MathsICSE Class 10 · Linear Inequations

Representing a Solution Set on a Number Line: Rules and Solved Questions (ICSE Class 10)

Q: When do I use dots and when do I shade on the number line?

Use dots when x is a natural number, whole number or integer. Shade a continuous stretch when x is a real number.

Q: What is the difference between a filled and a hollow circle?

A filled circle means the end value is included (≤ or ≥). A hollow circle means it is not included ( ).

Updated 15 October 2026 · Fact-checked

On a number line, show a solution set in N, W or Z as solid dots on each member. For x in R, darken the interval: a filled circle marks an included end (≤, ≥), a hollow circle an excluded end (<, >), and shading runs to the arrowhead if the set has no end.

For integers mark separate solid dots; for real numbers draw one thick line, with a hollow circle for < or > and a solid circle for ≤ or ≥.

Illustration drawn for QuestionBuddy.

Understand Solution Set on a Number Line

Almost every ICSE inequation question ends with 'represent the solution set on the real number line'. It carries a mark of its own, and the examiners' comments for both 2024 Q7(ii) and 2025 Q4(i) list number-line errors: no arrows on the line, extra integers marked beyond the ends, the end value $8 \frac{1}{3}$ not clearly marked, and no shading.

Step 1: draw a proper number line. A straight line with arrowheads at both ends (it continues for ever), equally spaced marks, and the integers written below them. Include a little beyond your solution on each side, so the reader can see where it starts and stops.

**Step 2a: replacement set $N$ , $W$ or $Z$ . The solution set is a list of separate numbers. Put a solid dot** on each member and nothing between them. For example ${- 2, - 1, 0, 1, 2, 3}$ is six dots. If the set is infinite, such as ${3, 4, 5, \dots}$ , put dots on the first few members and then three dots or a dotted continuation towards the arrow, to show it goes on.

**Step 2b: replacement set $R$ . Every real number in the interval is a solution, so the solution is a continuous** stretch of the line:

Draw a thick (darkened) line over the interval.
At an end that is included ( $\leq$ or $\geq$ ), draw a filled (solid) circle.
At an end that is excluded ( $<$ or $>$ ), draw a hollow (open) circle.
If the interval has no end on one side (e.g. $x \geq - 2$ ), continue the thick line to the arrowhead.

An end-point that is not an integer, such as $\frac{25}{3}$ , must be marked between 8 and 9 and labelled ( $8 \frac{1}{3}$ ).

Reading a number line back. The reverse question gives a diagram and asks for the set or the inequation. Dots mean an integer (or natural/whole) replacement set; a shaded line means $R$ . A filled circle gives $\leq$ or $\geq$ ; a hollow circle gives $<$ or $>$ . With integer dots, the same set can be written several ways; for example dots at $- 3, \dots, 1$ is $- 3 \leq x \leq 1$ or $- 4 < x < 2$ , $x \in Z$ .

Intersections. To find $A \cap B$ , draw both solution sets above the same number line (or shade them in different styles); the overlap is the intersection.

Solution Set on a Number Line: formulas to remember

Integer, natural or whole replacement set: $x \in N, W, Z : solid dots on each member only$; No shading between dots.
Real replacement set: $x \in R : thick line over the interval$; Every point of the interval is a solution.
Included end-point: $\leq, \geq \to ∙ (filled circle)$; The end value belongs to the set.
Excluded end-point: $<, > \to \circ (hollow circle)$; The end value does not belong.
Unbounded interval: $x \geq a : shade from a to the right arrowhead$; Similarly $x < a$ shades to the left arrowhead.

How to solve Solution Set on a Number Line questions

Draw the line first, then decide dots or shading from the replacement set.

1Solve the inequation and write the solution set.
2Draw a straight line with arrowheads at both ends, equal spacing, and integers written below.
3If $x \in N, W$ or $Z$ : put solid dots on exactly the members of the solution set.
4If $x \in R$ : darken the interval; filled circle for an included end, hollow circle for an excluded end.
5Mark and label any fractional end-point (for example $8 \frac{1}{3}$ ) in its correct place.
6For an infinite set, continue the dots or shading to the arrowhead.
7Check: an end value is in the set only if its circle is filled.

