MathsICSE Class 8 · Linear Inequations

Combined (Double) Inequations and the Number Line (ICSE Class 8 Maths)

Q: Can I split a double inequation?

Yes, into two inequations joined by "and", then take the overlap.

Q: What if the middle has x on both sides?

Split it into two separate inequations.

Updated 17 October 2026 · Fact-checked

In a double inequation such as $-7 < 2x + 3 \le 9$, perform every step on all three parts. For two separate inequations, solve each and keep the common part of the solutions. On the number line use an open circle for $<, >$ and a filled circle for $\le, \ge$.

Understand Combined Inequations on the Number Line

A combined (double) inequation puts two limits on $x$ at once, for example $- 7 < 2 x + 3 \leq 9$ . It means $2 x + 3 > - 7$ and $2 x + 3 \leq 9$ .

Method 1: operate on all three parts. Subtract $3$ : $- 10 < 2 x \leq 6$ . Divide by $2$ : $- 5 < x \leq 3$ . If you multiply or divide by a negative number, both signs reverse, and the order of the end values swaps: $- 8 < - 2 x \leq 12 \Rightarrow 4 > x \geq - 6$ , which is best written $- 6 \leq x < 4$ .

Method 2: two separate inequations. Solve $2 x - 3 < 7$ and $3 x + 1 \geq - 8$ separately: $x < 5$ and $x \geq - 3$ . The common solution is $- 3 \leq x < 5$ . If there is no overlap, the solution set is empty.

Number line. Draw a line, mark the ends, use an open circle for a strict end (not included) and a filled circle for an included end, and shade between them.

Listing members. From $- 5 < x \leq 3$ with $x \in Z$ : ${- 4, - 3, - 2, - 1, 0, 1, 2, 3}$ (the left end $- 5$ is excluded, the right end $3$ is included).

Special replacement sets. If $x$ must also be, say, not natural, remove those members from the list.

Counting and summing. The solutions form an arithmetic list, so you can count and add them: from $- 6$ to $3$ there are $10$ integers with sum $- 15$ .

Combined Inequations on the Number Line: formulas to remember

Double inequation: $a < f (x) \leq b ⟺ a < f (x) and f (x) \leq b$; Solution set is the overlap.
Negative divisor: $a < c x \leq b, c < 0 \Rightarrow \frac{b}{c} \leq x < \frac{a}{c}$; Both signs reverse, ends swap.
Common solution: $x < m and x \geq n \Rightarrow n \leq x < m$; Empty if $n \geq m$ .

How to solve Combined Inequations on the Number Line questions

Work on all three parts together, or split into two and take the overlap.

1Note the replacement set.
2If $x$ is only in the middle, operate on all three parts together.
3If dividing by a negative number, reverse both signs and rewrite with the smaller end on the left.
4For two inequations, solve each and find the overlap.
5Show the number line with correct circles and list members from the replacement set.

Combined Inequations on the 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.

Allowed temperature range

Suppose a storage room must stay between 2 and 8 degrees Celsius, both included, and a sensor reads $T$ where the real temperature is $\frac{T}{2} + 1$ .

Find the range of sensor readings $T$ that are acceptable.

Try it first, then show the solutionSolution

$2 \leq \frac{T}{2} + 1 \leq 8$ .
Subtract $1$ : $1 \leq \frac{T}{2} \leq 7$ ; multiply by $2$ : $2 \leq T \leq 14$ .

Answer: $2 \leq T \leq 14$ .

Memory hook: Whatever you do to the middle, do to both ends.

Tricks to remember Combined Inequations on the Number Line

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

Sandwich
Do every step to all three parts of the sandwich.
Use it when: Double inequations.
Negative: flip both
Dividing by a negative reverses both signs; then rewrite with the small end first.
Use it when: Questions with -2x in the middle.

Combined Inequations on the Number Line: solved examples

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

Example 1: List the members · 1 mark · MCQBoard level

The solution set of $- 2 < x \leq 3$ , $x \in Z$ , is:
(a) ${- 1, 0, 1, 2, 3}$ (b) ${- 2, - 1, 0, 1, 2, 3}$ (c) ${- 1, 0, 1, 2}$ (d) ${0, 1, 2, 3}$

Try it first, then show the solutionSolution

$- 2$ is excluded, $3$ is included.

Answer: (a)

Example 2: x only in the middle · 3 marks · Short answerBoard level

Solve $2 \leq 3 x - 1 < 11$ , $x \in Z$ .

Try it first, then show the solutionSolution

Add $1$ : $3 \leq 3 x < 12$ .
Divide by $3$ : $1 \leq x < 4$ .
Solution set ${1, 2, 3}$ .

Answer: ${1, 2, 3}$

Example 3: Real replacement set · 3 marks · Short answerHarder board level

Solve $- 7 < 2 x + 3 \leq 9$ , $x \in R$ , and show the solution on a number line.

