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Graphing Linear Inequalities in Two Variables

A free Algebra I lesson from the “Functions, Linear Relationships, and Rate of Change” unit, with a worked example and practice problems including step-by-step solutions.

A linear inequality in two variables (like y > 2x + 1) has infinitely many solution points that form a half-plane. The boundary line is solid for <= or >= and dashed for < or >. Shade above the line for y > or y >= and below for y < or y <=. When unsure, test a point (the origin is easiest if the line does not pass through it).

What you'll learn

Why it matters: Budget constraints (x + y <= 100 dollars), shipping weight limits, and any 'at most' or 'at least' production planning question creates a shaded half-plane on a graph.

Worked example

Problem. Graph y <= 2x + 1.

  1. Boundary line is y = 2x + 1, drawn solid because the inequality uses <=.
  2. Shade below the line because y is less than or equal to the line.
  3. Test (0, 0): is 0 <= 2(0) + 1 = 1? Yes, so (0, 0) lies in the shaded region.
  4. Connect the result back to Graphing Linear Inequalities in Two Variables so the method and meaning are both clear.

Answer: Solid line through (0, 1) with slope 2; shade below.

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. For y > x + 2, is the boundary line solid or dashed?

Choices: Solid · Dashed

Show solution
  1. A strict > or < makes the boundary dashed.
  2. Points on the line are not solutions.

Answer: Dashed

2. For y <= 3x - 1, is the boundary line solid or dashed?

Choices: Solid · Dashed

Show solution
  1. <= or >= makes the boundary solid.
  2. Points on the line ARE solutions.

Answer: Solid

3. For y > x, shade above or below the line?

Choices: Above · Below

Show solution
  1. y greater than the line means points where y-values are higher.
  2. Shade above.

Answer: Above

4. For y < -2x + 5, shade above or below the line?

Choices: Above · Below

Show solution
  1. y less than the line means lower y-values.
  2. Shade below.

Answer: Below

5. Is (0, 0) a solution of y < x + 3?

Choices: Yes · No

Show solution
  1. Test: is 0 < 0 + 3 = 3? Yes.
  2. So (0, 0) lies in the solution set.

Answer: Yes

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