2.6Graphing Inequalities on a Number Line
A number line graph turns an algebraic solution into a picture — open circles, closed circles, and shaded rays tell the complete story of which values satisfy an inequality.
Why This Matters
Number line graphs are the visual language of inequalities — they appear on every standardized test and are essential for communicating solution sets clearly.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do open circles, closed circles, and shaded regions on a number line communicate the complete solution set of an inequality?
Lesson Overview
A number line graph is the standard visual representation of an inequality's solution set. Every graph uses three elements: a circle at each boundary point, a shaded region showing which values are included, and arrows when the solution extends to infinity. An open circle (hollow dot) means the boundary point is not included — used with strict inequalities (< or >). A closed circle (filled dot) means the boundary point is included — used with ≤ or ≥. For a simple inequality like x > 2, shade a single ray to the right. For a compound AND inequality like −3 < x ≤ 4, shade the segment between the two endpoints. For a compound OR inequality like x < −2 OR x ≥ 5, shade two separate rays pointing outward. Each graph connects directly to interval notation: open circles correspond to parentheses ( ), closed circles correspond to square brackets [ ].
Visual Reference Card
| Symbol | Circle Type | Shade Direction | Interval Notation |
|---|---|---|---|
| x > a | Open (hollow) | Right → | (a, ∞) |
| x ≥ a | Closed (filled) | Right → | [a, ∞) |
| x < a | Open (hollow) | ← Left | (−∞, a) |
| x ≤ a | Closed (filled) | ← Left | (−∞, a] |
| a < x < b | Open at both | Segment between | (a, b) |
| a ≤ x ≤ b | Closed at both | Segment between | [a, b] |
| x < a OR x > b | Open at both | Two outward rays | (−∞, a) ∪ (b, ∞) |
Worked Examples
Graph x > 2 on a number line and write in interval notation.
Identify the boundary point: x = 2.
Symbol is > (strict) → use an open circle at 2.
x > 2 means values greater than 2 → shade to the right.
The ray extends to positive infinity.
Interval notation: (2, ∞) — parenthesis because 2 is not included.
Check: Test x = 3: 3 > 2 ✓ — in solution (shaded region).
Check: Test x = 1: 1 > 2 ✗ — not in solution (correct).
Graph: x > 2
Open circle at 2, shade right → (2, ∞)
Graph x ≤ −1 on a number line and write in interval notation.
Identify the boundary point: x = −1.
Symbol is ≤ (non-strict) → use a closed circle at −1.
x ≤ −1 means values less than or equal to −1 → shade to the left.
The ray extends to negative infinity.
Interval notation: (−∞, −1] — square bracket because −1 IS included.
Check: Test x = −3: −3 ≤ −1 ✓ — in solution.
Check: Test x = 0: 0 ≤ −1 ✗ — not in solution (correct).
Graph: x ≤ −1
Closed circle at −1, shade left → (−∞, −1]
Graph −3 < x ≤ 4 on a number line and write in interval notation.
This is a compound AND inequality (double inequality).
Left boundary: x = −3 with strict < → open circle at −3.
Right boundary: x = 4 with ≤ → closed circle at 4.
Shade the segment between −3 and 4.
Interval notation: (−3, 4] — parenthesis at −3 (open), bracket at 4 (closed).
Check: Test x = 0: −3 < 0 ≤ 4 ✓.
Check: Test x = 4: −3 < 4 ≤ 4 ✓ (4 is included).
Check: Test x = −3: −3 < −3 ✗ (−3 is not included — correct).
Graph: −3 < x ≤ 4
Open at −3, closed at 4 · segment → (−3, 4]
Graph x < −2 OR x ≥ 5 on a number line and write in interval notation.
This is a compound OR inequality — two separate rays.
Left part: x < −2 → open circle at −2, shade left.
Right part: x ≥ 5 → closed circle at 5, shade right.
The two regions do not overlap.
Interval notation: (−∞, −2) ∪ [5, ∞)
The ∪ symbol means union — the combination of both regions.
Check: Test x = −4: −4 < −2 ✓ — in solution.
Check: Test x = 0: 0 < −2 ✗ and 0 ≥ 5 ✗ — not in solution (correct).
Graph: x < −2 OR x ≥ 5
Open at −2, closed at 5 · two rays → (−∞, −2) ∪ [5, ∞)
Reading a graph: A number line shows a closed circle at −4, an open circle at 2, and the segment between them is shaded. Write the inequality and interval notation.
Closed circle at −4 → the endpoint −4 IS included → use ≤.
Open circle at 2 → the endpoint 2 is NOT included → use <.
Shaded segment between → compound AND inequality.
Inequality: −4 ≤ x < 2
Interval notation: [−4, 2) — bracket at −4 (closed), parenthesis at 2 (open).
