Linear Inequations ICSE Class 10: Rules, Solution Sets and Number Lines

What Linear Inequations is about

Weightage: CISCE publishes no chapter or unit weightage for ICSE Mathematics. Linear Inequations is in the 2. Algebra (i) Linear Inequations part of the 2027 syllabus and can be asked in both Section A and Section B of the 80-mark paper.Source: CISCE ICSE Mathematics syllabus, examination year 2027

Key concepts

Replacement set and solution set

The replacement set is the set from which $x$ may be chosen ($N$, $W$, $Z$ or $R$). The solution set is the set of all members of the replacement set that make the inequation true.

The same inequation can have different solution sets for different replacement sets. For example, $x<3$ gives $\{1,2\}$ in $N$, $\{0,1,2\}$ in $W$, $\{\dots ,-1,0,1,2\}$ in $Z$, and $\{x:x\in R,x<3\}$ in $R$.

Write the solution set in roster form (listing elements in braces) for $N$, $W$ and $Z$, and in set-builder form for $R$, because real numbers cannot be listed.

Worked example

Solve $3x-4<11$ for (a) $x\in N$, (b) $x\in W$.

Try it first, then show the solutionSolution

1. $3x<15$, so $x<5$.
2. (a) Natural numbers less than 5: $\{1,2,3,4\}$.
3. (b) Whole numbers less than 5: $\{0,1,2,3,4\}$.

Answer: (a) {1, 2, 3, 4} (b) {0, 1, 2, 3, 4}

Rules for solving: when the sign reverses

You may do these without changing the inequality sign:

• add or subtract the same number (or expression) on both sides;
• multiply or divide both sides by the same positive number.

If you multiply or divide both sides by a negative number, the sign reverses: $<$ becomes $>$, $\le$ becomes $\ge$, and so on.

Why? $2<5$ is true, but $-2<-5$ is false; $-2>-5$ is true. Multiplying by a negative number flips the order on the number line.

A safe habit: collect the $x$ terms on the side where the coefficient will be positive. Then you never divide by a negative number.

Worked example

Solve $-2x+5\ge 11$, $x\in Z$.

Try it first, then show the solutionSolution

1. Subtract 5: $-2x\ge 6$.
2. Divide by $-2$ and reverse the sign: $x\le -3$.
3. Solution set $=\{\dots ,-5,-4,-3\}$.

Answer: {…, −5, −4, −3}

Inequations with fractions

Clear fractions by multiplying both sides by the LCM of the denominators. The LCM is positive, so the sign does not change. Use brackets so that every term is multiplied, including the whole numerators: $\frac{x+2}{4}$ times 12 is $3(x+2)$, not $3x+2$.

Then collect like terms and divide by the coefficient of $x$, reversing the sign only if that coefficient is negative.

Worked example

Solve $\frac{x}{3}+2>\frac{x}{2}-1$, $x\in R$.

Try it first, then show the solutionSolution

1. Multiply by 6: $2x+12>3x-6$.
2. Collect: $18>x$, i.e. $x<18$.
3. Solution set $=\{x:x\in R,x<18\}$.

Answer: {x : x ∈ R, x < 18}

Double (combined) inequations

A double inequation such as $a\le px+q<b$ says two things at once. If the unknown appears only in the middle, do the same operation to all three parts. For example $-4\le 2x+2<8$ becomes $-6\le 2x<6$, then $-3\le x<3$.

If $x$ appears in more than one part, for example $x-1\le \frac{3x+5}{2}<x+6$, split it into two separate inequations, solve each, and keep only the values that satisfy both (the common part, or intersection).

Worked example

Solve $-4\le 2x+2<8$, $x\in Z$.

Try it first, then show the solutionSolution

1. Subtract 2 throughout: $-6\le 2x<6$.
2. Divide by 2: $-3\le x<3$.
3. Integers from $-3$ up to, but not including, 3: $\{-3,-2,-1,0,1,2\}$.

Answer: {−3, −2, −1, 0, 1, 2}

Representing the solution on a number line

Draw a straight line with arrows at both ends (it goes on for ever), mark a few equally spaced integers, and include at least one integer beyond each end of the solution.

• For $x\in N$, $W$ or $Z$: mark each solution with a solid dot. Do not shade between dots.
• For $x\in R$: shade (darken) the whole interval. Use a filled circle at an end point that is included ($\le$ or $\ge$) and an open (hollow) circle at an end point that is not included ($<$ or $>$). If the solution continues for ever, darken the line up to the arrow.

