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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.

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

Practice problems

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

Choices: Solid · Dashed

Show solution
  1. Warm-up: First identify exactly what the question is asking: For y > x + 2, is the boundary line solid or dashed?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A strict > or < makes the boundary dashed.
  4. Points on the line are not solutions.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Dashed

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

Choices: Solid · Dashed

Show solution
  1. Warm-up: First identify exactly what the question is asking: For y <= 3x - 1, is the boundary line solid or dashed?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. <= or >= makes the boundary solid.
  4. Points on the line ARE solutions.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Solid

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

Choices: Above · Below

Show solution
  1. Warm-up: First identify exactly what the question is asking: For y > x, shade above or below the line?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. y greater than the line means points where y-values are higher.
  4. Shade above.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Above

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

Choices: Above · Below

Show solution
  1. Core Practice: First identify exactly what the question is asking: For y < -2x + 5, shade above or below the line?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. y less than the line means lower y-values.
  4. Shade below.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Below

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

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: Is (0, 0) a solution of y < x + 3?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Test: is 0 < 0 + 3 = 3? Yes.
  4. So (0, 0) lies in the solution set.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

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