Linear Inequation Class 10 ICSE — Bansal Chapter 4
Linear inequation class 10 ICSE — this is Chapter 4 of the Bansal Mathematics book. The chapter teaches you to compare two quantities using inequality signs. You will learn the four order relations, the six algebraic solving rules, and how to graph solution sets on a real number line. The sections below follow the book’s own order and numbering.
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ICSE Class 10 Mathematics Bansal Chapter 4 Linear Inequations PDF
What is inside Chapter 4
13 pages · 6 sections · 10 worked examples · 35 exercise questions
| Section | Title |
|---|---|
| 4.1 | Introduction |
| 4.2 | Linear Inequations In One Variable |
| 4.3 | Solving a Linear Inequation Algebraically |
| 4.4 | Replacement Set and Solution Set |
| 4.5 | Representation of the solution on the number line |
| 4.7 | Important |
- EXERCISE 4(B): 35 questions
4.1 Introduction
An inequation is a statement of order between two quantities. If x and y are two quantities, exactly one of four relations holds. Either x \gt y, x \geq y, x \lt y, or x \leq y. Each of these four relations is an inequation.
Many expressions fit this pattern. The book lists x \lt 8, x \geq 5, -x + 4 \leq 3, and x + 8 \gt 4 as further examples. Each one compares a quantity with a fixed value or with another expression.
The key idea is that an inequation states an order, not an equality. An equation tells you two expressions are exactly equal. An inequation tells you one side is larger, smaller, or possibly equal. This difference drives every rule in the chapter.
Read each sign carefully. The symbol \gt means “greater than” and \geq adds “or equal to”. The symbol \lt means “less than” and \leq adds “or equal to”. The direction of the sign tells you which side is bigger.
4.2 Linear Inequations In One Variable
If a, b and c are real numbers, then each of the following is a linear inequation in one variable:
- ax + b \gt c: read as “ax plus b is greater than c”.
- ax + b \lt c: read as “ax plus b is less than c”.
- ax + b \geq c: read as “ax plus b is greater than or equal to c”.
- ax + b \leq c: read as “ax plus b is less than or equal to c”.
The signs \gt, \lt, \geq and \leq are called the signs of inequality.
The form is ax + b, so the variable x appears only to the first power. There is no x^2 term and no x in a denominator. That is what makes the inequation linear.
For example, 3x + 5 \gt 8 is a linear inequation in x. So is 8 - 5y \leq 3, this time in y. Solving either one means finding the value or values of the variable that make the statement true.
4.3 Solving a Linear Inequation Algebraically
To solve a linear inequation means to find the value or values of the variable used in it. The book gives six working rules. The table below summarises them.
Rule 1: a positive term transferred to the other side becomes negative. So 2x + 3 \gt 7 becomes 2x \gt 7 - 3. Rule 2: a negative term transferred becomes positive. So 2x - 3 \gt 7 becomes 2x \gt 7 + 3.
Rules 3 and 4 are the critical pair. Multiplying or dividing by a positive number keeps the sign. Multiplying or dividing by a negative number reverses it.
Rule 5: changing the sign of every term on both sides reverses the inequality. Rule 6: taking reciprocals of two positive or two negative quantities reverses the sign.
| Rule | Operation | Sign reverses? | Example |
|---|---|---|---|
| 1 | Transfer a positive term | No | 2x + 3 \gt 7 \Rightarrow 2x \gt 4 |
| 2 | Transfer a negative term | No | 2x - 3 \gt 7 \Rightarrow 2x \gt 10 |
| 3 | Multiply or divide by positive p | No | x \leq 6 \Rightarrow 4x \leq 24 |
| 4 | Multiply or divide by negative p | Yes | x \leq 6 \Rightarrow -4x \geq -24 |
| 5 | Change the sign of every term | Yes | -x \gt 5 \Rightarrow x \lt -5 |
| 6 | Take reciprocals of same-sign sides | Yes | x \gt y \Rightarrow \frac{1}{x} \lt \frac{1}{y} |
Worked pattern: solve 7 - 2x \geq 3. Transfer 7 to the right: -2x \geq 3 - 7, so -2x \geq -4. Divide by -2: the sign reverses, giving x \leq 2.
