Graphing Inequalities on Number Lines and Grids
A free Pre-Algebra lesson from the “Coordinate Plane, Functions, and Slope” unit, with a worked example and practice problems including step-by-step solutions.
Inequality graphs show many solutions at once. On a number line, use an open circle for < or > and a closed circle for <= or >=. For simple two-variable inequalities, a boundary line separates the coordinate plane into regions.
What you'll learn
- Graph one-variable inequalities on a number line
- Use open and closed boundary points
- Connect simple two-variable inequalities to shaded regions
Worked example
Problem. Describe the graph of x <= 4 on a number line.
- The symbol <= includes the boundary value.
- Use a closed circle at 4.
- Values less than 4 are to the left, so shade left.
Answer: Closed circle at 4, shade left
Practice problems
1. How should x < 3 be graphed?
Choices: Open circle at 3, shade left · Closed circle at 3, shade left · Open circle at 3, shade right · Closed circle at 3, shade right
Show solution
- Warm-up: First identify exactly what the question is asking: How should x < 3 be graphed?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- The symbol < does not include 3.
- Values less than 3 are to the left.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Open circle at 3, shade left
2. How should x >= -2 be graphed?
Choices: Open circle at -2, shade left · Closed circle at -2, shade right · Open circle at -2, shade right · Closed circle at -2, shade left
Show solution
- Warm-up: First identify exactly what the question is asking: How should x >= -2 be graphed?
- For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
- >= includes the boundary.
- Values greater than -2 are to the right.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Closed circle at -2, shade right
3. Which value satisfies x <= 5?
Choices: 6 · 5 · 8 · 10
Show solution
- Warm-up: First identify exactly what the question is asking: Which value satisfies x <= 5?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Less than or equal includes 5.
- 5 satisfies x <= 5.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 5
4. How should x > 0 be graphed?
Choices: Open circle at 0, shade right · Closed circle at 0, shade right · Open circle at 0, shade left · Closed circle at 0, shade left
Show solution
- Core Practice: First identify exactly what the question is asking: How should x > 0 be graphed?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- > does not include 0.
- Values greater than 0 are to the right.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Open circle at 0, shade right
5. Which inequality matches a closed circle at 7 shaded left?
Choices: x < 7 · x <= 7 · x > 7 · x >= 7
Show solution
- Core Practice: First identify exactly what the question is asking: Which inequality matches a closed circle at 7 shaded left?
- For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
- Closed circle means the boundary is included.
- Shading left means less than.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x <= 7
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