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hscscience Maths Std · Y11
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Module 1 · L15 of 20 ~45 min ⚡ +90 XP available

Graphing Inequalities on a Number Line

Represent inequality solutions visually using open circles (strict: $<$ or $>$) and closed circles (non-strict: $\leq$ or $\geq$), with an arrow showing the direction of the solution set. Learn to read graphs and write the matching inequality.

Today's hook — A speed limit sign shows "60 km/h". You must travel at most 60 km/h — so the legal speed is $v \leq 60$. How do you draw that on a number line? The circle at 60 tells you everything.
0/5QUESTS
01
Recall — your gut answer first
+5 XP warm-up

Draw $x \geq 3$ and $x < 3$ on a number line. Before reading on — how do the two graphs differ? What tells you whether the boundary value 3 is included?

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Open circle, closed circle, and direction
+5 XP to read

A number line graph shows the solution set of an inequality. The endpoint is drawn as a circle, and the arrow shows which direction the solutions extend.

Open circle (hollow) — the endpoint is NOT included. Use for $<$ and $>$. Closed circle (filled) — the endpoint IS included. Use for $\leq$ and $\geq$.

OPEN CIRCLE → strict (< or >) endpoint NOT in solution set CLOSED CIRCLE → non-strict (≤ or ≥) endpoint IS in solution set
$x > 2$: open circle at 2, arrow right. $x \leq 2$: closed circle at 2, arrow left.
Circle type = inclusion
Open circle = not included ($<$, $>$). Closed (filled) circle = included ($\leq$, $\geq$). The circle answers "is the endpoint in the set?"
Arrow = direction of solutions
Arrow pointing right means the solutions get larger ($>$ or $\geq$). Arrow pointing left means the solutions get smaller ($<$ or $\leq$).
Reading graphs
Note the endpoint value, decide open or closed, then read the direction. Write the inequality using $x$ and the appropriate sign.
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What you'll master
Know

Key facts

  • Open circle = strict inequality ($<$, $>$) — boundary not included.
  • Closed circle = non-strict ($\leq$, $\geq$) — boundary included.
  • Arrow direction shows which values are in the solution set.
Understand

Concepts

  • How to interpret a number line graph as an inequality statement.
  • Why a single graph can represent infinitely many solutions.
  • How to verify whether a value is in the solution set from a graph.
Can do

Skills

  • Draw the number line graph for any single-variable inequality.
  • Write the inequality shown by a given graph.
  • Identify specific values as in or outside the solution set.
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Key terms
Number lineA horizontal line with numbers increasing left to right, used to show the position of values and solution sets.
Open circleA hollow dot on a number line graph indicating the endpoint is NOT part of the solution set (used for $<$ and $>$).
Closed circleA filled dot on a number line graph indicating the endpoint IS included in the solution set (used for $\leq$ and $\geq$).
Solution regionThe set of all values satisfying an inequality, shown as a shaded/arrowed portion of the number line.
Strict inequalityAn inequality using $<$ or $>$ where the boundary value is excluded.
Boundary valueThe value at the endpoint of the solution region — the value where the inequality changes from true to false.
05
Drawing inequalities on a number line
core concept

To graph an inequality on a number line, follow three steps: (1) mark the boundary value, (2) draw the correct circle (open or closed), (3) draw an arrow in the correct direction.

$x > 2$ -2 0 2 4 open — 2 not included $x \leq -1$ -3 -1 1 3 closed — -1 included
Quick check: which graph shows $x < -1$?
What to write in your book
  • Three steps: mark boundary, choose circle (open or closed), draw arrow in direction of solutions.
  • $<$ and $>$ use open circles. $\leq$ and $\geq$ use closed circles.
  • Arrow right = larger values ($>$, $\geq$). Arrow left = smaller values ($<$, $\leq$).
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Reading a graph and writing the inequality
core concept

To write an inequality from a number line graph, identify: the boundary value, whether it uses an open or closed circle, and the direction of the arrow. Then write the inequality accordingly.

Example: if the graph shows a closed circle at 5 with an arrow pointing right, the inequality is $x \geq 5$.