Solution Set on a Number Line in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Whole-number solutions shown as dots

Suppose a swimming class accepts children who are older than 6 years and at most 10 years old, with ages counted in whole years.

Write the inequation, list the possible ages, and describe the number line.

Try it first, then show the solutionSolution

$6 < a \leq 10$ , $a \in N$ .
Possible ages: ${7, 8, 9, 10}$ .
Number line: solid dots at 7, 8, 9 and 10 only; no dot at 6.

Answer: ${7, 8, 9, 10}$ , shown by four separate dots.

Memory hook: Whole-number answers are stepping stones: dots, not a road.

A real interval with open ends

Imagine a room's temperature $T$ (in °C) must stay above 18 and below 24 for a science experiment.

Write the inequation and describe its number line.

Try it first, then show the solutionSolution

$18 < T < 24$ , $T \in R$ .
Hollow circles at 18 and 24; thick line between them.

Answer: ${T : 18 < T < 24, T \in R}$ : hollow circles at both ends with the line between them darkened.

Memory hook: Strictly between: hollow at both ends.

Tricks to remember Solution Set on a Number Line

Short memory aids. Say each one aloud once, then use it on a question the same day.

Filled = 'or equal'
The little line under ≤ fills the circle. No line under < means an empty circle.
Use it when: Choosing filled or hollow circles.
Dots for counting numbers, roads for reals
N, W, Z: separate stepping-stones (dots). R: a continuous road (thick line).
Use it when: Deciding how to draw.
Arrows on both ends, always
A number line never stops, so it always has two arrowheads.
Use it when: Avoiding the error the examiners named in 2024 and 2025.

Solution Set on a Number Line: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Reading a number line · 1 mark · MCQBoard level

A real number line shows a hollow circle at 2, a filled circle at 5, and the part between them darkened. The set shown is:
(a) ${x : 2 \leq x < 5, x \in R}$
(b) ${x : 2 < x \leq 5, x \in R}$
(c) ${3, 4, 5}$
(d) ${x : 2 < x < 5, x \in R}$

Try it first, then show the solutionSolution

Shaded line: real numbers. Hollow at 2: $2$ excluded. Filled at 5: $5$ included.

Answer: (b)

Example 2: Whole numbers as dots · 3 marks · Short answerBoard level

Solve $3 x - 2 \leq 10$ , $x \in W$ , and represent the solution set on a number line.

Try it first, then show the solutionSolution

$3 x \leq 12 \Rightarrow x \leq 4$ .
Solution set ${0, 1, 2, 3, 4}$ .
Number line with arrowheads at both ends, marks from $- 2$ to $6$ ; solid dots at 0, 1, 2, 3, 4 only.

Answer: ${0, 1, 2, 3, 4}$ , five solid dots.

Example 3: A real interval, one end open · 3 marks · Short answerHarder board level

Solve $- 1 < \frac{2 x + 3}{5} \leq 3$ , $x \in R$ , and represent the solution on the number line.

Try it first, then show the solutionSolution

Multiply by 5: $- 5 < 2 x + 3 \leq 15$ ; subtract 3: $- 8 < 2 x \leq 12$ ; divide by 2: $- 4 < x \leq 6$ .
Solution set ${x : - 4 < x \leq 6, x \in R}$ .
Number line: hollow circle at $- 4$ , filled circle at $6$ , thick line between them.

Answer: ${x : - 4 < x \leq 6, x \in R}$ ; hollow at −4, filled at 6, shaded between.

Example 4: From a diagram to an inequation · 3 marks · Short answerHarder board level

A number line has solid dots at $- 3, - 2, - 1, 0$ and $1$ only. (a) Write the solution set. (b) Write it as a double inequation with $x \in Z$ , using $\leq$ at both ends. (c) Write it again using $<$ at both ends.

