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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

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 $\geq$.

$x > 2$: 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.

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.

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.

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.

Quick check: which graph shows $x < -1$?

Number line graph: open circle (○) for strict inequality (< or >, boundary not included); closed circle (●) for ≤ or ≥ (boundary included). Arrow points in the direction of the solution set from the boundary.

Pause, copy the number-line graph rules: open circle (○) for strict < or > (boundary not included); closed circle (●) for ≤ or ≥ (boundary included); arrow points toward all values that satisfy the inequality into your book.

Reading a graph and writing the inequality

core concept

We just saw drawing inequalities on a number line: open circle for strict inequalities (> or <), closed circle for non-strict (≥ or ≤), with an arrow pointing in the direction of the solution set. That raises a question: given a completed number-line graph, how do you reverse the process and write the inequality it represents? This card answers it → read the boundary value, identify open (strict) or closed (non-strict) circle, and check which way the arrow points to determine the inequality symbol.

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?

To write an inequality from a number-line graph: read the boundary value, identify open or closed circle → choose strict or non-strict symbol, check arrow direction → choose < or >. Write as: variable symbol boundary (e.g. x ≥ 3).

Pause, copy the three-step process for reading an inequality from a number-line graph: (1) read the boundary value; (2) open or closed circle → strict or non-strict symbol; (3) arrow direction → choose < or ≤ (arrow left) or > or ≥ (arrow right) into your book.

Checking whether values are in the solution set

core concept

We just saw how to read an inequality back from a number-line graph by identifying the boundary value, circle type, and arrow direction. That raises a question: knowing the inequality and its graph, how do you check whether a specific value is actually a solution? This card answers it → substitute the value into the original inequality; if it satisfies the inequality, the value is in the solution set, and for boundary points, check whether the circle is open or closed to determine inclusion.

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.

Testing a point against a graph: if the point falls in the shaded region, it is a solution; if in the unshaded region, it is not. For boundary points: closed circle → included; open circle → not included.

Pause, copy the region-membership rule (a value in the shaded region satisfies the inequality; a value in the unshaded region does not) and the boundary-point inclusion check (closed circle → boundary value is a solution; open circle → boundary value is not) into your book.

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

PROBLEM 1 · DRAW THE GRAPH FOR x > 2

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

The boundary value is 2. The symbol is $>$ (strict), so use an open circle at 2.

Open circle = endpoint not included.

PROBLEM 2 · WRITE THE INEQUALITY FROM A GRAPH

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

Closed circle → the endpoint $-3$ is included → use $\leq$ or $\geq$ (non-strict).

Filled dot = "or equal to".

PROBLEM 3 · SOLVE AND GRAPH

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

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).

Quick-fire practice

Draw the number line graph for $x \leq -2$.
Write the inequality for: open circle at 4, arrow pointing right.
Solve $3x - 6 > 9$ and draw the solution on a number line.
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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Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidencethat tells the system what to drill next.

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].

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

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.

← Lesson 14 · Introducing Linear Inequalities Lesson 16 · Compound Inequalities →

Module overview · Mathematics Standard