MathsICSE Class 8 · Linear Inequations

Solving Linear Inequations in One Variable with Replacement Sets (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

Solve a linear inequation like an equation, but reverse the sign when you multiply or divide both sides by a negative number. Then pick the solutions that belong to the replacement set ($\mathbb{N}$, $\mathbb{W}$, $\mathbb{Z}$ or $\mathbb{R}$).

Understand Solving Linear Inequations in One Variable

The CISCE curriculum asks for properties of inequalities and the solution of inequalities in one variable.

Properties.

Adding or subtracting the same number on both sides keeps the sign: $x - 4 > 3 \Rightarrow x > 7$ .
Multiplying or dividing both sides by a positive number keeps the sign.
Multiplying or dividing both sides by a negative number reverses the sign: $- 2 x < 10 \Rightarrow x > - 5$ .

Replacement set. The set from which $x$ may be chosen ( $N$ , $W$ , $Z$ , $R$ ) decides what the solution set looks like. Solve first; then keep only the values in the replacement set. The solution set for $x > 7$ is ${8, 9, 10, \dots}$ in $N$ , but "all real numbers greater than 7" in $R$ .

Fractions. Multiply both sides by the (positive) LCM of the denominators to clear them: $\frac{2 x - 1}{3} - \frac{x + 2}{4} < 1$ becomes $4 (2 x - 1) - 3 (x + 2) < 12$ .

Word problems. Translate phrases: "at most" is $\leq$ , "at least" is $\geq$ , "less than" is $<$ , "more than" is $>$ . Define the variable, write the inequation, solve, then apply the realistic conditions (whole numbers, positive lengths, and so on).

Common errors. Forgetting to reverse the sign, and including an end value when the sign is strict.

Solving Linear Inequations in One Variable: formulas to remember

Adding/subtracting: $a < b \Rightarrow a \pm c < b \pm c$; Sign unchanged.
Multiplying by positive: $a < b, c > 0 \Rightarrow a c < b c$; Sign unchanged.
Multiplying by negative: $a < b, c < 0 \Rightarrow a c > b c$; Sign reverses.

How to solve Solving Linear Inequations in One Variable questions

Isolate x as for an equation, reverse the sign for a negative multiplier, then use the replacement set.

1Clear fractions with the LCM (positive).
2Collect $x$ terms on one side and numbers on the other.
3Divide by the coefficient of $x$ ; reverse the sign if it is negative.
4Write the solution as an inequality.
5List the members of the solution set from the replacement set.

Solving Linear Inequations in One Variable 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.

Taxi budget

Suppose a taxi charges Rs 40 plus Rs 12 per km, and Riya has Rs 300 to spend.

What is the greatest whole number of kilometres she can travel?

Try it first, then show the solutionSolution

$40 + 12 k \leq 300 \Rightarrow 12 k \leq 260$ .
$k \leq 21.67$ ; whole kilometres: $21$ .

Answer: 21 km.

Memory hook: At most means less than or equal to.

Tricks to remember Solving Linear Inequations in One Variable

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

Negative flips
Whenever you multiply or divide by a negative number, flip the sign.
Use it when: Coefficients like -3x.
Strict means skip the end
< and > exclude the end value; ≤ and ≥ include it.
Use it when: Listing solution sets.

Solving Linear Inequations in One Variable: solved examples

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

Example 1: Reading a strict inequality · 1 mark · MCQBoard level

The solution set of $x - 4 > 3$ , $x \in N$ , is:
(a) ${1, 2, \dots, 7}$ (b) ${7, 8, 9, \dots}$ (c) ${8, 9, 10, \dots}$ (d) ${4, 5, 6}$

Try it first, then show the solutionSolution

$x > 7$ ; natural numbers greater than $7$ .

Answer: (c)

Example 2: Replacement set N · 3 marks · Short answerBoard level

Solve $3 x - 5 \leq 10$ , $x \in N$ .

Try it first, then show the solutionSolution

$3 x \leq 15 \Rightarrow x \leq 5$ .
Solution set: ${1, 2, 3, 4, 5}$ .

Answer: ${1, 2, 3, 4, 5}$

Example 3: Negative coefficient · 3 marks · Short answerHarder board level

Solve $- 3 x + 7 > - 5$ , $x \in W$ .

Try it first, then show the solutionSolution

$- 3 x > - 12$ .
Divide by $- 3$ and reverse the sign: $x < 4$ .
Whole numbers less than $4$ : ${0, 1, 2, 3}$ .

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

Example 4: Fractions · 4 marks · Short answerHarder board level

Solve $\frac{2 x - 1}{3} - \frac{x + 2}{4} < 1$ , $x \in N$ .