Try it first, then show the solutionSolution

(a) ${- 3, - 2, - 1, 0, 1}$ .
(b) $- 3 \leq x \leq 1$ , $x \in Z$ .
(c) $- 4 < x < 2$ , $x \in Z$ .

Answer: (a) ${- 3, - 2, - 1, 0, 1}$ (b) $- 3 \leq x \leq 1$ (c) $- 4 < x < 2$ , with $x \in Z$ .

Example 5: Intersection drawn on one line · 4 marks · Short answerHOTS / competency

Solve $2 (x - 1) \geq x - 4$ and $4 x - 7 < 2 x + 3$ for $x \in R$ . Represent the values satisfying both on a number line, and state how many integers they include.

Try it first, then show the solutionSolution

First: $2 x - 2 \geq x - 4 \Rightarrow x \geq - 2$ .
Second: $2 x < 10 \Rightarrow x < 5$ .
Both: $- 2 \leq x < 5$ . Filled circle at $- 2$ , hollow circle at $5$ , thick line between.
Integers: $- 2, - 1, 0, 1, 2, 3, 4$ , that is 7 integers.

Answer: ${x : - 2 \leq x < 5, x \in R}$ ; 7 integers.

Example 6: Case study: travel time within speed limits · 4 marks · Case-basedHOTS / competency

On a highway the speed must be at least 40 km/h and at most 80 km/h. A car travels 120 km at a steady speed $v$ , taking $t$ hours.
(a) Write a double inequation in $v$ , and use $v = \frac{120}{t}$ to write one in $t$ .
(b) Solve for $t$ .
(c) Describe the number line for $t$ .

Try it first, then show the solutionSolution

(a) $40 \leq v \leq 80$ , so $40 \leq \frac{120}{t} \leq 80$ with $t > 0$ .
(b) $40 \leq \frac{120}{t} \Rightarrow t \leq 3$ ; $\frac{120}{t} \leq 80 \Rightarrow t \geq 1.5$ . So $1.5 \leq t \leq 3$ .
(c) Filled circle at 1.5 (between 1 and 2, labelled), filled circle at 3, thick line between.

Answer: (a) $40 \leq \frac{120}{t} \leq 80$ (b) $1.5 \leq t \leq 3$ (c) Filled circles at 1.5 and 3 with the interval darkened.

Where students lose marks in Solution Set on a Number Line

The slips examiners see most often, and what to write instead.

Mistake: No arrowheads on the number line
A number line extends without end in both directions.
Wrong working
A line segment from −3 to 5 with no arrows.
Right working
A line with arrowheads at both ends.
Do this: Draw the arrows first.
Usually costs: The number-line mark; reported in 2024 and 2025.
Mistake: Shading between integer dots
For $Z$ only the integers are solutions.
Wrong working
${- 2, \dots, 3}$ shown as a darkened segment from −2 to 3.
Right working
Separate solid dots at −2, −1, 0, 1, 2, 3.
Do this: Check the replacement set before drawing.
Usually costs: The representation mark.
Mistake: Extra dots beyond the ends
Only members of the solution set may be marked.
Wrong working
Dots from −3 to 4 for ${- 2, \dots, 3}$ .
Right working
Dots only from −2 to 3.
Do this: Count your dots against your solution set.
Usually costs: Reported for 2024 Q7(ii).
Mistake: Fractional end-point not marked
The reader must see exactly where the interval stops.
Wrong working
Shading stops vaguely between 8 and 9.
Right working
Hollow circle at $8 \frac{1}{3}$ , labelled.
Do this: Label every end-point value.
Usually costs: Reported for 2025 Q4(i).

Exam tips

Use a ruler, equal spacing and integers written below the line.
In 2025 Q4(i) the expected number line had a closed circle at −5, an open circle at $8 \frac{1}{3}$ and shading between.
In 2024 Q7(ii) ( $x \in I$ ) the expected number line had dots at −2, −1, 0, 1, 2, 3 only.
Mark one or two integers beyond each end of your solution so the reader can see where it stops.