If an end point is a fraction, such as $\frac{11}{3}$, mark and label its position between 3 and 4.

Worked example

Solve $x-1\le \frac{3x+5}{2}<x+6$, $x\in R$, and describe the number line.

Try it first, then show the solutionSolution

1. First part: $2x-2\le 3x+5$, so $-7\le x$.
2. Second part: $3x+5<2x+12$, so $x<7$.
3. Both: $-7\le x<7$.
4. Number line: filled circle at $-7$, open circle at $7$, darken the line between them; arrows at both ends of the line.

Answer: {x : x ∈ R, −7 ≤ x < 7}

Combining solution sets: intersection

Questions sometimes give two inequations, define their solution sets $A$ and $B$, and ask for $A\cap B$: the values that satisfy both. Solve each separately, then take the overlap. On a number line, $A\cap B$ is the part shaded twice.

If one set is from $Z$ and the other from $R$, the intersection contains only integers.

Worked example

$A=\{x:2x-3>1,x\in R\}$ and $B=\{x:3x-1\le 14,x\in R\}$. Find $A\cap B$.

Try it first, then show the solutionSolution

1. $A$: $2x>4$, so $x>2$.
2. $B$: $3x\le 15$, so $x\le 5$.
3. $A\cap B=\{x:x\in R,2<x\le 5\}$.

Answer: {x : x ∈ R, 2 < x ≤ 5}

Maths in real life: Linear Inequations

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.

Topics in this chapter

• Solving Linear Inequations
• Double (Combined) Inequations
• Solution Set on a Number Line

Linear Inequations formula sheet

Number sets

$N=\{1,2,3,\dots \};W=\{0,1,2,\dots \};Z=\{\dots ,-1,0,1,\dots \}$

R is the set of all real numbers.

Adding or subtracting

$a<b\Rightarrow a\pm c<b\pm c$

Sign unchanged.

Multiplying by a positive number

$a<b,k>0\Rightarrow ka<kb$

Sign unchanged; same for division.

Multiplying by a negative number

$a<b,k<0\Rightarrow ka>kb$

Sign reverses; same for division.

Double inequation

$a\le px+q<b⟺\frac{a-q}{p}\le x<\frac{b-q}{p}(p>0)$

If p < 0, both signs reverse.

Set-builder form for R

$\{x:x\in R,a\le x<b\}$

Use roster form for N, W and Z.

How the board asks Linear Inequations

The question patterns to expect, and the method that works for each.

Tricks to remember Linear Inequations

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

• Negative flips the sign

  Multiply or divide by a negative: flip the sign, like turning the number line round.

  Use it when: Every step where you divide by a negative coefficient.

• Solid for 'or equal', hollow for strict

  ≤ and ≥ have an extra line underneath, so the circle is filled in; < and > have none, so the circle is empty.

  Use it when: Number lines for x ∈ R.

• Dots for counting numbers, a line for reals

  N, W, Z: dots you can count. R: a darkened stretch you cannot count.

  Use it when: Choosing how to show the solution.

• Split, solve, share

  Split a double inequation, solve both parts, keep what they share.

  Use it when: Double inequations with x in more than one part.

Linear Inequations: board-level solved questions

Ordered from board level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section B style)

Read the situation, then answer every part. In the board paper this is a 10-mark question split into 3 + 3 + 4 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Linear Inequations

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

• ✗Mistake: Not reversing the sign when dividing by a negative number

  CISCE examiners' comments, 2024 and 2025: errors in transposing terms gave $x\le -2$ instead of $x\ge -2$, and candidates did not reverse the sign on multiplying or dividing by a negative number.

  Wrong working

  $-3x\le 6\Rightarrow x\le -2$

  Right working

  $-3x\le 6\Rightarrow x\ge -2$

  ✓Do this: Arrange terms so the coefficient of $x$ is positive, or write 'dividing by −3, sign reverses' next to the step.

  Usually costs: Every mark after that line.

• ✗Mistake: Including an end value that is excluded

  CISCE examiners' comments, 2024 and 2025: candidates included 4 in the solution set although $x<4$, and included end values in a double inequation MCQ.

  ✓Do this: Before listing, write the final range (e.g. $-2\le x<4$) and tick which ends have '='.