Try it first, then show the solutionSolution

Multiply by $12$ : $4 (2 x - 1) - 3 (x + 2) < 12$ .
$8 x - 4 - 3 x - 6 < 12 \Rightarrow 5 x < 22 \Rightarrow x < 4.4$ .
Solution set: ${1, 2, 3, 4}$ . Check $x = 4$ : $\frac{7}{3} - \frac{6}{4} \approx 0.83 < 1$ ; $x = 5$ : $3 - 1.75 = 1.25$ , not less than $1$ .

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

Example 5: Triangle with a perimeter limit · 4 marks · Case-basedHOTS / competency

The sides of a triangle are $x$ , $x + 3$ and $2 x - 1$ (in cm, $x$ a whole number) and its perimeter is at most 28 cm. Find the possible values of $x$ .

Try it first, then show the solutionSolution

Perimeter $= 4 x + 2 \leq 28 \Rightarrow x \leq 6.5$ .
Triangle inequality: $x + (2 x - 1) > x + 3 \Rightarrow 2 x > 4 \Rightarrow x > 2$ . The other two conditions hold for all positive $x$ .
So $2 < x \leq 6.5$ with $x$ whole: $x \in {3, 4, 5, 6}$ .
Check $x = 3$ : sides $3, 6, 5$ (valid); $x = 2$ : $2, 5, 3$ fails since $2 + 3 = 5$ .

Answer: $x \in {3, 4, 5, 6}$

Where students lose marks in Solving Linear Inequations in One Variable

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

Mistake: Not reversing the sign on dividing by a negative number
Multiplying or dividing by a negative reverses the order.
Wrong working
$- 3 x > - 12 \Rightarrow x > 4$ .
Right working
$x < 4$ .
Do this: Circle the negative divisor and flip.
Usually costs: Answer and solution set.
Mistake: Ignoring the replacement set
The solution set depends on $N$ , $W$ , $Z$ or $R$ .
Wrong working
$x \geq 5$ written as the answer when $x \in N$ , $x \leq 8$ .
Right working
${5, 6, 7, 8}$ .
Do this: End with a set in the replacement set.
Usually costs: Final mark.

Exam tips

Mention the replacement set in the last line.
List elements in increasing order inside braces.
In a word problem, check the answer is practical (whole kilometres, positive lengths).

Practice questions on Solving Linear Inequations in One Variable

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
If $- 2 x < 10$ , then:
(a) $x < - 5$ (b) $x > - 5$ (c) $x < 5$ (d) $x > 5$
Hint
Dividing by a negative reverses the sign.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
Solve $x + 6 \geq 11$ , $x \in N$ .
Hint
Subtract 6.
Answer
$x \geq 5$ ; solution set ${5, 6, 7, \dots}$ .
Q3 · 3 marks · Short answerHarder board level
Solve $7 - 3 x \leq 1$ , $x \in R$ .
Hint
Subtract 7, then divide by -3.
Answer
$- 3 x \leq - 6 \Rightarrow x \geq 2$ ; ${x : x \geq 2, x \in R}$ .
Q4 · 3 marks · Short answerHarder board level
Solve $\frac{x}{2} + 3 > \frac{x}{3} + 4$ , $x \in N$ , $x \leq 10$ .
Hint
Multiply by 6.
Answer
$3 x + 18 > 2 x + 24 \Rightarrow x > 6$ . With $x \leq 10$ : ${7, 8, 9, 10}$ .
Q5 · 4 marks · Case-basedHOTS / competency
A taxi charges Rs 40 plus Rs 12 per km. Riya has Rs 300. Form an inequation and find the greatest whole number of kilometres she can travel. What is the change left?
Hint
Total cost must not exceed 300.
Answer
$40 + 12 k \leq 300 \Rightarrow k \leq 21 \frac{2}{3}$ , so $k = 21$ . Cost $= 40 + 252 = 292$ ; change $=$ Rs 8.

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.When must the inequality sign be reversed?Show answer
On multiplying or dividing by a negative number.
2.Solution of x > 7 in natural numbers?Show answer
{8, 9, 10, ...}.
3."At most" means?Show answer
Less than or equal to.
4.Does adding a number change the sign?Show answer
No.

Solving Linear Inequations in One Variable with Replacement Sets (ICSE Class 8)

Understand Solving Linear Inequations in One Variable

Solving Linear Inequations in One Variable: formulas to remember

How to solve Solving Linear Inequations in One Variable questions

Solving Linear Inequations in One Variable in real life

Tricks to remember Solving Linear Inequations in One Variable

Solving Linear Inequations in One Variable: solved examples

Where students lose marks in Solving Linear Inequations in One Variable

Exam tips

Practice questions on Solving Linear Inequations in One Variable

Test your recall

Solving Linear Inequations in One Variable: frequently asked questions

Is the solution always a single number?

What if no value in the replacement set works?