Common error: Confusing which direction is "greater than". Remember — numbers increase to the right on a number line, so arrow right means the value is getting larger (greater than).
Which symbol always uses an open circle on a number line?
What to write in your book
  • To write the inequality from a graph: boundary value + circle type + arrow direction.
  • Open circle + arrow right → $x >$ boundary. Closed circle + arrow right → $x \geq$ boundary.
  • Open circle + arrow left → $x <$ boundary. Closed circle + arrow left → $x \leq$ boundary.
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Checking whether values are in the solution set
core concept

From a number line graph of $x \geq 3$, you can read off whether any specific value is a solution. Values shaded (in the arrow direction, including a closed boundary) are solutions. Values on the unshaded side are not.

For $x \geq 3$: $x = 5$ is a solution (shaded). $x = 3$ is a solution (closed circle — included). $x = 2$ is NOT a solution (unshaded).

Fill the blank: a $\leq$ symbol is drawn with a _______ circle on the number line.
What to write in your book
  • A value is a solution if it lies in the shaded/arrowed region, including a closed boundary.
  • A value is NOT a solution if it lies outside the shaded region, or at an open boundary.
  • Substitute the value into the original inequality to verify: the inequality should be true.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · DRAW THE GRAPH FOR x > 2

Draw the number line representation of $x > 2$.

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The boundary value is 2. The symbol is $>$ (strict), so use an open circle at 2.
Open circle = endpoint not included.
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PROBLEM 2 · WRITE THE INEQUALITY FROM A GRAPH

A graph shows a closed circle at $-3$ with an arrow pointing left. Write the inequality.

1
Closed circle → the endpoint $-3$ is included → use $\leq$ or $\geq$ (non-strict).
Filled dot = "or equal to".
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PROBLEM 3 · SOLVE AND GRAPH

Solve $2x - 1 \geq 5$ and represent the solution on a number line.

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Add 1 to both sides: $2x \geq 6$. Divide by 2: $x \geq 3$.
Standard two-step solving — no flip needed (dividing by positive 2).
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What to write in your book
  • Draw the graph after solving: boundary value → circle type → arrow direction.
  • Verify by checking at least two values: one inside and one outside the solution set.
  • The boundary value itself is only included if the circle is closed.
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Quick-fire practice
  1. Draw the number line graph for $x \leq -2$.
  2. Write the inequality for: open circle at 4, arrow pointing right.
  3. Solve $3x - 6 > 9$ and draw the solution on a number line.
  4. Is $x = -2$ in the solution set of $x \leq -2$? Explain.
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Revisit the speed limit

The speed limit of 60 km/h means $v \leq 60$: closed circle at 60, arrow pointing left (all speeds from 0 up to and including 60 are legal). $v = 60$ is legal (closed circle), $v = 61$ is not.

Earlier you described how the two graphs $x \geq 3$ and $x < 3$ differ. Now write a precise statement explaining what the circle type and arrow tell you, using the speed limit as an example.

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01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence — that tells the system what to drill next.

02
Short answer
ApplyBand 33 marks

Q1. Solve $4x + 3 > 11$ and draw the solution on a number line. Describe the circle and arrow. (3 marks)

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ApplyBand 32 marks

Q2. A number line graph has a closed circle at $-4$ and an arrow pointing left. Write the inequality and check one value. (2 marks)

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UnderstandBand 32 marks

Q3. Explain the difference between open and closed circles on a number line graph. Give one example of each. (2 marks)

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📖 Comprehensive answers (click to reveal)

Practice 1: Closed circle at $-2$, arrow left. Practice 2: $x > 4$. Practice 3: $x > 5$, open circle at 5, arrow right. Practice 4: Yes — closed circle means $-2$ is included.

Q1 (3 marks): $4x > 8 \Rightarrow x > 2$ [1]. Open circle at 2, arrow right [1]. Check: $x = 3$: $4(3)+3=15 > 11$ ✓ [1].

Q2 (2 marks): $x \leq -4$ [1]. Check: $x = -5$: $-5 \leq -4$ ✓ [1].

Q3 (2 marks): Open circle = endpoint not included, used for strict inequalities ($<$, $>$), e.g. $x > 3$ [1]. Closed circle = endpoint included, used for $\leq$ or $\geq$, e.g. $x \leq 3$ [1].

01
Boss battle · Number Line Challenge
earn bronze · silver · gold

Match inequalities to their number line graphs and vice versa. Beat the boss to bank a tier.

⚔ Enter the arena
02
Science Jump · platform challenge

Climb platforms by answering number line inequality questions. Pool: lesson 15.

Mark lesson as complete

Tick when you've finished the practice and review.

Module overview · Mathematics Standard