  Usually costs: 1 mark for the solution set, and the MCQ mark.

• ✗Mistake: Writing the wrong form of solution set

  CISCE examiners' comments, 2025: for $x\in R$, candidates listed integers in roster form; in 2024 others wrote no solution set at all.

  Wrong working

  $x\in R$, $-2<x\le 8\frac{1}{3}$: $\{-1,0,1,\dots ,8\}$

  Right working

  $\{x:x\in R,-2<x\le 8\frac{1}{3}\}$

  ✓Do this: N, W, Z → list the elements. R → set-builder form.

  Usually costs: 1 mark.

• ✗Mistake: Incomplete or wrong number line

  CISCE examiners' comments, 2024 and 2025: number lines without arrows, with extra integers marked beyond the ends, without a clearly marked fractional end point such as $8\frac{1}{3}$, or without shading.

  ✓Do this: Arrows on both ends; dots for integers, darkened segment with filled/open circles for reals; label any fractional end point.

  Usually costs: 1 mark in a 3-mark question.

• ✗Mistake: Multiplying only some terms by the LCM

  Students clear fractions but forget to multiply the whole-number terms, or drop the brackets around numerators.

  Wrong working

  $\frac{x+2}{4}+1<x$, ×4: $x+2+1<4x$

  Right working

  $x+2+4<4x$

  ✓Do this: Put brackets round each side before multiplying by the LCM.

  Usually costs: 1 to 2 marks.

Practice questions on Linear Inequations

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

1. Q1 · 1 mark · MCQBoard level

   The solution set of $2x+3\le 9$, $x\in W$, is:
   (a) $\{1,2,3\}$
   (b) $\{0,1,2,3\}$
   (c) $\{0,1,2\}$
   (d) $\{3,4,5,\dots \}$

   Hint

   W includes 0.

   Answer

   (b), since $x\le 3$.

2. Q2 · 1 mark · MCQBoard level

   The solution of $x>-2$, $x\in R$, is shown on a number line by:
   (a) a filled circle at $-2$ and the line darkened to the left
   (b) an open circle at $-2$ and the line darkened to the right
   (c) dots at $-1,0,1,\dots$
   (d) a filled circle at $-2$ and the line darkened to the right

   Hint

   Strict inequality, real numbers.

   Answer

   (b)

3. Q3 · 1 mark · MCQHarder board level

   The solution set of $-1<-\frac{x}{2}\le 2$, $x\in Z$, is:
   (a) $\{-4,-3,-2,-1,0,1\}$
   (b) $\{-3,-2,-1,0,1\}$
   (c) $\{-4,-3,-2,-1,0,1,2\}$
   (d) $\{-1,0,1,2,3,4\}$

   Hint

   Multiply all parts by −2 and reverse both signs.

   Answer

   (a): $2>x\ge -4$.

4. Q4 · 1 mark · Assertion-reasonBoard level

   Assertion (A): The inequation $x<1$, $x\in N$, has no solution.
   Reason (R): The smallest natural number is 1.
   (a) (A) is true and (R) is false.
   (b) (A) is false and (R) is true.
   (c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
   (d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).

   Hint

   List N.

   Answer

   (c). No natural number is less than 1, precisely because 1 is the smallest.

5. Q5 · 4 marks · Short answerHarder board level

   Solve $-3\le \frac{2x-3}{5}-1<1$, $x\in Z$. Write the solution set and represent it on a number line.

   Hint

   Add 1 throughout, then multiply by 5.

   Answer

   $-2\le \frac{2x-3}{5}<2\Rightarrow -10\le 2x-3<10\Rightarrow -3.5\le x<6.5$. Solution set $\{-3,-2,-1,0,1,2,3,4,5,6\}$; solid dots at these integers.

6. Q6 · 4 marks · Short answerHarder board level

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

   Hint

   Solve both parts; then think carefully about which condition is stronger.

   Answer

   Left: $3x-18\le 4x+2\Rightarrow x\ge -20$. Right: $2x+1<3x+6\Rightarrow x>-5$. Both hold when $x>-5$. Solution set $\{x:x\in R,x>-5\}$; open circle at $-5$, darkened to the right up to the arrow.

7. Q7 · 3 marks · Long answerBoard level

   Solve $3(x-2)+1<5x+7\le 2(x+8)$, $x\in Z$. Write the solution set and represent it on a number line.