Practice questions on Solution Set on a Number Line

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
On a real number line, $x \geq - 1$ is shown by:
(a) a hollow circle at −1 and shading to the right
(b) a filled circle at −1 and shading to the right
(c) a filled circle at −1 and shading to the left
(d) dots at −1, 0, 1, 2, …
Hint
≥ includes the end; real numbers need shading.
Answer
(b)
Q2 · 3 marks · Short answerBoard level
Solve $5 - x > 2$ , $x \in N$ , and describe its number line.
Hint
$x < 3$ ; natural numbers start at 1.
Answer
${1, 2}$ : solid dots at 1 and 2 only.
Q3 · 3 marks · Short answerHarder board level
Solve $- 2 \leq 1 - x < 3$ , $x \in Z$ , and represent the solution set.
Hint
Subtract 1, then multiply by $- 1$ and reverse both signs.
Answer
$- 3 \leq - x < 2 \Rightarrow - 2 < x \leq 3$ : ${- 1, 0, 1, 2, 3}$ , five solid dots.
Q4 · 1 mark · MCQHarder board level
A real number line shows a hollow circle at −3 with the line darkened to the left arrowhead. The inequation is:
(a) $x \leq - 3$
(b) $x > - 3$
(c) $x < - 3$
(d) $x \geq - 3$
Hint
Hollow means excluded; left means smaller.
Answer
(c) $x < - 3$
Q5 · 3 marks · Short answerHarder board level
Solve $\frac{x}{3} - 1 \leq 1 < \frac{x}{2}$ , $x \in R$ , and represent the solution.
Hint
Split into $\frac{x}{3} - 1 \leq 1$ and $1 < \frac{x}{2}$ .
Answer
$x \leq 6$ and $x > 2$ : ${x : 2 < x \leq 6, x \in R}$ ; hollow at 2, filled at 6, darkened between.
Q6 · 4 marks · Short answerHOTS / competency
A real number line shows a hollow circle at −2, a filled circle at 3 and the part between darkened. Find $a$ and $b$ so that this is the solution of $a < 2 x - 1 \leq b$ .
Hint
The diagram shows $- 2 < x \leq 3$ ; multiply by 2 and subtract 1.
Answer
$- 4 < 2 x \leq 6 \Rightarrow - 5 < 2 x - 1 \leq 5$ , so $a = - 5$ , $b = 5$ .
Q7 · 4 marks · Short answerHarder board level
$P = {x : - 2 < x \leq 4, x \in Z}$ and $Q = {x : 1 \leq x < 6, x \in Z}$ . List $P$ , $Q$ and $P \cap Q$ , and represent $P \cap Q$ on a number line.
Hint
List each set first.
Answer
$P = {- 1, 0, 1, 2, 3, 4}$ , $Q = {1, 2, 3, 4, 5}$ , $P \cap Q = {1, 2, 3, 4}$ : solid dots at 1, 2, 3, 4.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Filled or hollow circle for $x < 4$ ?Show answer
Hollow.
2.How is ${1, 2, 3}$ shown?Show answer
Three separate solid dots.
3.Write the set shown by a filled circle at 0 and shading to the right, $x \in R$ .Show answer
${x : x \geq 0, x \in R}$ .
4.Solve $2 x < 6$ , $x \in W$ , and count the dots.Show answer
${0, 1, 2}$ : 3 dots.
5.How many integers lie in $- 1.5 < x \leq 2$ ?Show answer
4 (−1, 0, 1, 2).

Solution Set on a Number Line: frequently asked questions

When do I use dots and when do I shade on the number line?

Use dots when x is a natural number, whole number or integer. Shade a continuous stretch when x is a real number.

What is the difference between a filled and a hollow circle?

A filled circle means the end value is included (≤ or ≥). A hollow circle means it is not included (< or >).

Do I need to draw arrows on the number line in ICSE?

Yes. The examiners' reports for 2024 and 2025 both mention number lines drawn without arrows as an error.