Check: Test x = 0: −4 ≤ 0 < 2 ✓.
Check: Test x = −4: −4 ≤ −4 < 2 ✓ (−4 is included).
Check: Test x = 2: −4 ≤ 2 < 2 ✗ (2 is not included — correct).
Graph: −4 ≤ x < 2
Closed at −4, open at 2 · segment → [−4, 2)
Guided Practice
Guided Practice Video: Graphing Inequalities on a Number Line
Watch the guided practice walkthrough for graphing inequalities on a number line, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Graph x ≥ −3 on a number line and write in interval notation.
Hint: Symbol is ≥ → closed circle at −3. Values ≥ −3 → shade right. Interval: [−3, ∞).
Graph your answer (x ≥ −3):
Graph x < 4 on a number line and write in interval notation.
Hint: Symbol is < (strict) → open circle at 4. Values < 4 → shade left. Interval: (−∞, 4).
Graph your answer (x < 4):
Graph 1 ≤ x < 6 on a number line and write in interval notation.
Hint: Closed circle at 1 (≤), open circle at 6 (<), shade the segment between. Interval: [1, 6).
Graph your answer (1 ≤ x < 6):
Graph x ≤ −1 OR x > 3 on a number line and write in interval notation.
Hint: Closed circle at −1 shade left; open circle at 3 shade right. Two rays. Interval: (−∞, −1] ∪ (3, ∞).
Graph your answer (x ≤ −1 OR x > 3):
Reading a graph: open circle at 0, shade right. Write the inequality and interval notation.
Hint: Open circle → strict inequality. Shade right → greater than. Inequality: x > 0. Interval: (0, ∞).
Key Vocabulary
Open Circle
A hollow dot on a number line graph indicating the boundary point is NOT included in the solution. Used with strict inequalities (< or >).
Example: x > 3 → open circle at 3
Closed Circle
A filled dot on a number line graph indicating the boundary point IS included in the solution. Used with non-strict inequalities (≤ or ≥).
Example: x ≤ 3 → closed circle at 3
Ray
A shaded half-line extending from a boundary point to infinity. Used for simple inequalities.
x > 2 → ray to the right
Interval Notation
A compact way to write a solution set using parentheses ( ) for open endpoints and brackets [ ] for closed endpoints.
Example: (−3, 4] means −3 < x ≤ 4
Bounded vs. Unbounded
A bounded solution set has two finite endpoints (a segment). An unbounded solution set extends to ±∞ (a ray or two rays).
Example: [−2, 5] is bounded; (3, ∞) is unbounded
Interval Notation — Visual Reference
Interactive Practice — 5 Questions
Which circle type is used when the inequality symbol is ≤ or ≥?
The graph of x > 2 uses:
What is the interval notation for the graph: closed circle at −1, shade left?
The graph of −3 < x ≤ 4 shows:
Which interval notation matches: open circle at −2 shade left, closed circle at 5 shade right?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Graph x > −4 and write in interval notation.
Graph x ≤ 2 and write in interval notation.
Graph −2 ≤ x < 5 and write in interval notation.
Graph x < 0 OR x ≥ 4 and write in interval notation.
Reading a graph: closed circle at −3, open circle at 1, segment shaded between. Write the inequality and interval notation.
ChallengeProblem 1 graph:
Problem 2 graph:
Problem 3 graph:
Problem 4 graph:
Problem 5 graph:
Common Mistakes
Using an open circle for ≤ or ≥ — forgetting that 'or equal to' means the endpoint is included.
≤ and ≥ always use a closed (filled) circle. Only strict < and > use an open (hollow) circle.
Shading in the wrong direction — shading left for x > a or right for x < a.
x > a and x ≥ a shade to the RIGHT (values larger than a). x < a and x ≤ a shade to the LEFT (values smaller than a).
Writing interval notation with a bracket at infinity — writing [3, ∞] instead of [3, ∞).
Infinity is never a real number and is never included. Always use a parenthesis next to ∞ or −∞: [3, ∞).
Reading a compound OR graph as a single segment — writing (−2, 5) instead of (−∞, −2) ∪ (5, ∞).
Two separate shaded regions always indicate an OR (union) solution. Use the ∪ symbol between the two intervals.
Math Tips
Circle memory trick: "Open = Out" — an open circle means the point is left out. "Closed = Captured" — a closed circle means the point is captured in the solution.
Shade direction: the inequality symbol points toward the shaded region. x > 2 points right → shade right. x < 5 points left → shade left.
Infinity rule: always use a parenthesis next to ∞ or −∞ in interval notation. Infinity is a concept, not a number — it is never "reached" or included.
SAT/ACT tip: when reading a number line graph, identify the circle type first (open or closed), then the shading direction. This gives you the inequality symbol and the interval notation directly.