitute it: if the inequality holds, shade the side containing the test point; otherwise shade the opposite side.
4. Read solutions as every point in the shaded half-plane.

Worked Example

Graph 2x + 3y ≤ 12. The boundary 2x + 3y = 12 is a solid line through (6, 0) and (0, 4) — check: 2(6) + 3(0) = 12 and 2(0) + 3(4) = 12. Test (0, 0): 0 ≤ 12 is true, so shade the half-plane containing the origin. Verify a shaded point, (1, 1): 2 + 3 = 5 ≤ 12 holds. Verify an unshaded point, (10, 10): 20 + 30 = 50, and 50 ≤ 12 is false, so it is correctly excluded. Every point below-left of the line is a solution.

Common Misconceptions

• Shading the wrong side. Never guess the side — always substitute a test point and follow what it tells you.
• A solid line for a strict inequality. For 2x + 3y < 12 the boundary is dashed, because points on the line itself are not solutions.
• Thinking the solution is just the line. The line is only the boundary; the solution is the entire shaded half-plane.

Real-World Applications

• Setting up linear programming problems in business and operations research.
• Production planning: machine hours and labour hours shared between two products.
• Diet and mixture problems balancing two nutritional constraints.
• Budget allocation between two spending categories.

A linear inequality in two variables describes a half-plane bounded by its line. Draw the line solid or dashed as the sign demands, test (0, 0), shade the correct side, and the whole shaded region is your solution set.