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.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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?
Work through the core explanation before applying it.
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 $\geq$.
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Work through the core explanation before applying it.
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.
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.
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 $x \geq 5$.
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.
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).
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.
We just saw how to test whether a particular value belongs to a solution set by reading its position on the number line. That raises a question: is there a way to write that whole solution set compactly, without drawing anything? This card answers it → interval notation records the two endpoints and uses the bracket shape to say whether each endpoint is included.
Interval notation is a shorthand for a solution set. You write the smallest value, a comma, then the largest value. The shape of each bracket carries the same information as the shape of the circle on your number line.
A square bracket means the endpoint is included, which matches a closed circle and a $\leq$ or $\geq$ sign. A round bracket means the endpoint is not included, which matches an open circle and a $<$ or $>$ sign.
The three representations say exactly the same thing, so you should be able to move between any two of them:
Interval notation writes a solution set as (smallest, largest). A square bracket includes the endpoint and matches a closed circle; a round bracket excludes it and matches an open circle. Infinity always takes a round bracket.
Pause, copy the bracket rule (square bracket includes the endpoint, round bracket excludes it), the infinity rule (always a round bracket), and one worked translation such as $-2 \leq x < 4$ becoming $[-2, 4)$ into your book.
Worked examples · 3 in a row, reveal as you go
Follow the reasoning through complete worked solutions.
Draw the number line representation of $x > 2$.
A graph shows a closed circle at $-3$ with an arrow pointing left. Write the inequality.
Solve $2x - 1 \geq 5$ and represent the solution on a number line.
Work through the core explanation before applying it.
Run the quick drill and copy the summary into your book.
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.
Answer the drill bank and rate your confidence.
Pick your answer, then rate your confidence, that tells the system what to drill next.
Write full responses, then check them against the model answers.
Q1. Solve $4x + 3 > 11$, draw the solution on a number line, describe the circle and arrow, and write the solution in interval notation. (4 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)
Q3. Explain the difference between open and closed circles on a number line graph. Give one example of each. (2 marks)
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 (4 marks): $4x > 8 \Rightarrow x > 2$ [1]. Open circle at 2, arrow right [1]. Check: $x = 3$: $4(3)+3=15 > 11$ ✓ [1]. Interval notation: $(2, \infty)$, a round bracket at 2 because the inequality is strict, and a round bracket at infinity because infinity is never included [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].
Take the module quiz if you are ready, then mark the lesson complete.
Match inequalities to their number line graphs and vice versa. Beat the boss to bank a tier.
⚔ Enter the arenaTick when you've finished the practice and review.
Work through this topic 1-on-1 with an experienced HSC tutor.
Book a free session →