The common mistake is writing x \geq 2. Check with x = 1: 7 - 2(1) = 5, and 5 \geq 3 is true. So x \leq 2 is correct.
Mnemonic: negative flips. A negative multiplier or divisor flips the inequality sign.
4.4 Replacement Set and Solution Set
The replacement set is the set from which the value of the variable x is chosen. The solution set is the subset of the replacement set whose elements satisfy the given inequation.
For x \lt 3, the solution set changes with the replacement set. With N (natural numbers), it is {1, 2}. With W (whole numbers), it is {0, 1, 2}. With Z (integers), it is {…, -2, -1, 0, 1, 2}. With R (real numbers), you cannot list every value, so you use set-builder form: \{x : x \in \mathbb{R} \text{ and } x \lt 3\}.
Worked pattern 1: solve 2x + 5 \lt 13 with replacement set N. Transfer 5: 2x \lt 8. Divide by 2: x \lt 4. The natural numbers below 4 are {1, 2, 3}.
Worked pattern 2: solve 9 - x \leq 3x - 3 with replacement set N. Gather terms: -x - 3x \leq -3 - 9, so -4x \leq -12. Divide by -4: the sign reverses, x \geq 3. The solution set is {3, 4, 5,…}.
Fraction boundary case: solve 2x - 7 \lt 2 with replacement set Z. Transfer: 2x \lt 9, so x \lt 4.5. The integers below 4.5 are {…, -1, 0, 1, 2, 3, 4}.
The book shows an alternative method too. Since x \leq y and y \geq x mean the same thing, you can read an inequation from either side. Sometimes that makes the algebra simpler.
Common mistake: writing \{x \lt 4\} instead of listing actual elements. When the replacement set is N, the answer must be a list like {1, 2, 3}.
4.5 Representation of the solution on the number line
A real number line can represent the solution set of an inequation. The book uses a two-circle convention. A hollow circle \circ marks the end of a range with a strict inequality (\lt or \gt). A darkened circle \bullet marks an endpoint that includes equality (\leq or \geq).
For x \lt 2, the circle at 2 is hollow because 2 is not included. For x \leq 2, the circle is darkened because 2 is part of the set.
Graph \{x : x \geq 4, x \in \mathbb{R}\}: a darkened circle at 4 with the arrow pointing right. Graph \{x : x \lt -3, x \in \mathbb{R}\}: a hollow circle at -3 with the arrow pointing left.
The same interval differs for integers and reals. For -2 \lt x \leq 4 with Z, you mark the integer points -1, 0, 1, 2, 3, 4. For R, you shade the whole segment between -2 and 4.
Combining inequations
The book’s section 4.6 sits inside this page span. A double inequation like -2 \leq 4 + 3x \lt 19 splits into two parts: -2 \leq 4 + 3x and 4 + 3x \lt 19. Solve each: -6 \leq 3x gives -2 \leq x; 3x \lt 15 gives x \lt 5. Combined: -2 \leq x \lt 5.
For “and”, take the points common to both graphs — the intersection. For “or”, take the union of both graphs. Example: x \geq 1 and x \lt 3 graphs to the overlap between 1 and 3. Example: x \gt 2 or x \leq -4 graphs to both arms.
Set operations follow the same idea. P \cap Q is the overlap of two solution sets. P \cup Q is everything in either set. P - Q is what lies in P but not in Q.
Exam pattern note: board questions in this chapter carry year tags from 2004 through 2015. Nearly all ask the same thing — solve an inequation and graph the solution set on a real number line. Practise that exact pattern. For the current ICSE Mathematics syllabus and examination updates, check the official CISCE website.
4.7 Important
The product of two factors method handles inequations of the form (x - a)(x + b) \lt 0 or \gt 0. Two facts do all the work.