Try it first, then show the solutionSolution

Subtract $3$ : $- 10 < 2 x \leq 6$ .
Divide by $2$ : $- 5 < x \leq 3$ .
Number line: open circle at $- 5$ , filled circle at $3$ , shade between.

Answer: ${x : - 5 < x \leq 3, x \in R}$

Example 4: Two separate inequations · 4 marks · Short answerHarder board level

Solve $2 x - 3 < 7$ and $3 x + 1 \geq - 8$ for $x \in Z$ , and give the common solution set.

Try it first, then show the solutionSolution

First: $2 x < 10 \Rightarrow x < 5$ .
Second: $3 x \geq - 9 \Rightarrow x \geq - 3$ .
Common: $- 3 \leq x < 5$ .
${- 3, - 2, - 1, 0, 1, 2, 3, 4}$ .

Answer: ${- 3, - 2, - 1, 0, 1, 2, 3, 4}$

Example 5: Dividing by a negative number · 4 marks · Long answerHOTS / competency

Solve $- 1 < \frac{3 - 2 x}{5} \leq 3$ , $x \in R$ . Find the integers in the solution set and their sum.

Try it first, then show the solutionSolution

Multiply by $5$ : $- 5 < 3 - 2 x \leq 15$ .
Subtract $3$ : $- 8 < - 2 x \leq 12$ .
Divide by $- 2$ , reversing both signs: $4 > x \geq - 6$ , i.e. $- 6 \leq x < 4$ .
Integers: $- 6, \dots, 3$ (10 numbers); sum $= \frac{10 ( - 6 + 3 )}{2} = - 15$ .

Answer: $- 6 \leq x < 4$ ; sum of integers $= - 15$ .

Where students lose marks in Combined Inequations on the Number Line

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

Mistake: Changing only one sign when dividing by a negative number
Both inequality signs reverse.
Wrong working
$- 8 < - 2 x \leq 12 \Rightarrow 4 < x \leq - 6$ .
Right working
$4 > x \geq - 6$ , i.e. $- 6 \leq x < 4$ .
Do this: Rewrite with the smaller value on the left.
Usually costs: Answer mark.
Mistake: Including a strict end value
$<$ and $>$ exclude the end.
Wrong working
$- 2 < x \leq 3$ , $x \in Z$ : ${- 2, - 1, \dots, 3}$ .
Right working
${- 1, 0, 1, 2, 3}$ .
Do this: Mark each end as in or out before listing.
Usually costs: Final mark.

Exam tips

Always state the replacement set before listing members.
Use open and filled circles correctly on the number line.
Write combined results as a single double inequality.

Practice questions on Combined Inequations on the 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
The solution set of $1 \leq x < 4$ , $x \in N$ , is:
(a) ${2, 3}$ (b) ${1, 2, 3}$ (c) ${1, 2, 3, 4}$ (d) ${2, 3, 4}$
Hint
1 is included, 4 is not.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
Solve $- 4 < x - 1 \leq 2$ , $x \in Z$ .
Hint
Add 1 to all parts.
Answer
$- 3 < x \leq 3$ ; ${- 2, - 1, 0, 1, 2, 3}$ .
Q3 · 3 marks · Short answerHarder board level
Solve $5 < 2 x - 1 \leq 11$ , $x \in N$ .
Hint
Add 1 then halve.
Answer
$6 < 2 x \leq 12 \Rightarrow 3 < x \leq 6$ ; ${4, 5, 6}$ .
Q4 · 3 marks · Short answerHarder board level
Solve $3 x - 7 < 5$ and $2 x + 1 > - 5$ for $x \in W$ and give the common solution set.
Hint
Solve both, then overlap.
Answer
$x < 4$ and $x > - 3$ . For whole numbers: ${0, 1, 2, 3}$ .
Q5 · 4 marks · Long answerHOTS / competency
$2 x + 1$ lies strictly between $- 5$ and $9$ , and $x$ is an integer that is not a natural number. Find the possible values of $x$ .
Hint
First solve the double inequation for integers.
Answer
$- 5 < 2 x + 1 < 9 \Rightarrow - 6 < 2 x < 8 \Rightarrow - 3 < x < 4$ . Integers: ${- 2, - 1, 0, 1, 2, 3}$ . Removing the natural numbers $1, 2, 3$ leaves ${- 2, - 1, 0}$ .

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.Solve 2 < x + 3 <= 6 in integers.Show answer
-1 < x <= 3: {0, 1, 2, 3}.
2.What does an open circle mean?Show answer
The end value is not included.
3.If no overlap?Show answer
Solution set is empty.
4.Number of integers from -6 to 3?Show answer
10.

Combined Inequations on the Number Line: frequently asked questions

Can I split a double inequation?

Yes, into two inequations joined by "and", then take the overlap.

What if the middle has x on both sides?

Split it into two separate inequations.