   Hint

   Expand the brackets, then split into two inequations.

   Answer

   Left: $3x-5<5x+7\Rightarrow -12<2x\Rightarrow x>-6$. Right: $5x+7\le 2x+16\Rightarrow 3x\le 9\Rightarrow x\le 3$. Solution set $\{-5,-4,-3,-2,-1,0,1,2,3\}$; solid dots at these integers, arrows at both ends.

8. Q8 · 4 marks · Long answerHarder board level

   Solve $-2\le \frac{1}{2}-\frac{2x}{3}\le 1\frac{5}{6}$, $x\in N$, and represent the solution set on a number line.

   Hint

   Subtract ½ throughout, then multiply by $-\frac{3}{2}$ and reverse both signs.

   Answer

   $-\frac{5}{2}\le -\frac{2x}{3}\le \frac{4}{3}\Rightarrow \frac{15}{4}\ge x\ge -2$. With $x\in N$: $\{1,2,3\}$; solid dots at 1, 2, 3.

9. Q9 · 4 marks · Long answerHOTS / competency

   Find the set of real values of $x$ that satisfy $3x-7>x+1$ but do not satisfy $2x+5>4x-9$. Represent it on a number line.

   Hint

   'Does not satisfy' $a>b$ means $a\le b$.

   Answer

   First: $x>4$. Second: $14>2x$, i.e. $x<7$; not satisfying it means $x\ge 7$. Both: $x\ge 7$. $\{x:x\in R,x\ge 7\}$; filled circle at 7, darkened to the right.

10. Q10 · 4 marks · Long answerHarder board level

    $A=\{x:11x-5>7x+3,x\in R\}$ and $B=\{x:18x-9\ge 15+12x,x\in R\}$. Find $A\cap B$ and represent it on a number line.

    Hint

    Solve each separately.

    Answer

    $A$: $x>2$. $B$: $6x\ge 24$, $x\ge 4$. $A\cap B=\{x:x\in R,x\ge 4\}$; filled circle at 4, darkened to the right.

11. Q11 · 1 mark · MCQHOTS / competency

    Which pair of inequations has the same solution set for $x\in N$ but different solution sets for $x\in Z$?
    (a) $x<3$ and $0<x<3$
    (b) $x\le 2$ and $x<3$
    (c) $2x<6$ and $x<3$
    (d) $x>1$ and $x\ge 2$

    Hint

    Write each solution set in N and in Z.

    Answer

    (a). In N both give {1, 2}; in Z, $x<3$ includes 0, −1, −2, … but $0<x<3$ gives only {1, 2}. The other pairs are equal in both sets.

12. Q12 · 3 marks · Short answerHarder board level

    Twice a natural number decreased by 7 is at least 5 more than the number, and three times the number is less than 45. Form the inequations and find all such numbers.

    Hint

    'At least' means ≥.

    Answer

    $2x-7\ge x+5\Rightarrow x\ge 12$; $3x<45\Rightarrow x<15$. So $\{12,13,14\}$.

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. 1.When does the sign of an inequation reverse?Show answer

   When both sides are multiplied or divided by a negative number.

2. 2.What is the difference between N and W?Show answer

   W includes 0; N starts at 1.

3. 3.How is the solution set written for x ∈ R?Show answer

   In set-builder form, e.g. {x : x ∈ R, x < 5}.

4. 4.On a number line for x ∈ R, what does a filled circle mean?Show answer

   The end point is included (≤ or ≥).

5. 5.How do you solve a double inequation with x in all three parts?Show answer

   Split it into two inequations, solve each, and take the common values.

6. 6.Solve −x > 3.Show answer

   x < −3

7. 7.How are solutions shown when x ∈ Z?Show answer

   As separate solid dots, without shading between them.

8. 8.What is A ∩ B for two solution sets?Show answer

   The values that satisfy both inequations.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Do you reverse the sign whenever you multiply or divide by a negative number?
• ☐Can you clear fractions with the LCM, multiplying every term?
• ☐Can you split a double inequation and combine the two answers?
• ☐Do you write roster form for N, W, Z and set-builder form for R?
• ☐Do your number lines have arrows, the right dots or circles, and shading only for R?
• ☐Can you find the intersection of two solution sets?
• ☐Can you read off the greatest or smallest value, or count the solutions?

Linear Inequations: frequently asked questions