If the product of two numbers is negative, one factor is positive and the other is negative. If the product is positive, both factors have the same sign — both positive or both negative.
Worked pattern with original numbers: solve (x - 2)(x + 4) \gt 0. Both factors negative: x - 2 \lt 0 and x + 4 \lt 0, so x \lt 2 and x \lt -4. The tighter condition is x \lt -4. Both factors positive: x - 2 \gt 0 and x + 4 \gt 0, so x \gt 2.
Solution: x \lt -4 or x \gt 2.
Now solve (x - 2)(x + 4) \lt 0. The factors must have opposite signs. Case 1: x - 2 \gt 0 and x + 4 \lt 0, so x \gt 2 and x \lt -4. That is impossible. Case 2: x - 2 \lt 0 and x + 4 \gt 0, so x \lt 2 and x \gt -4.
That gives -4 \lt x \lt 2.
The book graphs these solutions on number lines for N, Z, and R, so the same interval takes different forms.
Common mistake: expanding (x - 3)(x + 5) \lt 0 like a quadratic and solving for roots. That loses the case logic. Another mistake: assuming both factors are negative without testing both cases.
What to practise from this chapter
EXERCISE 4(B) has 35 questions. The table maps each group to the section that teaches the needed skill.
| Questions | Skill | Section |
|---|---|---|
| Q1-2 | Represent inequalities on number lines; read an inequation from a graph | 4.5 |
| Q3-4 | Graph solution sets | 4.5 |
| Q5-11 | Solve single and double inequations with stated replacement sets | 4.4 |
| Q12-13 | Combine inequations with and/or | 4.5 |
| Q14-19 | Set operations on number lines | 4.5 |
| Q20-27 | Fractions, mixed numbers, restricted replacement sets | 4.4 |
| Q28 | Consecutive-number word problem | 4.4 |
| Q29-35 | Board-pattern solve-and-graph questions (tagged 2004-2015) | 4.5 |
The 15 solved examples inside sections 4.4-4.5 serve as worked models for these questions. Q22-24 also carry board year tags, so give them the same treatment as Q29-35.
The algebra you practise here feeds into later chapters. Shares and Dividends (Chapter 3) and Quadratic Equations (Chapter 5) both rely on the same manipulation skills. You can also browse the Class 10 book collection, Class 10 resources, or the wider ICSE resources on this site.
Frequently asked questions
When does the inequality sign reverse while solving a linear inequation?
The sign reverses in three situations: multiplying or dividing both sides by a negative number (Rule 4), changing the sign of every term on both sides (Rule 5), and taking reciprocals of two quantities that share the same sign (Rule 6).
What is the difference between the replacement set and the solution set of an inequation?
The replacement set is the set from which you choose the value of x. The solution set is the subset of the replacement set whose elements actually satisfy the inequation. For example, with replacement set N and x \lt 4, the solution set is {1, 2, 3}.
Why does the solution set change when the replacement set changes from natural numbers to whole numbers to integers to real numbers?
Because the replacement set decides which values are allowed. For x \lt 3, N gives {1, 2}, W gives {0, 1, 2}, Z adds all negative integers, and R needs set-builder form because you cannot list infinitely many real numbers.
When do I use a hollow circle and when a darkened circle on the number line?
A hollow circle marks the end of a range with a strict inequality (\lt or \gt), so the endpoint is not included. A darkened circle marks an endpoint that is included (\leq or \geq).
How do I solve a combined inequation, and what do “and” and “or” mean for the graph?
A combined inequation like -2 \leq 4 + 3x \lt 19 splits into two parts: -2 \leq 4 + 3x and 4 + 3x \lt 19. Solving each gives -2 \leq x and x \lt 5, so the answer is -2 \leq x \lt 5. “And” means take the points common to both graphs; “or” means take the union of both graphs.
Reference: ICSE Class 10 Mathematics textbook material, chapter 4: Linear Inequations — CISCE-aligned.
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