Question 1

If $x$ is a natural number and $x<4$, the solution set is:

Accepted Answer

**Answer: (a) $\{1,2,3\}$.** Natural numbers start at $1$, and $x<4$ excludes $4$, so $x\in\{1,2,3\}.$

Question 2

The solution of $2x>6$, where $x\in\mathbb{R}$, is:

Accepted Answer

**Answer: (a) $x>3$.** Dividing both sides by the positive number $2$ keeps the sign: $x>3.$

Question 3

The solution of $-3x\ge9$, where $x\in\mathbb{R}$, is:

Accepted Answer

**Answer: (a) $x\le-3$.** Dividing by $-3$ reverses the sign: $x\le\dfrac{9}{-3}=-3.$

Question 4

The solution set of $3\le2x-1<7$, where $x\in\mathbb{Z}$, is:

Accepted Answer

**Answer: (a) $\{2,3\}$.** $3\le2x-1\Rightarrow4\le2x\Rightarrow x\ge2$; and $2x-1<7\Rightarrow2x<8\Rightarrow x<4$. So $2\le x<4$, giving integers $\{2,3\}.$

Question 5

**Assertion (A):** Multiplying both sides of $-x>3$ by $-1$ gives $x>-3$. **Reason (R):** When both sides of an inequation are multiplied by a negative number, the inequality sign reverses.

Accepted Answer

**Answer: (d)** R is true, but A is false: multiplying $-x>3$ by $-1$ reverses the sign to give $x<-3$, not $x>-3$.

Question 6

Solve the inequation $5x-3<3x+7$, where $x\in\mathbb{R}$.

Accepted Answer

$5x-3<3x+7$ $5x-3x<7+3$ $2x<10\Rightarrow x<5.$ Solution set $=\{x:x<5,\ x\in\mathbb{R}\}.$

Question 7

Given the replacement set $\{-2,-1,0,1,2,3\}$, find the solution set of $2x-1<3$.

Accepted Answer

$2x-1<3\Rightarrow2x<4\Rightarrow x<2.$ From the replacement set, the values less than $2$ are $-2,-1,0,1$. Solution set $=\{-2,-1,0,1\}.$

Question 8

Solve $-2\le x+3<5$, where $x\in\mathbb{R}$, and represent the solution on a number line.

Accepted Answer

Subtract $3$ throughout: $-2-3\le x<5-3$, i.e. $-5\le x<2.$ Solution set $=\{x:-5\le x<2,\ x\in\mathbb{R}\}$ — a solid dot at $-5$ (included) and an open dot at $2$ (excluded). ```svg ```

Question 9

Solve $\dfrac{x}{2}+5\le\dfrac{x}{3}+6$, where $x\in\mathbb{R}$.

Accepted Answer

Multiply every term by $6$ (the LCM): $6	imes\dfrac{x}{2}+6	imes5\le6	imes\dfrac{x}{3}+6	imes6$ $3x+30\le2x+36$ $3x-2x\le36-30\Rightarrow x\le6.$ Solution set $=\{x:x\le6,\ x\in\mathbb{R}\}.$

Question 10

If $x$ is an integer and $-5<2x-1\le5$, find the solution set and state its greatest value.

Accepted Answer

$-5<2x-1\Rightarrow-4<2x\Rightarrow x>-2.$ $2x-1\le5\Rightarrow2x\le6\Rightarrow x\le3.$ So $-2<x\le3$. For integers, the solution set $=\{-1,0,1,2,3\}$, and its greatest value is $3$.

Question 11

Solve the pair of inequations $3x-2>2(x-1)$ and $2x+5\le15$ simultaneously, where $x\in\mathbb{R}$, and represent the solution on a number line.

Accepted Answer

First inequation: $3x-2>2x-2\Rightarrow3x-2x>-2+2\Rightarrow x>0.$ Second inequation: $2x+5\le15\Rightarrow2x\le10\Rightarrow x\le5.$ Common solution: $005 ```

Question 12

Solve the following inequation, write the solution set and represent it on a number line: $-3(x-7)\ge15-7x>\dfrac{x+1}{3}$, where $x\in\mathbb{R}$.

Accepted Answer

Split into two inequations. **Left part:** $-3(x-7)\ge15-7x\Rightarrow-3x+21\ge15-7x\Rightarrow-3x+7x\ge15-21\Rightarrow4x\ge-6\Rightarrow x\ge-\dfrac{3}{2}.$ **Right part:** $15-7x>\dfrac{x+1}{3}$. Multiply by $3$: $45-21x>x+1\Rightarrow45-1>x+21x\Rightarrow44>22x\Rightarrow x<2.$ Combining: $-\dfrac{3}{2}\le x<2.$ Solution set $=\left\{x:-1.5\le x<2,\ x\in\mathbb{R}\right\}$ — solid dot at $-1.5$, open dot at $2$. ```svg ```

Question 13

The replacement set for the variable $x$ is the set of integers $\mathbb{Z}$ . Consider the inequation $-4\le3x+2<11$ .

(i) Solve the inequation for $x$ .

(ii) Write the solution set.

(iii) State the number of elements in the solution set.

(iv) Represent the solution on a number line.

Accepted Answer

**(i)** $-4\le3x+2\Rightarrow-6\le3x\Rightarrow x\ge-2$; and $3x+2<11\Rightarrow3x<9\Rightarrow x<3$. So $-2\le x<3.$ **(ii)** For integers, solution set $=\{-2,-1,0,1,2\}.$ **(iii)** It has $5$ elements. **(iv)** Mark the five integer points $-2$ to $2$ with solid dots. ```svg ```

Question 14

Are these Linear Inequations (In One Variable) important questions free?

Accepted Answer

Yes. All 13 ICSE Class 10 Maths important questions for Linear Inequations (In One Variable) are free, with full model answers and no login required.

Question 15

Do these Linear Inequations (In One Variable) questions follow the latest ICSE syllabus?

Accepted Answer

Yes — they are aligned to the CISCE latest syllabus syllabus for ICSE Class 10 Maths, so nothing here is outside the current course.

Question 16

How should I practise the Linear Inequations (In One Variable) important questions?

Accepted Answer

Attempt each question on paper first, then reveal the model answer to check your method — not just the final result. Re-do anything you got wrong the same day.

Question 17

What types of questions are covered for Linear Inequations (In One Variable)?

Accepted Answer

A full mix — multiple-choice questions, assertion–reason questions, very short answer questions, short answer questions, long answer questions, case-based questions — so every format in the ICSE paper is covered.

Question type	Count	Marks
MCQ	4	1
Assertion–Reason	1	1
Very Short	2	2
Short Answer	3	3
Long Answer	2	5
Case-based	1	4

Linear Inequations (In One Variable) — Important Questions

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