Linear Inequalities in Two Variables: Complete Review

A linear inequality in two variables is a statement like 2x + 3y < 12, where x and y are unknowns and the relationship uses <, >, ≤ or ≥ instead of an equals sign. An equation graphs as a single line, but an inequality graphs as a whole region — a half-plane — because infinitely many (x, y) pairs satisfy it. Every point in that shaded region is a solution.

Inequalities in two variables are the language of limits and choices. Machine hours, budgets and nutrition minimums are all inequalities, not equations. They are the foundation of linear programming, where the best solution sits at the corners of the feasible region.

Key Concepts

• Boundary line: the line ax + by = c, found by replacing the sign with =; it divides the plane into two half-planes.
• Strict vs non-strict: < and > give a dashed boundary line (points on the line are excluded); ≤ and ≥ give a solid boundary line (points on the line are included).
• Half-plane: one of the two regions created by the boundary line; the inequality is true for all points in exactly one of them.
• Test point: a point off the boundary line, usually (0, 0), tested in the inequality to pick the correct half-plane.
• Solution set: all ordered pairs (x, y) satisfying the inequality — the entire shaded region.

How It Works

• Step 1 — Draw the boundary line: graph ax + by = c using its intercepts.
• Step 2 — Choose solid or dashed: use a solid line for ≤ or ≥ and a dashed line for < or >.
• Step 3 — Pick a test point: (0, 0) is the easiest, provided it is not on the line.
• Step 4 — Test and shade: shade the half-plane containing the test point if it satisfies the inequality, otherwise the opposite one.

Worked Example

Graph the solution set of 2x + 3y < 12.

1. Boundary line: 2x + 3y = 12. Setting y = 0 gives x = 6, so one intercept is (6, 0). Setting x = 0 gives y = 4, so the other intercept is (0, 4).
2. Draw the line through (6, 0) and (0, 4) as a dashed line, because the sign is < (strict).
3. Test (0, 0): 2(0) + 3(0) = 0, and 0 < 12 is true.
4. The origin works, so shade the half-plane containing (0, 0) — the region below-left of the dashed line. Every point there, e.g. (2, 2) since 10 < 12, is a solution; points on the line itself are not.

Common Misconceptions

• Shading the wrong side: always use a test point — the origin side is not automatically correct.
• Using a solid line for strict inequalities: strict inequalities (<, >) need a dashed line; ≤ and ≥ need a solid one.
• Testing a point on the boundary line: a point on the line makes both sides equal and tells you nothing. Always test a point off the line.

Real-World Applications

• Budgeting: combinations of two products a shopper can afford under a fixed budget.
• Production planning: machine-hour and labour limits for two products.
• Nutrition: minimum protein and vitamin requirements from two food items.
• Mixture problems: alloy or chemical blends that must stay within specification limits.

An inequality in two variables is a half-plane bounded by ax + by = c — solid for ≤ or ≥, dashed for < or >. Draw the boundary, test a point off it, and shade the side that works.